Skip to main content
Log in

Modeling Complex Quantum Dynamics: Evolution of Numerical Algorithms in the HPC Context

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

Due to complexity of the systems and processes it addresses, the development of computational quantum physics is influenced by the progress in computing technology. Here we overview the evolution, from the late 1980s to the current year 2020, of the algorithms used to simulate dynamics of quantum systems. We put the emphasis on implementation aspects and computational resource scaling with the model size and propagation time. Our mini-review is based on a literature survey and our experience in implementing different types of algorithms on supercomputers ‘‘Lobachevskii’’ (at Lobachevskii State University of Nizhny Novgorod) and ‘‘Lomonosov 2’’ (at Moscow State University).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

REFERENCES

  1. K. B. Davis et al., ‘‘Bose-Einstein condensation in a gas of sodium atoms,’’ Phys. Rev. Lett. 75, 3969–3973 (1995).

    Google Scholar 

  2. R. Barendes et al., ‘‘Coherent Josephson qubit suitable for scalable quantum integrated circuits,’’ Phys. Rev. Lett. 11, 080502 (2013).

  3. Ali W. Elshaari, W. Pernice, K. Srinivasan, O. Benson, and V. Zwiller, ‘‘Hybrid integrated quantum photonic circuits,’’ Nat. Photon. (2020, in press). https://doi.org/10.1038/s41566-020-0609-x

  4. C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, ‘‘Trapped-ion quantum computing: Progress and challenges,’’ Appl. Phys. Rev. 6, 021314 (2019).

  5. J. Yoneda et al., ‘‘A quantum-dot spin qubit with coherence limited by chargenoise and fidelity higher than 99.9%,’’ Nat. Nanotechnol. 13, 102–106 (2018).

    Google Scholar 

  6. N. Friis et al., ‘‘Observation of entangled states of a fully controlled 20-qubit system,’’ Phys. Rev. X 8, 021012 (2018).

  7. Gartner: Hype Cycle Research Methodology. https://www.gartner.com/en/research/methodologies/gart- ner-hype-cycle. Accessed 2020.

  8. ‘‘Towards quantum attractors: computational physics of open quantum nanosystems far from equlibrium,’’ Russ. Sci. Foundation Project no. 15-12-20029 (2015–2017). https://www.rscf.ru/en/contests/search-projects/15-12-20029/.

  9. ‘‘Dissipative quantum chaos: categorization with high performance computing,’’ Russ. Sci. Foundation Project no. 19-72-20086 (2019–2022). https://www.rscf.ru/en/contests/search-projects/19-72-20086/.

  10. V. I. Voevodin, A. Antonov, D. Nikitenko, P. Shvets, S. Sobolev, I. Sidorov, K. Stefanov, Vad. Voevodin, and S. Zhumatiy, ‘‘Supercomputer Lomonosov-2: Large scale, deep monitoring and fine analytics for the user community,’’ Supercomput. Front. Innov. 6, 4–11 (2019).

    Google Scholar 

  11. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford Univ. Press, Oxford, 2002).

    MATH  Google Scholar 

  12. S. Kohler, J. Lehmann, and P. Hänggi, ‘‘Driven quantum transport on the nanoscale,’’ Phys. Rep. 406, 379–443 (2005).

    Article  Google Scholar 

  13. R. Kosloff, ‘‘Propagation methods for quantum molecular dynamics,’’ Ann. Rev. Phys. Chem. 45, 145–178 (1994).

    Article  Google Scholar 

  14. R. Bellman, Dynamic Programmings (Princeton Univ. Press, Princeton, 1957).

    Google Scholar 

  15. I. V. Oseledets and E. E. Tyrtyshnikov, ‘‘Breaking the curse of dimensionality, or how to use SVD in many dimensions,’’ SIAM J. Sci. Comput. 31, 3744–3759 (2009).

    Article  MathSciNet  Google Scholar 

  16. G. Vidal, ‘‘Efficient classical simulation of slightly entangled quantum computations,’’ Phys. Rev. Lett. 91, 147902 (2003).

  17. J. Haegeman et al., ‘‘Time-dependent variational principle for quantum lattices,’’ Phys. Rev. Lett. 107, 070601 (2011).

  18. J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, ‘‘Unifying time evolution and optimization with matrix product states,’’ Phys. Rev. B 94, 165116 (2016).

  19. B. Kloss, Y. Bar Lev, and D. Reichman, ‘‘Time-dependent variational principle in matrix-product state manifolds: Pitfalls and potential,’’ Phys. Rev. B 97, 024307 (2018).

