Papers
Topics
Authors
Recent
Search
2000 character limit reached

Hyper-optimized approximate contraction of tensor networks with arbitrary geometry

Published 14 Jun 2022 in quant-ph, cond-mat.stat-mech, and cond-mat.str-el | (2206.07044v2)

Abstract: Tensor network contraction is central to problems ranging from many-body physics to computer science. We describe how to approximate tensor network contraction through bond compression on arbitrary graphs. In particular, we introduce a hyper-optimization over the compression and contraction strategy itself to minimize error and cost. We demonstrate that our protocol outperforms both hand-crafted contraction strategies in the literature as well as recently proposed general contraction algorithms on a variety of synthetic and physical problems on regular lattices and random regular graphs. We further showcase the power of the approach by demonstrating approximate contraction of tensor networks for frustrated three-dimensional lattice partition functions, dimer counting on random regular graphs, and to access the hardness transition of random tensor network models, in graphs with many thousands of tensors.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (88)
  1. I. Markov and Y. Shi, “Simulating Quantum Computation by Contracting Tensor Networks,” SIAM Journal on Computing, vol. 38, pp. 963–981, Jan. 2008.
  2. E. Pednault, J. A. Gunnels, G. Nannicini, L. Horesh, T. Magerlein, E. Solomonik, and R. Wisnieff, “Breaking the 49-Qubit Barrier in the Simulation of Quantum Circuits,” arXiv:1710.05867 [quant-ph], Oct. 2017.
  3. S. Boixo, S. V. Isakov, V. N. Smelyanskiy, and H. Neven, “Simulation of low-depth quantum circuits as complex undirected graphical models,” arXiv:1712.05384 [quant-ph], Dec. 2017.
  4. J. Chen, F. Zhang, C. Huang, M. Newman, and Y. Shi, “Classical Simulation of Intermediate-Size Quantum Circuits,” arXiv:1805.01450 [quant-ph], May 2018.
  5. B. Villalonga, S. Boixo, B. Nelson, C. Henze, E. Rieffel, R. Biswas, and S. Mandrà, “A flexible high-performance simulator for the verification and benchmarking of quantum circuits implemented on real hardware,” arXiv:1811.09599 [quant-ph], Nov. 2018.
  6. J. Gray and S. Kourtis, “Hyper-optimized tensor network contraction,” Quantum, vol. 5, p. 410, Mar. 2021.
  7. C. Huang, F. Zhang, M. Newman, J. Cai, X. Gao, Z. Tian, J. Wu, H. Xu, H. Yu, B. Yuan, M. Szegedy, Y. Shi, and J. Chen, “Classical Simulation of Quantum Supremacy Circuits,” arXiv:2005.06787 [quant-ph], May 2020.
  8. G. Kalachev, P. Panteleev, and M.-H. Yung, “Recursive Multi-Tensor Contraction for XEB Verification of Quantum Circuits,” arXiv:2108.05665 [quant-ph], Aug. 2021.
  9. F. Pan and P. Zhang, “Simulating the Sycamore quantum supremacy circuits,” Mar. 2021.
  10. S. R. White, “Density-matrix algorithms for quantum renormalization groups,” Physical Review B, vol. 48, pp. 10345–10356, Oct. 1993.
  11. S. R. White, “Density matrix formulation for quantum renormalization groups,” Physical Review Letters, vol. 69, pp. 2863–2866, Nov. 1992.
  12. V. Murg, F. Verstraete, and J. I. Cirac, “Variational study of hard-core bosons in a two-dimensional optical lattice using projected entangled pair states,” Physical Review A, vol. 75, p. 033605, Mar. 2007.
  13. F. Verstraete and J. I. Cirac, “Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions,” arXiv:cond-mat/0407066, July 2004.
  14. G. Vidal, “Classical Simulation of Infinite-Size Quantum Lattice Systems in One Spatial Dimension,” Physical Review Letters, vol. 98, p. 070201, Feb. 2007.
  15. H. C. Jiang, Z. Y. Weng, and T. Xiang, “Accurate Determination of Tensor Network State of Quantum Lattice Models in Two Dimensions,” Physical Review Letters, vol. 101, p. 090603, Aug. 2008.
  16. E. Stoudenmire and S. R. White, “Studying Two-Dimensional Systems with the Density Matrix Renormalization Group,” Annual Review of Condensed Matter Physics, vol. 3, no. 1, pp. 111–128, 2012.
