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  • Open Access

Algorithmic Cluster Expansions for Quantum Problems

Ryan L. Mann1,2,* and Romy M. Minko2

  • 1Centre for Quantum Computation and Communication Technology, Centre for Quantum Software and Information, School of Computer Science, Faculty of Engineering & Information Technology, University of Technology Sydney, NSW 2007, Australia
  • 2School of Mathematics, University of Bristol, Bristol BS8 1UG, United Kingdom

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PRX Quantum 5, 010305 – Published 16 January, 2024

DOI: https://doi.org/10.1103/PRXQuantum.5.010305

Abstract

We establish a general framework for developing approximation algorithms for a class of counting problems. Our framework is based on the cluster expansion of the abstract polymer model formalism of Kotecký and Preiss. We apply our framework to obtain efficient algorithms for (1) approximating probability amplitudes of a class of quantum circuits close to the identity, (2) approximating expectation values of a class of quantum circuits with operators close to the identity, (3) approximating partition functions of a class of quantum spin systems at high temperature, and (4) approximating thermal expectation values of a class of quantum spin systems at high temperature with positive-semidefinite operators. Further, we obtain hardness of approximation results for approximating probability amplitudes of quantum circuits and partition functions of quantum spin systems. This establishes a computational complexity transition for these problems and shows that our algorithmic conditions are optimal under complexity-theoretic assumptions. Finally, we show that our algorithmic condition is almost optimal for expectation values and optimal for thermal expectation values in the sense of zero freeness.

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Popular Summary

The classification of the computational complexity of quantum problems is a crucial aspect of quantum information science, essential for understanding the capabilities and limitations of quantum computing. Our research focuses on the exploration of specific conditions under which quantum problems can be efficiently approximated using classical computational techniques. This not only enhances our understanding of quantum computing but opens new possibilities for its application.

We establish a general framework for developing approximating algorithms for quantum problems based on the cluster expansion. We apply this framework to obtain efficient approximation algorithms under certain algorithmic conditions for probability amplitudes, expectation values, partition functions, and thermal expectation values. Further, we obtain hardness of approximation results for probability amplitudes and partition functions, establishing a computational complexity transition and demonstrating the optimality of our algorithmic conditions under complexity-theoretic assumptions.

Our work offers a deeper understanding of the computational complexity of quantum problems, offering new insights into the tractability and intractability of these problems. This has significant implications for quantum computing, indicating that some problems are more suited to classical approximation than previously believed.

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References (31)

  1. S. Bravyi, D. Gosset, and R. Movassagh, Classical algorithms for quantum mean values, Nat. Phys. 17, 337 (2021).
  2. T. Kuwahara, K. Kato, and F. G. Brandão, Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature, Phys. Rev. Lett. 124, 220601 (2020).
  3. D. S. Wild, and Á. M. Alhambra, Classical simulation of short-time quantum dynamics, PRX Quantum 4, 020340 (2023).
  4. R. Kotecký, and D. Preiss, Cluster expansion for abstract polymer models, Commun. Math. Phys. 103, 491 (1986).
  5. T. Helmuth, W. Perkins, and G. Regts, Algorithmic Pirogov–Sinai theory, Probab. Theory Relat. Fields 176, 851 (2020).
  6. C. Borgs, J. Chayes, T. Helmuth, W. Perkins, and P. Tetali, in Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (ACM, 2020), pp. 738–751.
  7. V. Patel, and G. Regts, Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials, SIAM J. Comput. 46, 1893 (2017).
  8. A. Barvinok, Combinatorics and Complexity of Partition Functions (Springer, 2016), Vol. 274.
  9. V. Patel, and G. Regts, Bull. EATCS 138 (2022).
  10. R. L. Graham, M. Grötschel, and L. Lovász, Handbook of Combinatorics (Elsevier, 1995), Vol. 2.
  11. S. Friedli, and Y. Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction (Cambridge University Press, 2017).
  12. R. Fernández, and A. Procacci, Cluster expansion for abstract polymer models. New bounds from an old approach, Commun. Math. Phys. 274, 123 (2007).
  13. J. B. Shearer, On a problem of spencer, Combinatorica 5, 241 (1985).
  14. A. D. Scott, and A. D. Sokal, The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma, J. Stat. Phys. 118, 1151 (2005).
  15. D. Galvin, G. McKinley, W. Perkins, M. Sarantis, and P. Tetali, On the zeroes of hypergraph independence polynomials, E-prints ArXiv:2211.00464 (2022).
  16. F. Bencs, and P. Buys, Optimal zero-free regions for the independence polynomial of bounded degree hypergraphs, E-prints ArXiv:2306.00122 (2023).
  17. S. Arora, and B. Barak, Computational Complexity: A Modern Approach (Cambridge University Press, 2009).
  18. C. Borgs, J. Chayes, J. Kahn, and L. Lovász, Left and right convergence of graphs with bounded degree, Random Struct. Algorithms 42, 1 (2013).
  19. A. Björklund, T. Husfeldt, P. Kaski, and M. Koivisto, in 49th Annual IEEE Symposium on Foundations of Computer Science (IEEE, 2008), p. 677.
  20. A. Galanis, L. A. Goldberg, and A. Herrera-Poyatos, The complexity of approximating the complex-valued Ising model on bounded degree graphs, SIAM J. Discrete Math. 36, 2159 (2022).
  21. R. L. Mann, and M. J. Bremner, Approximation algorithms for complex-valued Ising models on bounded degree graphs, Quantum 3, 162 (2019).
  22. A. Sinclair, P. Srivastava, and M. Thurley, Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs, J. Stat. Phys. 155, 666 (2014).
  23. A. Sly, and N. Sun, in 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS) (IEEE, 2012), p. 361.
  24. A. Galanis, D. Štefankovič, and E. Vigoda, Inapproximability of the partition function for the antiferromagnetic Ising and hard-core models, Comb. Probab. Comput. 25, 500 (2016).
  25. L. A. Goldberg, and H. Guo, The complexity of approximating complex-valued Ising and Tutte partition functions, Comput. Complex. 26, 765 (2017).
  26. A. W. Harrow, S. Mehraban, and M. Soleimanifar, in Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (ACM, 2020), p. 378.
  27. R. L. Mann, and T. Helmuth, Efficient algorithms for approximating quantum partition functions, J. Math. Phys. 62, 022201 (2021).
  28. T. Helmuth, and R. L. Mann, Efficient algorithms for approximating quantum partition functions at low temperature, Quantum 7, 1155 (2023).
  29. S. Bravyi, A. Chowdhury, D. Gosset, and P. Wocjan, Quantum Hamiltonian complexity in thermal equilibrium, Nat. Phys. 18, 1367 (2022).
  30. H. Apel, and T. Cubitt, A mathematical framework for quantum Hamiltonian simulation and duality, E-prints ArXiv:2208.11941 (2022).
  31. Z. Chen, A. Galanis, L. A. Goldberg, W. Perkins, J. Stewart, and E. Vigoda, in Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019) (Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019).

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PRX Quantum 5, 010305– Published 16 January, 2024
  • Received 6 July 2023
  • Revised 15 October 2023
  • Accepted 1 December 2023
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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