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The Quantum Supremacy Tsirelson Inequality

Author William Kretschmer

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Author Details

William Kretschmer
  • University of Texas at Austin, TX, USA

Funding

  • Kretschmer, William: Supported by a Vannevar Bush Fellowship and a National Defense Science and Engineering Graduate (NDSEG) Fellowship from the US Department of Defense.

Acknowledgements

Thanks to Scott Aaronson, Sabee Grewal, Sam Gunn, Robin Kothari, Daniel Liang, Patrick Rall, Andrea Rocchetto, and Justin Thaler for helpful discussions and illuminating insights. Thanks also to anonymous reviewers for helpful comments regarding the presentation of this work.

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William Kretschmer. The Quantum Supremacy Tsirelson Inequality. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ITCS.2021.13

Abstract

A leading proposal for verifying near-term quantum supremacy experiments on noisy random quantum circuits is linear cross-entropy benchmarking. For a quantum circuit C on n qubits and a sample z ∈ {0,1}ⁿ, the benchmark involves computing |⟨z|C|0ⁿ⟩|², i.e. the probability of measuring z from the output distribution of C on the all zeros input. Under a strong conjecture about the classical hardness of estimating output probabilities of quantum circuits, no polynomial-time classical algorithm given C can output a string z such that |⟨z|C|0ⁿ⟩|² is substantially larger than 1/(2ⁿ) (Aaronson and Gunn, 2019). On the other hand, for a random quantum circuit C, sampling z from the output distribution of C achieves |⟨z|C|0ⁿ⟩|² ≈ 2/(2ⁿ) on average (Arute et al., 2019).
In analogy with the Tsirelson inequality from quantum nonlocal correlations, we ask: can a polynomial-time quantum algorithm do substantially better than 2/(2ⁿ)? We study this question in the query (or black box) model, where the quantum algorithm is given oracle access to C. We show that, for any ε ≥ 1/poly(n), outputting a sample z such that |⟨z|C|0ⁿ⟩|² ≥ (2 + ε)/2ⁿ on average requires at least Ω((2^{n/4})/poly(n)) queries to C, but not more than O (2^{n/3}) queries to C, if C is either a Haar-random n-qubit unitary, or a canonical state preparation oracle for a Haar-random n-qubit state. We also show that when C samples from the Fourier distribution of a random Boolean function, the naive algorithm that samples from C is the optimal 1-query algorithm for maximizing |⟨z|C|0ⁿ⟩|² on average.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Oracles and decision trees
Keywords
  • quantum supremacy
  • quantum query complexity
  • random circuit sampling

