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Computing a Sparse Approximate Inverse on Quantum Annealing Machines (2310.02388v1)

Published 3 Oct 2023 in math.NA, cs.ET, and cs.NA

Abstract: Many engineering problems involve solving large linear systems of equations. Conjugate gradient (CG) is one of the most popular iterative methods for solving such systems. However, CG typically requires a good preconditioner to speed up convergence. One such preconditioner is the sparse approximate inverse (SPAI). In this paper, we explore the computation of an SPAI on quantum annealing machines by solving a series of quadratic unconstrained binary optimization (QUBO) problems. Numerical experiments are conducted using both well-conditioned and poorly-conditioned linear systems arising from a 2D finite difference formulation of the Poisson problem.

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References (35)
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Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. 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Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) G. Tosti Balducci, B. Chen, M. Möller, M. Gerritsma, R. De Breuker, Review and perspectives in quantum computing for partial differential equations in structural mechanics. Frontiers in Mechanical Engineering p. 75 (2022) [4] Y. Wang, J.E. Kim, K. Suresh, Opportunities and challenges of quantum computing for engineering optimization. Journal of Computing and Information Science in Engineering 23(6) (2023) [5] A.W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations. Physical review letters 103(15), 150502 (2009) [6] A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) Y. Wang, J.E. Kim, K. Suresh, Opportunities and challenges of quantum computing for engineering optimization. Journal of Computing and Information Science in Engineering 23(6) (2023) [5] A.W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations. Physical review letters 103(15), 150502 (2009) [6] A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations. Physical review letters 103(15), 150502 (2009) [6] A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  2. ACM Transactions on Mathematical Software (TOMS) 33(2), 10–es (2007) [3] G. Tosti Balducci, B. Chen, M. Möller, M. Gerritsma, R. De Breuker, Review and perspectives in quantum computing for partial differential equations in structural mechanics. Frontiers in Mechanical Engineering p. 75 (2022) [4] Y. Wang, J.E. Kim, K. Suresh, Opportunities and challenges of quantum computing for engineering optimization. Journal of Computing and Information Science in Engineering 23(6) (2023) [5] A.W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations. Physical review letters 103(15), 150502 (2009) [6] A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) G. Tosti Balducci, B. Chen, M. Möller, M. Gerritsma, R. De Breuker, Review and perspectives in quantum computing for partial differential equations in structural mechanics. Frontiers in Mechanical Engineering p. 75 (2022) [4] Y. Wang, J.E. Kim, K. Suresh, Opportunities and challenges of quantum computing for engineering optimization. Journal of Computing and Information Science in Engineering 23(6) (2023) [5] A.W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations. Physical review letters 103(15), 150502 (2009) [6] A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) Y. Wang, J.E. Kim, K. Suresh, Opportunities and challenges of quantum computing for engineering optimization. Journal of Computing and Information Science in Engineering 23(6) (2023) [5] A.W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations. Physical review letters 103(15), 150502 (2009) [6] A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations. Physical review letters 103(15), 150502 (2009) [6] A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  3. Frontiers in Mechanical Engineering p. 75 (2022) [4] Y. Wang, J.E. Kim, K. Suresh, Opportunities and challenges of quantum computing for engineering optimization. Journal of Computing and Information Science in Engineering 23(6) (2023) [5] A.W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations. Physical review letters 103(15), 150502 (2009) [6] A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) Y. Wang, J.E. Kim, K. Suresh, Opportunities and challenges of quantum computing for engineering optimization. Journal of Computing and Information Science in Engineering 23(6) (2023) [5] A.W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations. Physical review letters 103(15), 150502 (2009) [6] A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations. Physical review letters 103(15), 150502 (2009) [6] A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  4. Journal of Computing and Information Science in Engineering 23(6) (2023) [5] A.W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations. Physical review letters 103(15), 150502 (2009) [6] A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations. Physical review letters 103(15), 150502 (2009) [6] A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
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Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  6. A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv preprint arXiv:1010.4458 (2010) [7] A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.M. Childs, R. Kothari, R.D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  7. SIAM Journal on Computing 46(6), 1920–1950 (2017) [8] X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) X. Liu, H. Xie, Z. Liu, C. Zhao, Survey on the improvement and application of HHL algorithm. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  8. Journal of Physics: Conference Series 2333(1), 012023 (2022) [9] J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  9. J. Preskill, Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018) [10] K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Srinivasan, B.K. Behera, P.K. Panigrahi, Solving linear systems of equations by gaussian elimination method using grover’s search algorithm: an ibm quantum experience. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  10. arXiv preprint arXiv:1801.00778 (2017) [11] D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. An, L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  11. ACM Transactions on Quantum Computing 3(2), 1–28 (2022) [12] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  12. arXiv preprint arXiv:1909.05820 (2019) [13] S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, How quantum is the d-wave machine? arXiv preprint arXiv:1401.7087 (2014) [14] P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
