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Fewer measurements from shadow tomography with $N$-representability conditions (2312.11715v1)

Published 18 Dec 2023 in quant-ph, physics.chem-ph, and physics.comp-ph

Abstract: Classical shadow tomography provides a randomized scheme for approximating the quantum state and its properties at reduced computational cost with applications in quantum computing. In this Letter we present an algorithm for realizing fewer measurements in the shadow tomography of many-body systems by imposing $N$-representability constraints. Accelerated tomography of the two-body reduced density matrix (2-RDM) is achieved by combining classical shadows with necessary constraints for the 2-RDM to represent an $N$-body system, known as $N$-representability conditions. We compute the ground-state energies and 2-RDMs of hydrogen chains and the N$_{2}$ dissociation curve. Results demonstrate a significant reduction in the number of measurements with important applications to quantum many-body simulations on near-term quantum devices.

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References (48)
  1. S. Aaronson, Shadow tomography of quantum states, SIAM Journal on Computing 49, STOC18 (2020).
  2. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys 16, 1050 (2020).
  3. H.-Y. Huang, Learning quantum states from their classical shadows, Nat. Rev. Phys. 4, 81–81 (2022).
  4. H.-Y. Hu, S. Choi, and Y.-Z. You, Classical shadow tomography with locally scrambled quantum dynamics, Phys. Rev. Res. 5, 023027 (2023).
  5. G. H. Low, Classical Shadows of Fermions with Particle Number Symmetry, arXiv 10.48550/arxiv.2208.08964 (2022).
  6. A. Zhao, N. C. Rubin, and A. Miyake, Fermionic partial tomography via classical shadows, Phys. Rev. Lett. 127, 110504 (2021).
  7. T.-C. Yen, V. Verteletskyi, and A. F. Izmaylov, Measuring all compatible operators in one series of single-qubit measurements using unitary transformations, J Theor Comput Chem. 16, 2400–2409 (2020).
  8. M. Reagor et al., Demonstration of universal parametric entangling gates on a multi-qubit lattice, Sci. Adv. 4, eaao3603 (2018).
  9. R. O’Donnell and J. Wright, Efficient quantum tomography, in Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, STOC ’16 (ACM, 2016).
  10. S. E. Smart and D. A. Mazziotti, Quantum solver of contracted eigenvalue equations for scalable molecular simulations on quantum computing devices, Phys. Rev. Lett. 126, 070504 (2021).
  11. D. A. Mazziotti, Reduced-Density-Matrix Mechanics: With Application to Many-Electron Atoms and Molecule, Vol. 134 (Adv. Chem. Phys., Wiley, New York, 2007).
  12. A. Coleman and V. Yukalov, Reduced Density Matrices: Coulson’s Challenge, Lecture Notes in Chemistry (Springer Berlin Heidelberg, 2000).
  13. A. J. Coleman, Structure of Fermion Density Matrices, Rev. Mod. Phys. 35, 668 (1963).
  14. C. Garrod and J. K. Percus, Reduction of the N𝑁Nitalic_N‐Particle Variational Problem, J. Math. Phys. 5, 1756 (1964).
  15. H. Kummer, N𝑁Nitalic_N-representability problem for reduced density matrices, J. Math. Phys. 8, 2063 (1967).
  16. R. M. Erdahl, Representability, Int. J. Quantum Chem. 13, 697 (1978).
  17. D. A. Mazziotti and R. M. Erdahl, Uncertainty relations and reduced density matrices: Mapping many-body quantum mechanics onto four particles, Phys. Rev. A 63, 042113 (2001).
  18. D. A. Mazziotti, Variational reduced-density-matrix method using three-particle N𝑁Nitalic_N-representability conditions with application to many-electron molecules, Phys. Rev. A 74, 032501 (2006).
  19. D. A. Mazziotti, Structure of fermionic density matrices: Complete n𝑛nitalic_n-representability conditions, Phys. Rev. Lett. 108, 263002 (2012).
  20. D. A. Mazziotti, Quantum many-body theory from a solution of the N𝑁Nitalic_N-representability problem, Phys. Rev. Lett. 130, 153001 (2023).
  21. L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Review 38, 49 (1996).
  22. D. A. Mazziotti, Realization of quantum chemistry without wave functions through first-order semidefinite programming, Phys. Rev. Lett. 93, 213001 (2004).