  20. IBM Q Experience. https://www.ibm.com/quantum-computing/technology/experience/. Accessed 2020.

  21. J. Preskill, ‘‘Quantum computing in the NISQ era and beyond,’’ Quantum 2, 79 (2018).

    Google Scholar 

  22. D. W. Berry et al., ‘‘Simulating Hamiltonian dynamics with a truncated Taylor series,’’ Phys. Rev. Lett. 114, 090502 (2015).

  23. G. H. Low and I. L. Chuang, ‘‘Optimal Hamiltonian simulation by quantum signal processing,’’ Phys. Rev. Lett. 118, 010501 (2017).

  24. G. H. Low and I. L. Chuang, ‘‘Hamiltonian simulation by qubitization,’’ Quantum 3, 163 (2019).

    Google Scholar 

  25. S. Lloyd, ‘‘Universal quantum simulators,’’ Science (Washington, DC, U. S.) 273, 1073 (1996).

    MathSciNet  MATH  Google Scholar 

  26. G. Carleo et al., ‘‘Machine learning and the physical sciences,’’ Rev. Mod. Phys. 91, 045002 (2019).

  27. G. Carleo and M. Troyer, ‘‘Solving the quantum many-body problem with artificial neural networks,’’ Science (Washington, DC, U. S.) 355, 602–606 (2017).

    MathSciNet  MATH  Google Scholar 

  28. I. López-Gutiérrez and C. B. Mendl, ‘‘Real time evolution with neural-network quantum states,’’ arXiv:1912.08831 (2019).

  29. M. Scmitt and M. Heyl, ‘‘Quantum many-body dynamics in two dimensions with artificial neural networks,’’ arXiv:1912.08828 (2019).

  30. T. V. Laptyeva et al., ‘‘Calculating floquet states of large quantum systems: A parallelization strategy and its cluster implementation,’’ Comput. Phys. Commun. 201, 85–94 (2016).

    MathSciNet  MATH  Google Scholar 

  31. A. Liniov et al., ‘‘Unfolding a quantum master equation into a system of real-valued equations: Computationally effective expansion over the basis of SU (N) generators,’’ Phys. Rev. E 100, 053305 (2019).

  32. I. Meyerov et al., ‘‘Transforming the Lindblad equation into a system of linear equations: Performance optimization and parallelization,’’ arXiv: 1912.01491 (2019).

  33. M. Ẑnidarič, A. Scardicchio, and V. K. Varma, ‘‘Diffusive and subdiffusive spin transport in the ergodic phase of a many-body localizable system,’’ Phys. Rev. Lett. 117, 040601 (2016).

  34. V. Volokitin et al., ‘‘Propagating large open quantum systems towards their asymptotic states: Cluster implementation of the time-evolving block decimation scheme,’’ J. Phys.: Conf. Ser. 1392, 012061 (2019).

  35. N. Auer et al., ‘‘Magnus integrators on multicore CPUs and GPUs,’’ Comput. Phys. Commun. 228, 115–122 (2018).

    Article  Google Scholar 

  36. J. R. Johansson, P. D. Nation, and F. Nori, ‘‘QuTiP 2: A Python framework for the dynamics of open quantum systems,’’ Comput. Phys. Commun. 184, 1234–1240 (2013).

    Article  Google Scholar 

  37. K. Björnson, ‘‘TBTK: A quantum mechanics software development kit,’’ SoftwareX 9, 205–210 (2019).

    Article  Google Scholar 

  38. B. Schmidt and U. Lorenz, ‘‘WavePacket: A Matlab package for numerical quantum dynamics. I: Closed quantum systems and discrete variable representations,’’ Comput. Phys. Commun. 213, 223–234 (2017).

    Article  Google Scholar 

  39. B. Schmidt and C. Hartmann, ‘‘WavePacket: A Matlab package for numerical quantum dynamics. II: Open quantum systems, optimal control, and model reduction,’’ Comput. Phys. Commun. 228, 229–244 (2018).

    Article  MathSciNet  Google Scholar 

  40. C. S. Bederián and A. D. Dente, ‘‘Boosting quantum evolutions using Trotter-Suzuki algorithms on GPUs,’’ in Proceedings of HPCLatAm-11, 4th High-Performance Computing Symposium, Cordoba, Argentina, 2011.

  41. T. Auckenthaler et al., ‘‘Matrix exponentials and parallel prefix computation in a quantum control problem,’’ Parallel Comput. 36, 359–369 (2010).

    MathSciNet  MATH  Google Scholar 

  42. D. Jaschke, M. L. Wall, and L. D. Carr, ‘‘Open source matrix product states: Opening ways to simulate entangled many-body quantum systems in one dimension,’’ Comput. Phys. Commun. 225, 59–91 (2018).