  17. P. C. G. Vlaar and P. Corboz, “Simulation of three-dimensional quantum systems with projected entangled-pair states,” arXiv:2102.06715 [cond-mat, physics:quant-ph], Feb. 2021.
  18. T. Nishino and K. Okunishi, “Corner Transfer Matrix Renormalization Group Method,” Journal of the Physical Society of Japan, vol. 65, pp. 891–894, Apr. 1996.
  19. T. Nishino and K. Okunishi, “Corner Transfer Matrix Algorithm for Classical Renormalization Group,” Journal of the Physical Society of Japan, vol. 66, pp. 3040–3047, Oct. 1997.
  20. M. Levin and C. P. Nave, “Tensor renormalization group approach to two-dimensional classical lattice models,” Physical review letters, vol. 99, no. 12, p. 120601, 2007.
  21. R. Orus and G. Vidal, “Simulation of two dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction,” Physical Review B, vol. 80, p. 094403, Sept. 2009.
  22. Z. Y. Xie, H. C. Jiang, Q. N. Chen, Z. Y. Weng, and T. Xiang, “Second Renormalization of Tensor-Network States,” Physical Review Letters, vol. 103, p. 160601, Oct. 2009.
  23. L. Vanderstraeten, B. Vanhecke, and F. Verstraete, “Residual entropies for three-dimensional frustrated spin systems with tensor networks,” Physical Review E, vol. 98, p. 042145, Oct. 2018.
  24. H.-H. Zhao, Z.-Y. Xie, T. Xiang, and M. Imada, “Tensor network algorithm by coarse-graining tensor renormalization on finite periodic lattices,” Physical Review B, vol. 93, no. 12, p. 125115, 2016.
  25. J.-G. Liu, L. Wang, and P. Zhang, “Tropical tensor network for ground states of spin glasses,” Physical Review Letters, vol. 126, no. 9, p. 090506, 2021.
  26. A. J. Ferris and D. Poulin, “Tensor networks and quantum error correction,” Phys. Rev. Lett., vol. 113, p. 030501, Jul 2014.
  27. S. Bravyi, M. Suchara, and A. Vargo, “Efficient algorithms for maximum likelihood decoding in the surface code,” Phys. Rev. A, vol. 90, p. 032326, Sep 2014.
  28. D. K. Tuckett, A. S. Darmawan, C. T. Chubb, S. Bravyi, S. D. Bartlett, and S. T. Flammia, “Tailoring surface codes for highly biased noise,” Phys. Rev. X, vol. 9, p. 041031, Nov 2019.
  29. C. T. Chubb and S. T. Flammia, “Statistical mechanical models for quantum codes with correlated noise,” Annales de l’Institut Henri Poincaré D, vol. 8, no. 2, pp. 269–321, 2021.
  30. C. T. Chubb, “General tensor network decoding of 2d pauli codes,” arXiv preprint arXiv:2101.04125, 2021.
  31. J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, “The xzzx surface code,” Nature communications, vol. 12, no. 1, pp. 1–12, 2021.
  32. T. Farrelly, R. J. Harris, N. A. McMahon, and T. M. Stace, “Tensor-Network Codes,” Physical Review Letters, vol. 127, p. 040507, July 2021.
  33. N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, “Computational complexity of projected entangled pair states,” Physical review letters, vol. 98, no. 14, p. 140506, 2007.
  34. A. García-Sáez and J. I. Latorre, “An exact tensor network for the 3sat problem,” arXiv preprint arXiv:1105.3201, 2011.
  35. J. D. Biamonte, J. Morton, and J. Turner, “Tensor Network Contractions for #SAT,” Journal of Statistical Physics, vol. 160, pp. 1389–1404, Sept. 2015.
  36. S. Kourtis, C. Chamon, E. Mucciolo, and A. Ruckenstein, “Fast counting with tensor networks,” SciPost Physics, vol. 7, no. 5, p. 060, 2019.
  37. J. M. Dudek, L. Duenas-Osorio, and M. Y. Vardi, “Efficient contraction of large tensor networks for weighted model counting through graph decompositions,” arXiv preprint arXiv:1908.04381, 2019.
  38. J. M. Dudek and M. Y. Vardi, “Parallel weighted model counting with tensor networks,” arXiv preprint arXiv:2006.15512, 2020.
