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Sample complexity of matrix product states at finite temperature (2403.10018v2)

Published 15 Mar 2024 in cond-mat.stat-mech and quant-ph

Abstract: For quantum many-body systems in one dimension, computational complexity theory reveals that the evaluation of ground-state energy remains elusive on quantum computers, contrasting the existence of a classical algorithm for temperatures higher than the inverse logarithm of the system size. This highlights a qualitative difference between low- and high-temperature states in terms of computational complexity. Here, we describe finite-temperature states using the matrix product state formalism. Within the framework of random samplings, we derive an analytical formula for the required number of samples, which provides both quantitative and qualitative measures of computational complexity. At high and low temperatures, its scaling behavior with system size is linear and quadratic, respectively, demonstrating a distinct crossover between these numerically difficult regimes of quantitative difference.

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References (65)
  1. J. M. Deutsch, Eigenstate thermalization hypothesis, Reports on Progress in Physics 81, 082001 (2018).
  2. R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annual Review of Condensed Matter Physics 6, 15–38 (2015).
  3. S. Moudgalya, B. A. Bernevig, and N. Regnault, Quantum many-body scars and hilbert space fragmentation: a review of exact results, Reports on Progress in Physics 85, 086501 (2022).
  4. A. Kitaev, A. Shen, and M. Vyalyi, Classical and Quantum Computation, Graduate studies in mathematics (American Mathematical Society, 2002).
  5. R. Oliveira and B. Terhal, The complexity of quantum spin systems on a two-dimensional square lattice, Quantum Information and Computation 8, 900–924 (2008).
  6. D. Nagaj, Local hamiltonians in quantum computation, arXiv:0808.2117  (2008).
  7. S. Hallgren, D. Nagaj, and S. Narayanaswami, The local hamiltonian problem on a line with eight staes is qma-complete, Quantum Information and Computation 13, 721–750 (2013).
  8. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  9. S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993).
  10. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011), january 2011 Special Issue.
  11. Z. Landau, U. Vazirani, and T. Vidick, A polynomial time algorithm for the ground state of one-dimensional gapped local hamiltonians, Nature Physics 11, 566 (2015).
  12. C. T. Chubb and S. T. Flammia, Computing the degenerate ground space of gapped spin chains in polynomial time, Chicago Journal of Theoretical Computer Science 2016 (2016).
  13. B. Roberts, T. Vidick, and O. I. Motrunich, Implementation of rigorous renormalization group method for ground space and low-energy states of local hamiltonians, Phys. Rev. B 96, 214203 (2017).
  14. M. B. Hastings, An area law for one-dimensional quantum systems, Journal of Statistical Mechanics: Theory and Experiment 2007, P08024 (2007).
  15. J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010).
  16. T. S. Cubitt, D. Perez-Garcia, and M. M. Wolf, Undecidability of the spectral gap, Nature 528, 207 (2015).
  17. T. Kuwahara, A. M. Alhambra, and A. Anshu, Improved thermal area law and quasilinear time algorithm for quantum gibbs states, Phys. Rev. X 11, 011047 (2021).
  18. A. M. Alhambra and J. I. Cirac, Locally accurate tensor networks for thermal states and time evolution, PRX Quantum 2, 040331 (2021).
  19. T. Kuwahara, K. Kato, and F. G. S. L. Brandão, Clustering of conditional mutual information for quantum gibbs states above a threshold temperature, Phys. Rev. Lett. 124, 220601 (2020).
  20. D. Aharonov, I. Arad, and T. Vidick, Guest column: the quantum pcp conjecture, ACM SIGACT News 44, 47–79 (2013).
  21. A. W. Sandvik, Computational Studies of Quantum Spin Systems, AIP Conference Proceedings 1297, 135 (2010).
  22. C. Domb and M. S. Green, Phase Transitions and Critical Phenomena (Academic Press, New York, 1974).
  23. M. Rigol, T. Bryant, and R. R. P. Singh, Numerical linked-cluster approach to quantum lattice models, Phys. Rev. Lett. 97, 187202 (2006).
  24. M. Rigol, T. Bryant, and R. R. P. Singh, Numerical linked-cluster algorithms. ii. t−j𝑡𝑗t\text{$-$}jitalic_t - italic_j models on the square lattice, Phys. Rev. E 75, 061119 (2007b).
  25. M. Imada and M. Takahashi, Quantum transfer monte carlo method for finite temperature properties and quantum molecular dynamics method for dynamical correlation functions, Journal of the Physical Society of Japan 55, 3354 (1986).
  26. J. Jaklič and P. Prelovšek, Lanczos method for the calculation of finite-temperature quantities in correlated systems, Phys. Rev. B 49, 5065 (1994).
  27. A. Hams and H. De Raedt, Fast algorithm for finding the eigenvalue distribution of very large matrices, Phys. Rev. E 62, 4365 (2000).
  28. S. Sugiura and A. Shimizu, Thermal pure quantum states at finite temperature, Phys. Rev. Lett. 108, 240401 (2012).
  29. S. Sugiura and A. Shimizu, Canonical thermal pure quantum state, Phys. Rev. Lett. 111, 010401 (2013).