  23. D. A. Mazziotti, Large-scale semidefinite programming for many-electron quantum mechanics, Phys. Rev. Lett. 106, 083001 (2011).
  24. D. A. Mazziotti, Variational minimization of atomic and molecular ground-state energies via the two-particle reduced density matrix, Phys. Rev. A 65, 062511 (2002).
  25. E. Cancès, G. Stoltz, and M. Lewin, The electronic ground-state energy problem: A new reduced density matrix approach, J. Chem. Phys. 125, 064101 (2006).
  26. G. Gidofalvi and D. A. Mazziotti, Active-space two-electron reduced-density-matrix method: Complete active-space calculations without diagonalization of the N-electron Hamiltonian, J. Chem. Phys. 129, 134108 (2008).
  27. N. Shenvi and A. F. Izmaylov, Active-Space N-Representability Constraints for Variational Two-Particle Reduced Density Matrix Calculations, Phys. Rev. Lett. 105, 213003 (2010).
  28. T. Baumgratz and M. B. Plenio, Lower bounds for ground states of condensed matter systems, New J. Phys. 14, 023027 (2012).
  29. D. A. Mazziotti, Enhanced Constraints for Accurate Lower Bounds on Many-Electron Quantum Energies from Variational Two-Electron Reduced Density Matrix Theory, Phys. Rev. Lett. 117, 153001 (2016).
  30. D. A. Mazziotti, Dual-cone variational calculation of the two-electron reduced density matrix, Phys. Rev. A 102, 052819 (2020).
  31. R. R. Li, M. D. Liebenthal, and A. E. DePrince, Challenges for variational reduced-density-matrix theory with three-particle N-representability conditions, J. Chem. Phys. 155, 174110 (2021).
  32. M. Piris, Global Natural Orbital Functional: Towards the Complete Description of the Electron Correlation, Phys. Rev. Lett. 127, 233001 (2021), 2112.02119 .
  33. M. J. Knight, H. M. Quiney, and A. M. Martin, Reduced density matrix approach to ultracold few-fermion systems in one dimension, New J. Phys. 24, 053004 (2022), 2106.09187 .
  34. A. W. Schlimgen, C. W. Heaps, and D. A. Mazziotti, Entangled Electrons Foil Synthesis of Elusive Low-Valent Vanadium Oxo Complex, J. Phys. Chem. Lett. 7, 627 (2016).
  35. A. Haar, Der massbegriff in der theorie der kontinuierlichen gruppen, The Annals of Mathematics 34, 147 (1933).
  36. E. J. Candès and B. Recht, Exact Matrix Completion via Convex Optimization, Found. Comput. Math. 9, 717 (2009).
  37. J.-F. Cai, E. J. Cands, and Z. Shen, A Singular Value Thresholding Algorithm for Matrix Completion, SIAM J. Optim. 20, 1956 (2010).
  38. J. J. Foley and D. A. Mazziotti, Measurement-driven reconstruction of many-particle quantum processes by semidefinite programming with application to photosynthetic light harvesting, Phys. Rev. A 86, 012512 (2012).
  39. N. C. Rubin, R. Babbush, and J. McClean, Application of fermionic marginal constraints to hybrid quantum algorithms, New J. Phys. 20, 053020 (2018).
  40. S. E. Smart and D. A. Mazziotti, Quantum-classical hybrid algorithm using an error-mitigating N-representability condition to compute the Mott metal-insulator transition, Phys. Rev. A 100, 022517 (2019), 2004.07739 .
  41. S. E. Smart, J.-N. Boyn, and D. A. Mazziotti, Resolving correlated states of benzyne with an error-mitigated contracted quantum eigensolver, Phys. Rev. A 105, 022405 (2022), 2103.06876 .
  42. G. Gidofalvi and D. A. Mazziotti, Spin and symmetry adaptation of the variational two-electron reduced-density-matrix method, Phys. Rev. A 72, 052505 (2005).
  43. Maple (Maplesoft, Waterloo, 2023).
  44. S. Suhai, Electron correlation in extended systems: Fourth-order many-body perturbation theory and density-functional methods applied to an infinite chain of hydrogen atoms, Phys. Rev. B 50, 14791 (1994).
  45. W. J. Hehre, R. F. Stewart, and J. A. Pople, Self-consistent molecular-orbital methods. i. use of gaussian expansions of slater-type atomic orbitals, Chem. Phys. 51, 2657–2664 (1969).
  46. T. H. Dunning, Gaussian basis sets for use in correlated molecular calculations. i. the atoms boron through neon and hydrogen, Chem. Phys. 90, 1007–1023 (1989).
  47. C. J. Cramer, Essentials of Computational Chemistry: Theories and Models (John Wiley & Sons, 2013).
  48. The Supplemental Material (SM) presents additional data for longer hydrogen chains.
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