  39. N. de Beaudrap, A. Kissinger, and K. Meichanetzidis, “Tensor network rewriting strategies for satisfiability and counting,” arXiv preprint arXiv:2004.06455, 2020.
  40. A. J. Gallego and R. Orus, “Language design as information renormalization,” arXiv preprint arXiv:1708.01525, 2017.
  41. V. Pestun and Y. Vlassopoulos, “Tensor network language model,” arXiv preprint arXiv:1710.10248, 2017.
  42. T.-D. Bradley, E. M. Stoudenmire, and J. Terilla, “Modeling sequences with quantum states: a look under the hood,” Machine Learning: Science and Technology, vol. 1, no. 3, p. 035008, 2020.
  43. K. Meichanetzidis, A. Toumi, G. de Felice, and B. Coecke, “Grammar-aware question-answering on quantum computers,” arXiv preprint arXiv:2012.03756, 2020.
  44. F. Verstraete, V. Murg, and J. I. Cirac, “Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems,” Advances in physics, vol. 57, no. 2, pp. 143–224, 2008.
  45. R. Orús, “Tensor networks for complex quantum systems,” Nature Reviews Physics, vol. 1, no. 9, pp. 538–550, 2019.
  46. B. Dittrich, F. C. Eckert, and M. Martin-Benito, “Coarse graining methods for spin net and spin foam models,” New Journal of Physics, vol. 14, p. 035008, Mar. 2012.
  47. G. Evenbly and G. Vidal, “Tensor Network Renormalization,” Physical Review Letters, vol. 115, p. 180405, Oct. 2015.
  48. S. Yang, Z.-C. Gu, and X.-G. Wen, “Loop optimization for tensor network renormalization,” Physical Review Letters, vol. 118, p. 110504, Mar. 2017.
  49. M. Bal, M. Mariën, J. Haegeman, and F. Verstraete, “Renormalization Group Flows of Hamiltonians Using Tensor Networks,” Physical Review Letters, vol. 118, p. 250602, June 2017.
  50. Springer Nature, 2020.
  51. A. Jermyn, “Automatic contraction of unstructured tensor networks,” SciPost Physics, vol. 8, no. 1, p. 005, 2020.
  52. F. Pan, P. Zhou, S. Li, and P. Zhang, “Contracting arbitrary tensor networks: General approximate algorithm and applications in graphical models and quantum circuit simulations,” Phys. Rev. Lett., vol. 125, p. 060503, Aug 2020.
  53. There is a minor effect on memory.
  54. G. Evenbly and G. Vidal, “Tensor network renormalization,” Physical review letters, vol. 115, no. 18, p. 180405, 2015.
  55. M. Hauru, C. Delcamp, and S. Mizera, “Renormalization of tensor networks using graph-independent local truncations,” Physical Review B, vol. 97, no. 4, p. 045111, 2018.
  56. S. Yang, Z.-C. Gu, and X.-G. Wen, “Loop optimization for tensor network renormalization,” Physical review letters, vol. 118, no. 11, p. 110504, 2017.
  57. J. Gray and G. K.-L. Chan, “Supplementary information for “hyper-optimized approximate contraction of tensor networks with arbitrary geometry,” 2022.
  58. G. Evenbly, “Gauge fixing, canonical forms, and optimal truncations in tensor networks with closed loops,” Physical Review B, vol. 98, no. 8, p. 085155, 2018.
  59. M. Hauru, C. Delcamp, and S. Mizera, “Renormalization of tensor networks using graph independent local truncations,” Physical Review B, vol. 97, p. 045111, Jan. 2018.
  60. L. Wang and F. Verstraete, “Cluster update for tensor network states,” Oct. 2011.
  61. P. Corboz, T. M. Rice, and M. Troyer, “Competing states in the t-J model: Uniform d-wave state versus stripe state,” Physical Review Letters, vol. 113, p. 046402, July 2014.
  62. S. Iino, S. Morita, and N. Kawashima, “Boundary Tensor Renormalization Group,” Physical Review B, vol. 100, p. 035449, July 2019.
  63. S. Schlag, High-Quality Hypergraph Partitioning. PhD thesis, Karlsruhe Institute of Technology, Germany, 2020.
  64. S. Schlag, T. Heuer, L. Gottesbüren, Y. Akhremtsev, C. Schulz, and P. Sanders, “High-quality hypergraph partitioning,” ACM J. Exp. Algorithmics, mar 2022.
