Papers
Topics
Authors
Recent
Search
2000 character limit reached

Global optimization of MPS in quantum-inspired numerical analysis

Published 16 Mar 2023 in quant-ph | (2303.09430v2)

Abstract: This work discusses the solution of partial differential equations (PDEs) using matrix product states (MPS). The study focuses on the search for the lowest eigenstates of a Hamiltonian equation, for which five algorithms are introduced: imaginary-time evolution, steepest gradient descent, an improved gradient descent, an implicitly restarted Arnoldi method, and density matrix renormalization group (DMRG) optimization. The first four methods are engineered using a framework of limited-precision linear algebra, where operations between MPS and matrix product operators (MPOs) are implemented with finite resources. All methods are benchmarked using the PDE for a quantum harmonic oscillator in up to two dimensions, over a regular grid with up to $2{28}$ points. Our study reveals that all MPS-based techniques outperform exact diagonalization techniques based on vectors, with respect to memory usage. Imaginary-time algorithms are shown to underperform any type of gradient descent, both in terms of calibration needs and costs. Finally, Arnoldi like methods and DMRG asymptotically outperform all other methods, including exact diagonalization, as problem size increases, with an exponential advantage in memory and time usage.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (97)
  1. Román Orús. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of Physics, 349:117–158, oct 2014.
  2. Matrix product operators, matrix product states, and ab initio density matrix renormalization group algorithms, 2016.
  3. Hand-waving and interpretive dance: an introductory course on tensor networks. Journal of Physics A: Mathematical and Theoretical, 50(22):223001, may 2017.
  4. Time-evolution methods for matrix-product states. Annals of Physics, 411:167998, 2019.
  5. Tensor Network Contractions: Methods and Applications to Quantum Many-Body Systems. Lecture Notes in Physics. Springer International Publishing, 2020.
  6. Guifré Vidal. Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett., 93:040502, Jul 2004.
  7. Mixed-state dynamics in one-dimensional quantum lattice systems: A time-dependent superoperator renormalization algorithm. Phys. Rev. Lett., 93:207205, Nov 2004.
  8. Matrix product density operators: Simulation of finite-temperature and dissipative systems. Phys. Rev. Lett., 93:207204, Nov 2004.
  9. Steven R. White. Density matrix formulation for quantum renormalization groups. Physical Review Letters, 69(19):2863–2866, nov 1992.
  10. Steven R. White. Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B, 48:10345–10356, Oct 1993.
  11. Criticality, the area law, and the computational power of projected entangled pair states. Phys. Rev. Lett., 96:220601, Jun 2006.
  12. Classification of gapped symmetric phases in one-dimensional spin systems. Phys. Rev. B, 83:035107, Jan 2011.
  13. Classifying quantum phases using matrix product states and projected entangled pair states. Phys. Rev. B, 84:165139, Oct 2011.
  14. Contraction of fermionic operator circuits and the simulation of strongly correlated fermions. Phys. Rev. A, 80:042333, Oct 2009.
  15. Fermionic multiscale entanglement renormalization ansatz. Phys. Rev. B, 80:165129, Oct 2009.
  16. Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law. Physical Review B, 80:235127, December 2009.
  17. Simulation of interacting fermions with entanglement renormalization. Phys. Rev. A, 81:010303, Jan 2010.
  18. Simulation of strongly correlated fermions in two spatial dimensions with fermionic projected entangled-pair states. Phys. Rev. B, 81:165104, Apr 2010.
  19. Fermionic projected entangled pair states. Phys. Rev. A, 81:052338, May 2010.
  20. Fermionic implementation of projected entangled pair states algorithm. Phys. Rev. B, 81:245110, Jun 2010.
  21. Román Orús. Advances on tensor network theory: symmetries, fermions, entanglement, and holography. The European Physical Journal B, 87(11), nov 2014.
  22. Román Orús. Tensor networks for complex quantum systems. Nature Reviews Physics, 1(9):538–550, aug 2019.
  23. Supervised learning with tensor networks. Advances in Neural Information Processing Systems, 29, 2016.
  24. E Miles Stoudenmire. Learning relevant features of data with multi-scale tensor networks. Quantum Science and Technology, 3(3):034003, apr 2018.
  25. Machine learning by unitary tensor network of hierarchical tree structure. New Journal of Physics, 21(7):073059, 2019.