    Google Scholar 

  43. P. Wittek and F. M. Cucchietti, ‘‘A second-order distributed Trotter-Suzuki solver with a hybrid CPU-GPU kernel,’’ Comput. Phys. Commun. 184, 1165–1171 (2013).

    MathSciNet  Google Scholar 

  44. P. Wittek and L. Calderaro, ‘‘Extended computational kernels in a massively parallel implementation of the Trotter-Suzuki approximation,’’ Comput. Phys. Commun. 197, 339–340 (2015).

    Google Scholar 

  45. S. Blanes et al., ‘‘The Magnus expansion and some of its applications,’’ Phys. Rep. 470, 151–238 (2009).

    MathSciNet  Google Scholar 

  46. C. Moler and C. van Loan, ‘‘Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later,’’ SIAM Rev. 45, 3–49 (2003).

    MathSciNet  MATH  Google Scholar 

  47. Y. Saad, ‘‘Analysis of some Krylov subspace approximations to the matrix exponential operator,’’ SIAM J. Numer. Anal. 29, 209–228 (1992).

    MathSciNet  MATH  Google Scholar 

  48. D. Jaschke and L. D. Carr, ‘‘Open source matrix product states: Exact diagonalization and other entanglement-accurate methods revisited in quantum systems,’’ J. Phys. A: Math. Theor. 51, 465302 (2018).

  49. M. Brenes et al., ‘‘Massively parallel implementation and approaches to simulate quantum dynamics using Krylov subspace techniques,’’ Comput. Phys. Commun. 235, 477–488 (2019).

    Google Scholar 

  50. H. Samet, Foundations of Multidimensional and Metric Data Structures (Morgan Kaufmann, 2006).

    MATH  Google Scholar 

  51. U. Schollwoeck, ‘‘The density-matrix renormalization group in the age of matrix product states,’’ Ann. Phys. 326, 96 (2011).

    MathSciNet  Google Scholar 

  52. P. Secular, N. Gourianov, M. Lubasch, S. Dolgov, S. R. Clark, and D. Jaksch, ‘‘Parallel time-dependent variational principle algorithm for matrix product states,’’ arXiv:1912.06127 (2019).

  53. R. P. Feynman, ‘‘Simulating physics with computers,’’ Int. J. Theor. Phys. 21, 467–488 (1982).

    MathSciNet  Google Scholar 

  54. A. M. Childs et al., ‘‘Toward the first quantum simulation with quantum speedup,’’ Proc. Natl. Acad. Sci. U. S. A. 115, 9456 (2019).

    MathSciNet  MATH  Google Scholar 

  55. QuEST—Quantum Exact Simulation Toolkit. https://quest.qtechtheory.org/. Accessed 2020.

  56. T. Jones, A. Brown, I. Bush, and S. C. Benjami, ‘‘QuEST and high performance simulation of Quantum Computer,’’ Sci. Rep. 9, 1073 (2019).

    Google Scholar 

  57. Zhih-Ahn Jia et al., ‘‘Quantum neural network states: A brief review of methods and applications,’’ Adv. Quantum Technol., 1800077 (2019).

  58. I. Glasser, N. Pancotti, M. August, I. D. Rodriguez, and J. I. Cirac, ‘‘Neural-network quantum states, string-bond states, and chiral topological states,’’ Phys. Rev. X 8, 011006 (2018).

  59. List of QC simulators. https://quantiki.org/wiki/list-qc-simulators. Accessed April 2020.

  60. A. S. Green et al., ‘‘Quipper: A scalable quantum programming language,’’ in Proceedings of the 34th ACM SIGPLAN Conference on Programming Language Design and Implementation, 2013, pp. 333–342.

  61. A. W. Cross et al., ‘‘Open quantum assembly language,’’ arXiv:1707.03429 (2017).

  62. K. Svore et al., ‘‘Q# Enabling scalable quantum computing and development with a high-level DSL,’’ in Proceedings of the Real World Domain Specific Languages Workshop 2018 (2018), pp. 1–10.

  63. A. J. Abhari et al., ‘‘Scaffold: Quantum programming language,’’ TR-934-12 (2012).

  64. G. Guerreschi et al., ‘‘Intel Quantum Simulator: A cloud-ready high-performance simulator of quantum circuits,’’ arXiv:2001.10554 (2020).

  65. M. Smelyanskiy, N. P. Sawaya, and A. Aspuru-Guzik, ‘‘qHiPSTER: the quantum high performance software testing environment,’’ arXiv:1601.07195 (2016).

  66. G. Aleksandrowicz et al., ‘‘Qiskit: An open-source framework for quantum computing,’’ https://zenodo.org/record/2562111. Accessed April 2020.

