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Quantum Hamiltonian complexity in thermal equilibrium

Nature Physics volume 18, pages 1367–1370 (2022)Cite this article

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Abstract

The physical properties of a quantum many-body system in thermal equilibrium are determined by its partition function and free energy. Here we study the computational complexity of approximating these quantities for n-qubit local Hamiltonians. First, we report a classical algorithm with poly(n) runtime, which approximates the free energy of a given 2-local Hamiltonian provided that it satisfies a certain denseness condition. Our algorithm contributes to a body of work investigating the hardness of approximation for difficult optimization problems. Specifically, this extends existing efficient approximation algorithms for dense instances of the ground energy of 2-local quantum Hamiltonians and the free energy of classical Ising models. Second, we establish polynomial-time equivalence between the problem of approximating the free energy of local Hamiltonians and several other natural tasks ubiquitous in condensed-matter physics and quantum computing, such as the problem of approximating the number of input states accepted by a polynomial-size quantum circuit. These results suggest that the simulation of quantum many-body systems in thermal equilibrium may precisely capture the complexity of a broad family of computational problems that have yet to be defined or characterized in terms of known complexity classes. Finally, we summarize state-of-the-art classical and quantum algorithms for approximating the free energy and show how to improve their runtime and memory footprint.

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References

  1. Kitaev, A. Y., Shen, A. H. & Vyalyi, M. N. Classical and Quantum Computation (American Mathematical Society, 2002).

    Book  MATH  Google Scholar 

  2. Dinur, I. The PCP theorem by gap amplification. J. ACM 54, 12–es (2007).

    Article  MathSciNet  MATH  Google Scholar 

  3. Alimonti, P. & Kann, V. Hardness of approximating problems on cubic graphs. In Italian Conference on Algorithms and Complexity 288–298 (Springer, 1997).

  4. Suzuki, M. Quantum Monte Carlo Methods in Condensed Matter Physics (World Scientific, 1993).

  5. Barvinok, A. Combinatorics and Complexity of Partition Functions Vol. 9 (Springer, 2016).

  6. Harrow, A. W., Mehraban, S. & Soleimanifar, M. Classical algorithms, correlation decay, and complex zeros of partition functions of quantum many-body systems. In Proc. 52nd Annual ACM SIGACT Symposium on Theory of Computing 378–386 (ACM, 2020).

  7. Bravyi, S., Divincenzo, D. P., Oliveira, R. I. & Terhal, B. M. The complexity of stoquastic local Hamiltonian problems. Quant. Inf. Comp. vol. 8, no. 5, pp. 0361-0385 (2008).

  8. Crosson, E. & Harrow, A. W. Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians. Quantum 5, 395 (2021).

    Article  Google Scholar 

  9. Bravyi, S. & Gosset, D. Polynomial-time classical simulation of quantum ferromagnets. Phys. Rev. Lett. 119, 100503 (2017).

    Article  ADS  Google Scholar 

  10. Kuwahara, T., Kato, K. & Brandão, F. G. S. L. Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature. Phys. Rev. Lett. 124, 220601 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  11. Risteski, A. How to calculate partition functions using convex programming hierarchies: provable bounds for variational methods. In 29th Annual Conference on Learning Theory (eds. Feldman, V., Rakhlin, A. & Shamir, O.) vol. 49 of Proceedings of Machine Learning Research 1402–1416 (PMLR, 2016).

  12. Yoshida, Y. & Zhou, Y. Approximation schemes via Sherali-Adams hierarchy for dense constraint satisfaction problems and assignment problems. In Proc. 5th Conference on Innovations in Theoretical Computer Science ITCS ’14 423–438 (ACM, 2014).

  13. Bertsimas, D. & Vempala, S. Solving convex programs by random walks. J. ACM 51, 540–556 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  14. Grötschel, M., Lovász, L. & Schrijver, A. Geometric Algorithms and Combinatorial Optimization Vol. 2 (Springer Science & Business Media, 2012).

  15. Barak, B., Raghavendra, P. & Steurer, D. Rounding semidefinite programming hierarchies via global correlation. IEEE 52nd Annual Symposium on Foundations of Computer Science, pp. 472-481 (2011).

  16. Brandao, F. G. & Harrow, A. W. Product-state approximations to quantum ground states. In Proc. Forty-Fifth Annual ACM Symposium on Theory of Computing STOC ’13 871–880 (ACM, 2013).

  17. Gharibian, S. & Kempe, J. Approximation algorithms for QMA-complete problems. SIAM J. Comput. 41, 1028–1050 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  18. Brandão, F. G. S. L. Entanglement Theory and the Quantum Simulation of Many-Body Physics. PhD thesis, Imperial College of Science, Technology and Medicine (2008).

  19. Cade, C. & Montanaro, A. The quantum complexity of computing Schatten p-norms. In 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018) (Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik 2018).

  20. Chowdhury, A. N., Somma, R. D. & Subasi, Y. Computing partition functions in the one-clean-qubit model. Phys. Rev. A 103, 032422 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  21. Knill, E. & Laflamme, R. Power of one bit of quantum information. Phys. Rev. Lett. 81, 5672 (1998).