  26. Unsupervised generative modeling using matrix product states. Phys. Rev. X, 8:031012, Jul 2018.
  27. Tree tensor networks for generative modeling. Phys. Rev. B, 99:155131, Apr 2019.
  28. Generative modeling with projected entangled-pair states. arXiv preprint arXiv:2202.08177, 2022.
  29. Learning generative models for active inference using tensor networks. arXiv preprint arXiv:2208.08713, 2022.
  30. Self-correcting quantum many-body control using reinforcement learning with tensor networks. arXiv preprint arXiv:2201.11790, 2022.
  31. Towards quantum machine learning with tensor networks. Quantum Science and technology, 4(2):024001, 2019.
  32. Improvements to gradient descent methods for quantum tensor network machine learning, 2022.
  33. A quantum-inspired tensor network method for constrained combinatorial optimization problems. arXiv preprint arXiv:2203.15246, 2022.
  34. Juan José García-Ripoll. Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations. Quantum, 5:431, apr 2021.
  35. Density matrix renormalization group and periodic boundary conditions: A quantum information perspective. Phys. Rev. Lett., 93:227205, Nov 2004.
  36. U. Schollwöck. The density-matrix renormalization group. Rev. Mod. Phys., 77:259–315, Apr 2005.
  37. Ulrich Schollwöck. The density-matrix renormalization group in the age of matrix product states. Annals of Physics, 326(1):96–192, jan 2011.
  38. Time-dependent variational principle for quantum lattices. Physical Review Letters, 107(7), aug 2011.
  39. Tangent-space methods for uniform matrix product states. SciPost Physics Lecture Notes, jan 2019.
  40. Salvatore R. Manmana. Time evolution of one-dimensional quantum many body systems. In AIP Conference Proceedings. AIP, 2005.
  41. Juan José García-Ripoll. Time evolution of matrix product states. New Journal of Physics, 8(12):305, dec 2006.
  42. Dominic W. Berry. High-order quantum algorithm for solving linear differential equations. Journal of Physics A: Mathematical and Theoretical, 47(10):105301, feb 2014.
  43. Quantum algorithm for linear differential equations with exponentially improved dependence on precision. Communications in Mathematical Physics, 356(3):1057–1081, oct 2017.
  44. Quantum algorithms and the finite element method. Physical Review A, 93(3), mar 2016.
  45. High-precision quantum algorithms for partial differential equations. arXiv e-prints, page arXiv:2002.07868, February 2020.
  46. John Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2:79, aug 2018.
  47. Quantum fourier analysis for multivariate functions and applications to a class of schrödinger-type partial differential equations. Phys. Rev. A, 105:012433, Jan 2022.
  48. Variational quantum algorithms for nonlinear problems. Physical Review A, 101(1), jan 2020.
  49. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Information, 5(1), sep 2019.
  50. Solving nonlinear differential equations with differentiable quantum circuits. Phys. Rev. A, 103:052416, May 2021.
  51. Solving Differential Equations via Continuous-Variable Quantum Computers. arXiv e-prints, page arXiv:2012.12220, December 2020.
  52. Fast solution of parabolic problems in the tensor train/quantized tensor train format with initial application to the fokker–planck equation. SIAM Journal on Scientific Computing, 34(6):A3016–A3038, 2012.
  53. Solving high-dimensional parabolic pdes using the tensor train format, 2021.
  54. Polynomial chaos expansion of random coefficients and the solution of stochastic partial differential equations in the tensor train format. SIAM/ASA Journal on Uncertainty Quantification, 3(1):1109–1135, jan 2015.
  55. Adaptive stochastic galerkin FEM with hierarchical tensor representations. Numerische Mathematik, 136(3):765–803, nov 2016.
  56. Rank-adaptive tensor methods for high-dimensional nonlinear PDEs. Journal of Scientific Computing, 88(2), jun 2021.
  57. A continuous analogue of the tensor-train decomposition. Computer methods in applied mechanics and engineering, 347:59–84, 2019.
  58. Linear hamilton jacobi bellman equations in high dimensions. In 53rd IEEE Conference on Decision and Control. IEEE, dec 2014.
  59. Sequential alternating least squares for solving high dimensional linear hamilton-jacobi-bellman equation. In 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 3757–3764, 2016.
  60. High-dimensional stochastic optimal control using continuous tensor decompositions. The International Journal of Robotics Research, 37(2-3):340–377, 2018.
  61. Tensor decomposition methods for high-dimensional hamilton–jacobi–bellman equations. SIAM Journal on Scientific Computing, 43(3):A1625–A1650, 2021.