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References

  1. Scott Aaronson and Lijie Chen. Complexity-theoretic foundations of quantum supremacy experiments. In Proceedings of the 32nd Computational Complexity Conference, CCC ’17, Dagstuhl, DEU, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  2. Scott Aaronson and Sam Gunn. On the classical hardness of spoofing linear cross-entropy benchmarking. Theory of Computing, 16(11):1-8, 2020. URL: https://doi.org/10.4086/toc.2020.v016a011.
  3. Scott Aaronson, Robin Kothari, William Kretschmer, and Justin Thaler. Quantum Lower Bounds for Approximate Counting via Laurent Polynomials. In Shubhangi Saraf, editor, 35th Computational Complexity Conference (CCC 2020), volume 169 of Leibniz International Proceedings in Informatics (LIPIcs), pages 7:1-7:47, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.7.
  4. Dorit Aharonov, Alexei Kitaev, and Noam Nisan. Quantum circuits with mixed states. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC ’98, page 20–30, New York, NY, USA, 1998. Association for Computing Machinery. URL: https://doi.org/10.1145/276698.276708.
  5. Andris Ambainis. Understanding quantum algorithms via query complexity. In Proceedings of the 2018 International Congress of Mathematicians, volume 3, pages 3249-3270, 2018. Google Scholar
  6. Andris Ambainis, Loïck Magnin, Martin Roetteler, and Jérémie Roland. Symmetry-assisted adversaries for quantum state generation. In 2011 IEEE 26th Annual Conference on Computational Complexity, pages 167-177, 2011. Google Scholar
  7. Andris Ambainis, Ansis Rosmanis, and Dominique Unruh. Quantum attacks on classical proof systems: The hardness of quantum rewinding. In Proceedings of the 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, FOCS ’14, page 474–483, USA, 2014. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2014.57.
  8. Srinivasan Arunachalam, Aleksandrs Belovs, Andrew M. Childs, Robin Kothari, Ansis Rosmanis, and Ronald de Wolf. Quantum Coupon Collector. In Steven T. Flammia, editor, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020), volume 158 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1-10:17, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.TQC.2020.10.
  9. Frank Arute, Kunal Arya, Ryan Babbush, et al. Quantum supremacy using a programmable superconducting processor. Nature, 574(7779):505-510, 2019. URL: https://doi.org/10.1038/s41586-019-1666-5.
  10. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. J. ACM, 48(4):778–797, July 2001. URL: https://doi.org/10.1145/502090.502097.
  11. John Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1:195-200, November 1964. URL: https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195.
  12. Aleksandrs Belovs. Variations on quantum adversary, 2015. URL: http://arxiv.org/abs/1504.06943.
  13. Aleksandrs Belovs and Ansis Rosmanis. Approximate unitary t-designs by short random quantum circuits using nearest-neighbor and long-range gates, 2020. URL: http://arxiv.org/abs/2002.06879.
  14. Fernando G. S. L. Brandão, Aram W. Harrow, and Michał Horodecki. Local random quantum circuits are approximate polynomial-designs. Communications in Mathematical Physics, 346(2):397-434, 2016. URL: https://doi.org/10.1007/s00220-016-2706-8.
  15. Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation. In Quantum Computation and Quantum Information, volume 305 of Contemporary Mathematics, pages 53-74. American Mathematical Society, 2002. Google Scholar
  16. Gilles Brassard, Peter Høyer, and Alain Tapp. Quantum cryptanalysis of hash and claw-free functions. SIGACT News, 28(2):14–19, June 1997. URL: https://doi.org/10.1145/261342.261346.
  17. Mark Bun and Justin Thaler. Dual lower bounds for approximate degree and markov-bernstein inequalities. Inf. Comput., 243(C):2–25, August 2015. URL: https://doi.org/10.1016/j.ic.2014.12.003.
  18. Boris Cirel’son (Tsirelson). Quantum generalizations of Bell’s inequality. Letters in Mathematical Physics, 4(2):93-100, 1980. URL: https://doi.org/10.1007/BF00417500.
  19. John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23:880-884, October 1969. URL: https://doi.org/10.1103/PhysRevLett.23.880.
  20. Richard Cleve, Peter Høyer, Benjamin Toner, and John Watrous. Consequences and limits of nonlocal strategies. In Proceedings of the 19th IEEE Annual Conference on Computational Complexity, CCC ’04, page 236–249, USA, 2004. IEEE Computer Society. Google Scholar
  21. Aram Harrow and Saeed Mehraban. Approximate unitary t-designs by short random quantum circuits using nearest-neighbor and long-range gates, 2018. URL: http://arxiv.org/abs/1809.06957.
  22. Norman L. Johnson and Samuel Kotz. Urn models and their application: an approach to modern discrete probability theory. Wiley, 1977. Google Scholar
  23. Shelby Kimmel, Cedric Yen-Yu Lin, Guang Hao Low, Maris Ozols, and Theodore J. Yoder. Hamiltonian simulation with optimal sample complexity. npj Quantum Information, 3(1):13, 2017. URL: https://doi.org/10.1038/s41534-017-0013-7.
  24. Troy Lee, Rajat Mittal, Ben W. Reichardt, Robert Špalek, and Mario Szegedy. Quantum query complexity of state conversion. In Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS ’11, page 344–353, USA, 2011. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2011.75.
  25. Nathan Lindzey and Ansis Rosmanis. A Tight Lower Bound For Non-Coherent Index Erasure. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), volume 151 of Leibniz International Proceedings in Informatics (LIPIcs), pages 59:1-59:37, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.59.
  26. Frederic Magniez, Ashwin Nayak, Jeremie Roland, and Miklos Santha. Search via quantum walk. In Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, STOC '07, page 575–584, New York, NY, USA, 2007. Association for Computing Machinery. URL: https://doi.org/10.1145/1250790.1250874.
  27. Martin Raab and Angelika Steger. "Balls into bins" - a simple and tight analysis. In Proceedings of the Second International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM '98, pages 159-170, Berlin, Heidelberg, 1998. Springer-Verlag. Google Scholar
  28. Ben W. Reichardt. Reflections for quantum query algorithms. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '11, page 560–569, USA, 2011. Society for Industrial and Applied Mathematics. Google Scholar
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