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The Complexity of Simulating Local Measurements on Quantum Systems

Authors Sevag Gharibian, Justin Yirka

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Sevag Gharibian
Justin Yirka

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Sevag Gharibian and Justin Yirka. The Complexity of Simulating Local Measurements on Quantum Systems. In 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 73, pp. 2:1-2:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.TQC.2017.2

Abstract

An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, Ambainis defined the complexity class P^QMA[log], and motivated its study by showing that the physical task of estimating the expectation value of a local observable against the ground state of a local Hamiltonian is P^QMA[log]-complete. In this paper, we continue the study of P^QMA[log], obtaining the following results. The P^QMA[log]-completeness result of Ambainis requires O(log n)-local observ- ables and Hamiltonians. We show that simulating even a single qubit measurement on ground states of 5-local Hamiltonians is P^QMA[log]-complete, resolving an open question of Ambainis. We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarly P^QMA[log]-complete. P^QMA[log] is thought of as "slightly harder" than QMA. We justify this formally by exploiting the hierarchical voting technique of Beigel, Hemachandra, and Wechsung to show P^QMA[log] \subseteq PP. This improves the containment QMA \subseteq PP from Kitaev and Watrous. A central theme of this work is the subtlety involved in the study of oracle classes in which the oracle solves a promise problem. In this vein, we identify a flaw in Ambainis' prior work regarding a P^UQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a "query validation" technique, we build on his prior work to obtain P^UQMA[log]-hardness for estimating spectral gaps under polynomial-time Turing reductions.
Keywords
  • Complexity theory
  • Quantum Merlin Arthur (QMA)
  • local Hamiltonian
  • local measurement
  • spectral gap

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References

  1. D. Aharonov, M. Ben-Or, F. Brandão, and O. Sattath. The pursuit for uniqueness: Extending Valiant-Vazirani theorem to the probabilistic and quantum settings. Available at arXiv.org e-Print quant-ph/0810.4840v1, 2008. Google Scholar
  2. D. Aharonov and T. Naveh. Quantum NP - A survey. Available at arXiv.org e-Print quant-ph/0210077v1, 2002. Google Scholar
  3. A. Ambainis. On physical problems that are slightly more difficult than QMA. In Proceedings of 29th IEEE Conference on Computational Complexity (CCC 2014), pages 32-43, 2014. Google Scholar
  4. R. Beigel, L. A. Hemachandra, and G. Wechsung. On the power of probabilistic polynomial time: P^NP[log] ⊆ PP. In Proceedings of the 4th IEEE Conference on Structure in Complexity Theory, pages 225-227, 1989. Google Scholar
  5. A. D. Bookatz. QMA-complete problems. Quantum Information &Computation, 14(5-6), 2014. Google Scholar
  6. B. Brown, S. Flammia, and N. Schuch. Computational difficulty of computing the density of states. Physical Review Letters, 104:040501, 2011. Google Scholar
  7. A. Chailloux and O. Sattath. The complexity of the separable Hamiltonian problem. In Proceedings of 27th IEEE Conference on Computational Complexity (CCC 2012), pages 32-41, 2012. Google Scholar
  8. S. Cook. The complexity of theorem proving procedures. In Proceedings of the 3rd ACM Symposium on Theory of Computing (STOC 1972), pages 151-158, 1972. Google Scholar
  9. T. Cubitt and A. Montanaro. Complexity classification of local hamiltonian problems. Available at arXiv.org e-Print quant-ph/1311.3161, 2013. Google Scholar
  10. R. Feynman. Quantum mechanical computers. Optics News, 11:11, 1985. Google Scholar
  11. S. Gharibian. Approximation, proof systems, and correlations in a quantum world. PhD thesis, University of Waterloo, 2013. Available at arXiv.org e-Print quant-ph/1301.2632. Google Scholar
  12. S. Gharibian and J. Kempe. Hardness of approximation for quantum problems. In Proceedings of 39th International Colloquium on Automata, Languages and Programming (ICALP 2012), pages 387-398, 2012. Google Scholar
  13. S. Gharibian and J. Sikora. Ground state connectivity of local Hamiltonians. In Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP 2015), pages 617-628, 2015. Google Scholar
  14. Sevag Gharibian, Yichen Huang, Zeph Landau, and Seung Woo Shin. Quantum hamiltonian complexity. Foundations and Trends in Theoretical Computer Science, 10(3):159-282, 2015. URL: http://dx.doi.org/10.1561/0400000066.
  15. J. Gill. Computational complexity of probabilistic Turing machines. SIAM Journal on Computing, 6(4):675-695, 1977. Google Scholar
  16. O. Goldreich. On promise problems: A survey. Theoretical Computer Science, 3895:254-290, 2006. Google Scholar
  17. R. Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, pages 85-103. New York: Plenum, 1972. Google Scholar
  18. J. Kempe, A. Kitaev, and O. Regev. The complexity of the local Hamiltonian problem. SIAM Journal on Computing, 35(5):1070-1097, 2006. Google Scholar
  19. A. Kitaev, A. Shen, and M. Vyalyi. Classical and Quantum Computation. American Mathematical Society, 2002. Google Scholar
  20. A. Kitaev and J. Watrous. Parallelization, amplification, and exponential time simulation of quantum interactive proof systems. In Proceedings of the 32nd ACM Symposium on Theory of Computing (STOC 2000), pages 608-617, 2000. Google Scholar
  21. L. Levin. Universal search problems. Problems of Information Transmission, 9(3):265-266, 1973. Google Scholar
  22. Y.-K. Liu. Consistency of local density matrices is QMA-complete. In Lecture Notes in Computer Science, volume 4110, pages 438-449, 2006. Google Scholar
  23. C. Marriott and J. Watrous. Quantum Arthur-Merlin games. Computational Complexity, 14(2):122-152, 2005. Google Scholar
  24. S. Piddock and A. Montanaro. The complexity of antiferromagnetic interactions and 2d lattices. Available at arXiv.org e-Print quantph/1506.04014, 2015. Google Scholar
  25. Y. Shi and S. Zhang. Note on quantum counting classes. URL: http://www.cse.cuhk.edu.hk/~syzhang/papers/SharpBQP.pdf.
  26. M. Vyalyi. QMA=PP implies that PP contains PH. Electronic Colloquium on Computational Complexity, 2003. Google Scholar
  27. J. Watrous. Encyclopedia of Complexity and System Science, chapter Quantum Computational Complexity. Springer, 2009. Google Scholar
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