  65. T. Akiba, S. Sano, T. Yanase, T. Ohta, and M. Koyama, “Optuna: A next-generation hyperparameter optimization framework,” in Proceedings of the 25rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2019.
  66. J. Rapin and O. Teytaud, “Nevergrad - A gradient-free optimization platform.” https://GitHub.com/FacebookResearch/Nevergrad, 2018.
  67. T. Nishino and K. Okunishi, “Corner transfer matrix renormalization group method,” Journal of the Physical Society of Japan, vol. 65, no. 4, pp. 891–894, 1996.
  68. Z.-Y. Xie, J. Chen, M.-P. Qin, J. W. Zhu, L.-P. Yang, and T. Xiang, “Coarse-graining renormalization by higher-order singular value decomposition,” Physical Review B, vol. 86, no. 4, p. 045139, 2012.
  69. S. Bramwell and M. Harris, “Frustration in ising-type spin models on the pyrochlore lattice,” Journal of Physics: Condensed Matter, vol. 10, no. 14, p. L215, 1998.
  70. A. L. Passos, D. F. de Albuquerque, and J. B. S. Filho, “Representation and simulation for pyrochlore lattice via Monte Carlo technique,” Physica A: Statistical Mechanics and its Applications, vol. 450, pp. 541–545, May 2016.
  71. K. Soldatov, K. Nefedev, Y. Komura, and Y. Okabe, “Large-scale calculation of ferromagnetic spin systems on the pyrochlore lattice,” Physics Letters A, vol. 381, pp. 707–712, Feb. 2017.
  72. F. Wang and D. P. Landau, “Efficient, Multiple-Range Random Walk Algorithm to Calculate the Density of States,” Physical Review Letters, vol. 86, pp. 2050–2053, Mar. 2001.
  73. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of State Calculations by Fast Computing Machines,” The Journal of Chemical Physics, vol. 21, pp. 1087–1092, June 1953.
  74. J. Raymond, A. Sportiello, and L. Zdeborová, “Phase diagram of the 1-in-3 satisfiability problem,” Physical Review E, vol. 76, p. 011101, July 2007.
  75. L. Zdeborová and M. Mézard, “Constraint satisfaction problems with isolated solutions are hard,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2008, p. P12004, Dec. 2008.
  76. T. J. Schaefer, “The complexity of satisfiability problems,” in Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, STOC ’78, (New York, NY, USA), pp. 216–226, Association for Computing Machinery, May 1978.
  77. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. USA: W. H. Freeman & Co., 1979.
  78. B. Bollobás and B. D. McKay, “The number of matchings in random regular graphs and bipartite graphs,” Journal of Combinatorial Theory, Series B, vol. 41, no. 1, pp. 80–91, 1986.
  79. T. Grover and M. P. A. Fisher, “Entanglement and the sign structure of quantum states,” Physical Review A, vol. 92, p. 042308, Oct. 2015.
  80. L. A. Goldberg and H. Guo, “The complexity of approximating complex-valued Ising and Tutte partition functions,” Jan. 2017.
  81. A. Galanis, L. A. Goldberg, and A. Herrera-Poyatos, “The complexity of approximating the complex-valued Ising model on bounded degree graphs,” May 2021.
  82. P. Buys, A. Galanis, V. Patel, and G. Regts, “Lee–Yang zeros and the complexity of the ferromagnetic Ising model on bounded-degree graphs,” Forum of Mathematics, Sigma, vol. 10, 2022/ed.
  83. M. A. Beauchamp, “An improved index of centrality,” Behavioral science, vol. 10, no. 2, pp. 161–163, 1965.
  84. M. Marchiori and V. Latora, “Harmony in the small-world,” Physica A: Statistical Mechanics and its Applications, vol. 285, no. 3-4, pp. 539–546, 2000.
  85. L. Onsager, “Crystal statistics. i. a two-dimensional model with an order-disorder transition,” Physical Review, vol. 65, no. 3-4, p. 117, 1944.
  86. R. J. Baxter, Exactly solved models in statistical mechanics. Elsevier, 2016.
  87. F. Pan, P. Zhou, S. Li, and P. Zhang, “CATN.” https://github.com/panzhang83/catn, 2019.
  88. C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke, and T. E. Oliphant, “Array programming with NumPy,” Nature, vol. 585, pp. 357–362, Sept. 2020.
Citations (16)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up