  30. S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics, Nature Phys. 2, 754 (2006).
  31. P. Reimann, Typicality for generalized microcanonical ensembles, Phys. Rev. Lett. 99, 160404 (2007).
  32. A. Sugita, On the basis of quantum statistical mechanics, Nonlinear Phenom. Complex Syst 10, 192 (2007).
  33. M. Fannes, B. Nachtergaele, and R. F. Werner, Finitely correlated states on quantum spin chains, Commun. Math. Phys. 144, 443 (1992).
  34. G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett. 93, 040502 (2004).
  35. S. R. White and A. E. Feiguin, Real-time evolution using the density matrix renormalization group, Phys. Rev. Lett. 93, 076401 (2004).
  36. F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, Matrix product density operators: Simulation of finite-temperature and dissipative systems, Phys. Rev. Lett. 93, 207204 (2004).
  37. M. Zwolak and G. Vidal, Mixed-state dynamics in one-dimensional quantum lattice systems: A time-dependent superoperator renormalization algorithm, Phys. Rev. Lett. 93, 207205 (2004).
  38. A. E. Feiguin and S. R. White, Finite-temperature density matrix renormalization using an enlarged hilbert space, Phys. Rev. B 72, 220401 (2005).
  39. S. Garnerone and T. R. de Oliveira, Generalized quantum microcanonical ensemble from random matrix product states, Phys. Rev. B 87, 214426 (2013).
  40. S. Garnerone, Pure state thermodynamics with matrix product states, Phys. Rev. B 88, 165140 (2013).
  41. S. Goto, R. Kaneko, and I. Danshita, Matrix product state approach for a quantum system at finite temperatures using random phases and trotter gates, Phys. Rev. B 104, 045133 (2021).
  42. S. R. White, Minimally entangled typical quantum states at finite temperature, Phys. Rev. Lett. 102, 190601 (2009).
  43. E. M. Stoudenmire and S. R. White, Minimally entangled typical thermal state algorithms, New Journal of Physics 12, 055026 (2010).
  44. B. Bruognolo, J. von Delft, and A. Weichselbaum, Symmetric minimally entangled typical thermal states, Phys. Rev. B 92, 115105 (2015).
  45. M. Binder and T. Barthel, Minimally entangled typical thermal states versus matrix product purifications for the simulation of equilibrium states and time evolution, Phys. Rev. B 92, 125119 (2015).
  46. M. Binder and T. Barthel, Symmetric minimally entangled typical thermal states for canonical and grand-canonical ensembles, Phys. Rev. B 95, 195148 (2017).
  47. S. Goto and I. Danshita, Minimally entangled typical thermal states algorithm with trotter gates, Phys. Rev. Res. 2, 043236 (2020).
  48. J. Chen and E. M. Stoudenmire, Hybrid purification and sampling approach for thermal quantum systems, Phys. Rev. B 101, 195119 (2020).
  49. C.-M. Chung and U. Schollwöck, Minimally entangled typical thermal states with auxiliary matrix-product-state bases, arXiv:1910.03329  (2019).
  50. A. Iwaki and C. Hotta, Purity of thermal mixed quantum states, Phys. Rev. B 106, 094409 (2022).
  51. A. Iwaki, A. Shimizu, and C. Hotta, Thermal pure quantum matrix product states recovering a volume law entanglement, Phys. Rev. Research 3, L022015 (2021).
  52. M. Gohlke, A. Iwaki, and C. Hotta, Thermal pure matrix product state in two dimensions: Tracking thermal equilibrium from paramagnet down to the Kitaev honeycomb spin liquid state, SciPost Phys. 15, 206 (2023).
  53. H. N. Temperley and E. H. Lieb, Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 322, 251–280 (1971).
  54. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calculations, SciPost Phys. Codebases , 4 (2022).
  55. P. Pfeuty, The one-dimensional ising model with a transverse field, Annals of Physics 57, 79 (1970).
  56. H. Bethe, Zur theorie der metalle, Z. Phys. 71, 205 (1931).
  57. F. D. M. Haldane, Ground state properties of antiferromagnetic chains with unrestricted spin: Integer spin chains as realisations of the o(3) non-linear sigma model, arXiv:1612.00076  (1981).
  58. F. Haldane, Continuum dynamics of the 1-d heisenberg antiferromagnet: Identification with the o(3) nonlinear sigma model, Physics Letters A 93, 464–468 (1983).
  59. S. R. White and D. A. Huse, Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic s=1 heisenberg chain, Phys. Rev. B 48, 3844 (1993).
  60. R. Movassagh and P. W. Shor, Supercritical entanglement in local systems: Counterexample to the area law for quantum matter, Proceedings of the National Academy of Sciences 113, 13278–13282 (2016).
  61. K. Seki and S. Yunoki, Energy-filtered random-phase states as microcanonical thermal pure quantum states, Phys. Rev. B 106, 155111 (2022).
  62. L. Coopmans, Y. Kikuchi, and M. Benedetti, Predicting gibbs-state expectation values with pure thermal shadows, PRX Quantum 4, 010305 (2023).
  63. S. Goto, R. Kaneko, and I. Danshita, Evaluating thermal expectation values by almost ideal sampling with trotter gates, Phys. Rev. B 107, 024307 (2023).
  64. K. Mizukami and A. Koga, Quantum algorithm for the microcanonical thermal pure quantum state method, Phys. Rev. A 108, 012404 (2023).
  65. Z. Davoudi, N. Mueller, and C. Powers, Towards quantum computing phase diagrams of gauge theories with thermal pure quantum states, Phys. Rev. Lett. 131, 081901 (2023).

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