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Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) P. Hauke, H.G. Katzgraber, W. Lechner, H. Nishimori, W.D. Oliver, Perspectives of quantum annealing: Methods and implementations. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  14. Reports on Progress in Physics 83(5), 054401 (2020) [15] S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Yarkoni, E. Raponi, T. Bäck, S. Schmitt, Quantum annealing for industry applications: Introduction and review. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  15. Reports on Progress in Physics (2022) [16] D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) D. O’Malley, V.V. Vesselinov, B.S. Alexandrov, L.B. Alexandrov, Nonnegative/binary matrix factorization with a d-wave quantum annealer. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  16. PloS one 13(12), e0206653 (2018) [17] A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A. Borle, S.J. Lomonaco, in WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13 (Springer, 2019), pp. 289–301 [18] S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S.W. Park, H. Lee, B.C. Kim, Y. Woo, K. Jun, in 2021 International Conference on Information and Communication Technology Convergence (ICTC) (IEEE, 2021), pp. 1363–1367 [19] R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. 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Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. 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Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. 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Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  18. R. Conley, D. Choi, G. Medwig, E. Mroczko, D. Wan, P. Castillo, K. Yu, in Quantum Computing, Communication, and Simulation III, vol. 12446 (SPIE, 2023), pp. 53–63 [20] S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  19. S. Srivastava, V. Sundararaghavan, Box algorithm for the solution of differential equations on a quantum annealer. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  20. Physical Review A 99(5), 052355 (2019) [21] H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) H.P. Langtangen, S. Linge, Finite difference computing with PDEs: a modern software approach (Springer Nature, 2017) [22] M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  21. M. Bollhöfer, O. Schenk, R. Janalik, S. Hamm, K. Gullapalli, State-of-the-art sparse direct solvers. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  22. Parallel algorithms in computational science and engineering pp. 3–33 (2020) [23] O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) O. Axelsson, in Sparse Matrix Techniques: Copenhagen 1976 Advanced Course Held at the Technical University of Denmark Copenhagen, August 9–12, 1976 (Springer, 2007), pp. 1–51 [24] J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  23. J.R. Shewchuk, et al. An introduction to the conjugate gradient method without the agonizing pain (1994) [25] J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  24. J.L. Nazareth, Conjugate gradient method. Wiley Interdisciplinary Reviews: Computational Statistics 1(3), 348–353 (2009) [26] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  25. E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM Journal on Scientific Computing 21(5), 1804–1822 (2000) [27] M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  26. M. Benzi, Preconditioning techniques for large linear systems: a survey. Journal of computational Physics 182(2), 418–477 (2002) [28] A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  27. A.J. Wathen, Preconditioning. Acta Numerica 24, 329–376 (2015) [29] M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M.L. Rogers, R.L. Singleton Jr, Floating-point calculations on a quantum annealer: Division and matrix inversion. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  28. Frontiers in Physics 8, 265 (2020) [30] M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) M. Zaman, K. Tanahashi, S. Tanaka, Pyqubo: Python library for mapping combinatorial optimization problems to qubo form. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  29. IEEE Transactions on Computers 71(4), 838–850 (2021) [31] N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) N. Bell, L.N. Olson, J. Schroder, B. Southworth, PyAMG: Algebraic multigrid solvers in python. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  30. Journal of Open Source Software 8(87), 5495 (2023). 10.21105/joss.05495. URL https://doi.org/10.21105/joss.05495 [32] W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) W. Zuo, K. Saitou, Multi-material topology optimization using ordered simp interpolation. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  31. Structural and Multidisciplinary Optimization 55, 477–491 (2017) [33] K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
  32. K. Suresh, Efficient generation of large-scale pareto-optimal topologies. Structural and Multidisciplinary Optimization 47(1), 49–61 (2013) [34] E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014) [35] B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) B.D. Clader, B.C. Jacobs, C.R. Sprouse, Preconditioned quantum linear system algorithm. Physical review letters 110(25), 250504 (2013) [36] E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022) E. Pelofske, G. Hahn, H.N. Djidjev, Parallel quantum annealing. Scientific Reports 12(1), 4499 (2022)
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