    Article  ADS  Google Scholar 

  22. Stockmeyer, L. The complexity of approximate counting. In Proc. Fifteenth Annual ACM Symposium on Theory of Computing STOC ’83 118–126 (ACM, 1983).

  23. Goldberg, L. A. & Guo, H. The complexity of approximating complex-valued Ising and Tutte partition functions. comput. complex. 26, 765–833 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  24. Aharonov, D., Ben-Or, M., Brandao, F. G. & Sattath, O. The pursuit for uniqueness: extending Valiant-Vazirani theorem to the probabilistic and quantum settings. Quantum 6, 668 (2022).

  25. Cubitt, T. S., Montanaro, A. & Piddock, S. Universal quantum Hamiltonians. Proc. Natl Acad. Sci. USA 115, 9497–9502 (2018).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Zhou, L. & Aharonov, D. Strongly universal Hamiltonian simulators. Preprint at https://arxiv.org/abs/2102.02991 (2021).

  27. Kohler, T., Piddock, S., Bausch, J. & Cubitt, T. Translationally-invariant universal quantum Hamiltonians in 1D. Annales Henri Poincaré Vol. 23, pp. 223–254 (2022).

  28. Kohler, T., Piddock, S., Bausch, J. & Cubitt, T. General conditions for universality of quantum Hamiltonians. PRX Quantum 3, 010308 (2022).

  29. Gosset, D. & Smolin, J. A compressed classical description of quantum states. In 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019) (Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2019).

  30. Huang, H.-Y., Kueng, R. & Preskill, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020).

    Article  Google Scholar 

  31. Cleve, R., Leung, D., Liu, L. & Wang, C. Near-linear constructions of exact unitary 2-designs. Quant. Info. Comp. vol. 16, no. 9–10 pp. 721–756 (2016).

  32. Dankert, C., Cleve, R., Emerson, J. & Livine, E. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 80, 012304 (2009).

    Article  ADS  Google Scholar 

  33. Hutchinson, M. F. A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun. Stat. Simul. Comput. 18, 1059–1076 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  34. Meyer, R. A., Musco, C., Musco, C. & Woodruff, D. P. Hutch++: optimal stochastic trace estimation. In Symposium on Simplicity in Algorithms (SOSA) 142–155 (SIAM, 2021).

  35. Poulin, D. & Wocjan, P. Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. Phys. Rev. Lett. 103, 220502 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  36. Chowdhury, A. N. & Somma, R. D. Quantum algorithms for Gibbs sampling and hitting-time estimation. Quant. Inf. Comp. 17, 41–64 (2017).

    MathSciNet  Google Scholar 

  37. Gilyén, A., Su, Y., Low, G. H. & Wiebe, N. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proc. 51st Annual ACM SIGACT Symposium on Theory of Computing 193–204 (ACM, 2019).

  38. van Apeldoorn, J., Gilyén, A., Gribling, S. & de Wolf, R. Quantum SDP-solvers: better upper and lower bounds. Quantum 4, 230 (2020).

    Article  Google Scholar 

  39. Marriott, C. & Watrous, J. Quantum Arthur–Merlin games. Comput. Complex. 14, 122–152 (2005).

  40. Kempe, J., Kitaev, A. & Regev, O. The complexity of the local Hamiltonian problem. SIAM J. Comput. 35, 1070–1097 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  41. Brassard, G., Høyer, P., Mosca, M. and Tapp, A. Quantum amplitude amplification and estimation. In Quantum Computation and Information Vol. 305 of Contemporary Mathematics 53–74 (AMS, 2002).

  42. Aaronson, S. and Rall, P. Quantum approximate counting, simplified. In Symposium on Simplicity in Algorithms 24–32 (SIAM, 2020).

  43. Bravyi, S. & Maslov, D. Hadamard-free circuits expose the structure of the Clifford group. IEEE Trans. Inf. Theory 67, 4546–4563 (2021).

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

D.G. thanks B. Fefferman, S. Gharibian and R. Kothari for discussions about QAC. S.B., A.C. and D.G. are supported in part by the Army Research Office under grant no. W911NF-20-1-0014. D.G. is a CIFAR Fellow in the quantum information science program, and is also supported in part by IBM Research. Research at the Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities. S.B. is supported in part by the IBM Research Frontiers Institute.

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Authors and Affiliations

  1. IBM Quantum, IBM T.J. Watson Research Center, New York, NY, USA

    Sergey Bravyi & Pawel Wocjan

  2. Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada

    Anirban Chowdhury & David Gosset

  3. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada

    Anirban Chowdhury & David Gosset

  4. Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada

    Anirban Chowdhury & David Gosset

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  1. Sergey Bravyi
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  4. Pawel Wocjan
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All authors contributed equally to the technical content and writing of the manuscript.

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Correspondence to Anirban Chowdhury.

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Bravyi, S., Chowdhury, A., Gosset, D. et al. Quantum Hamiltonian complexity in thermal equilibrium. Nat. Phys. 18, 1367–1370 (2022). https://doi.org/10.1038/s41567-022-01742-5

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