  67. M. Amy and V. Gheorghiu, ‘‘staq–A full-stack quantum processing toolkit,’’ arXiv:1912.06070 (2019).

  68. A. B. de Avila et al., ‘‘State-of-the-art quantum computing simulators: Features, optimizations, and improvements for D-GM,’’ Neurocomputing (2019).

  69. T. Häner and D. S. Steiger, ‘‘5 petabyte simulation of a 45-qubit quantum circuit,’’ in Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, 2017, pp. 1–10.

  70. E. Pednault et al., ‘‘Breaking the 49-qubit barrier in the simulation of quantum circuits,’’ arXiv:1710.05867 (2017).

  71. E. Pednault et al., On ‘‘Quantum Supremacy.’’ https://www.ibm.com/blogs/research/2019/10/on-quantum-supremacy/. Accessed 2020.

  72. Z. Y. Chen, et al., ‘‘64-qubit quantum circuit simulation,’’ Sci. Bull. 63, 964-971 (2018).

    Google Scholar 

  73. I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning (MIT Press, Boston, MA, 2016).

    MATH  Google Scholar 

  74. V. Volokitin, A. Liniov, I. Meyerov, M. Hartmann, M. Ivanchenko, P. Hänggi, and S. Denisov, ‘‘Computation of the asymptotic states of modulated open quantum systems with a numerically exact realization of the quantum trajectory method,’’ Phys. Rev. E 96, 053313 (2017).

  75. A. Liniov, V. Volokitin, I. Meyerov, M. Ivanchenko, and S. Denisov, ‘‘Increasing performance of the quantum trajectory method by grouping trajectories,’’ Commun. Comput. Inform. Sci. 793, 136 (2017).

    Google Scholar 

  76. Increasing AI Performance and Efficiency with Intel DL Boost. https://www.intel.ai/increasing-ai-performance-intel-dlboost/#gs.117qh4. Accessed 2020.

  77. Intel Unveils New GPU Architecture with High-Performance Computing and AI Acceleration. https://newsroom.intel.com/news-releases/intel-unveils-new-gpu-architecture-optimized-for-hpc-ai-oneapi/#gs.11dtfx. Accessed 2020.

  78. Graphcore. https://www.graphcore.ai/. Accessed 2020.

  79. T. Häner, D. S. Steiger, K. Svore, and M. Troyer, ‘‘A software methodology for compiling quantum programs,’’ Quantum Sci. Technol. 3, 020501 (2018).

  80. S. Khatri et al., ‘‘Quantum-assisted quantum compiling,’’ Quantum 3, 140 (2019).

    Google Scholar 

  81. We use this word in its biological sense, as a cumulative inherited change in a population of organisms through time leading to the appearance of new forms, Dictionary, Merriam-Webster. https://www.merriam-webster.com/dictionary/evolution.

  82. S. Lloyd, ‘‘Universal quantum simulators,’’ Science (Washington, DC, U. S.) 273, 1073 (1996).

    MathSciNet  MATH  Google Scholar 

  83. B. Kloss, Y. Bar Lev, and D. Reichman, ‘‘Time-dependent variational principle in matrix-product state manifolds: Pitfalls and potential,’’ Phys. Rev. B 97, 024307 (2018).

  84. S. Goto, and I. Danshita, ‘‘Performance of the time-dependent variational principle for matrix product states in the long-time evolution of a pure state,’’ Phys. Rev. B 99, 054307 (2019).

  85. S. Paeckel et al., ‘‘Time-evolution methods for matrix-product states,’’ Ann. Phys. 411, 167998 (2019).

  86. J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac, ‘‘Classical simulation of infinite-size quantum lattice systems in two spatial dimensions,’’ Phys. Rev. Lett. 101, 250602 (2008).

  87. 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,’’ Phys. Rev. A 75, 033605 (2007).

  88. Ho N. Phien et al., ‘‘Infinite projected entangled pair states algorithm improved: Fast full update and gauge fixing,’’ Phys. Rev. B 92, 035142 (2015).

  89. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge, 2010).

    MATH  Google Scholar 

Download references

Funding

The work is supported by the Russian Science Foundation via Grant no. 19-72-20086. The research is carried out using the equipment of the shared research facilities of HPC computing resources at Moscow State University [10] and the Lobachevskii supercomputer at Lobachevskii University.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to I. Meyerov, A. Liniov, M. Ivanchenko or S. Denisov.

Additional information

(Submitted by E. E. Tyrtyshnikov)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meyerov, I., Liniov, A., Ivanchenko, M. et al. Modeling Complex Quantum Dynamics: Evolution of Numerical Algorithms in the HPC Context. Lobachevskii J Math 41, 1509–1520 (2020). https://doi.org/10.1134/S1995080220080120

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080220080120

Keywords:

Navigation