  62. Approximating the stationary bellman equation by hierarchical tensor products, 2019.
  63. Functional tensor network solving many-body schrödinger equation. Phys. Rev. B, 105:165116, Apr 2022.
  64. Solution of the fokker–planck equation by cross approximation method in the tensor train format. Frontiers in Artificial Intelligence, 4, 2021.
  65. Differentiable programming tensor networks. Phys. Rev. X, 9:031041, Sep 2019.
  66. Christof Zalka. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454(1969):313–322, jan 1998.
  67. Creating superpositions that correspond to efficiently integrable probability distributions. arXiv e-prints, pages quant–ph/0208112, August 2002.
  68. A hybrid Alternating Least Squares – TT Cross algorithm for parametric PDEs, July 2018.
  69. Single-Block Renormalization Group: Quantum Mechanical Problems. Nuclear Physics B, 601(3):569–590, May 2001.
  70. Javier Rodriguez-Laguna. Real Space Renormalization Group Techniques and Applications, July 2002.
  71. Time-evolving a matrix product state with long-ranged interactions. Phys. Rev. B, 91:165112, Apr 2015.
  72. Erwin Fehlberg. Classical fifth-, sixth-, seventh-, and eighth-order runge-kutta formulas with stepsize control. NASA Technical Report, 287, 1968.
  73. A comparison of different propagation schemes for the time dependent schrödinger equation. Journal of Computational Physics, 94(1):59–80, 1991.
  74. Time-dependent density-matrix renormalization-group using adaptive effective hilbert spaces. Journal of Statistical Mechanics: Theory and Experiment, 2004(04):P04005, apr 2004.
  75. Time-step targeting methods for real-time dynamics using the density matrix renormalization group. Phys. Rev. B, 72:020404, Jul 2005.
  76. Adaptive subgradient methods for online learning and stochastic optimization. Journal of Machine Learning Research, 12(61):2121–2159, 2011.
  77. Gradient-based optimization for regression in the functional tensor-train format. Journal of Computational Physics, 374:1219–1238, 2018.
  78. Anomaly detection with tensor networks, 2020.
  79. W. E. ARNOLDI. The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quarterly of Applied Mathematics, 9(1):17–29, 1951.
  80. Karen A. Hallberg. Density-matrix algorithm for the calculation of dynamical properties of low-dimensional systems. Phys. Rev. B, 52:R9827–R9830, Oct 1995.
  81. Adaptive lanczos-vector method for dynamic properties within the density matrix renormalization group. Phys. Rev. B, 83:161104, Apr 2011.
  82. Lanczos algorithm with matrix product states for dynamical correlation functions. Phys. Rev. B, 85:205119, May 2012.
  83. Chebyshev matrix product state approach for spectral functions. Phys. Rev. B, 83:195115, May 2011.
  84. ARPACK Users’ Guide. Society for Industrial and Applied Mathematics, 1998.
  85. PRIMME: PReconditioned Iterative MultiMethod Eigensolver: Methods and software description. ACM Transactions on Mathematical Software, 37(2):21:1–21:30, 2010.
  86. Primme_svds: A high-performance preconditioned SVD solver for accurate large-scale computations. SIAM Journal on Scientific Computing, 39(5):S248–S271, 2017.
  87. Juan José García-Ripoll. mps C++ GitHub repository, 2018.
  88. Chebyshev matrix product state impurity solver for dynamical mean-field theory. Phys. Rev. B, 90:115124, Sep 2014.
  89. Entanglement entropy for the long-range ising chain in a transverse field. Phys. Rev. Lett., 109:267203, Dec 2012.
  90. Unifying time evolution and optimization with matrix product states. Phys. Rev. B, 94:165116, Oct 2016.
  91. Variational optimization algorithms for uniform matrix product states. Phys. Rev. B, 97:045145, Jan 2018.
  92. M. Chandross and J. C. Hicks. Density-matrix renormalization-group method for excited states. Phys. Rev. B, 59:9699–9702, Apr 1999.
  93. Excited-state dmrg made simple with feast. Journal of Chemical Theory and Computation, 18(1):415–430, 2022. PMID: 34914392.
  94. Eric Jeckelmann. Dynamical density-matrix renormalization-group method. Phys. Rev. B, 66:045114, Jul 2002.
  95. Real-time evolution using the density matrix renormalization group. Phys. Rev. Lett., 93:076401, Aug 2004.
  96. Ulrich Schollwöck. Methods for time dependence in DMRG. In AIP Conference Proceedings. AIP, 2006.
  97. The pricing of options and corporate liabilities. Journal of Political Economy, 81(3):637–654, may 1973.
Citations (7)
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

Tweets

Sign up for free to view the 1 tweet with 0 likes about this paper.

Sign Up for Free