The Quantum Supremacy Tsirelson Inequality

William Kretschmer

Department of Computer Science, The University of Texas at Austin, Austin, TX 78712, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


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 $z \in \{0,1\}^n$, the benchmark involves computing $|\langle z|C|0^n \rangle|^2$, 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 $|\langle z|C|0^n\rangle|^2$ is substantially larger than $\frac{1}{2^n}$ (Aaronson and Gunn, 2019). On the other hand, for a random quantum circuit $C$, sampling $z$ from the output distribution of $C$ achieves $|\langle z|C|0^n\rangle|^2 \approx \frac{2}{2^n}$ 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 $\frac{2}{2^n}$? 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 $\varepsilon \ge \frac{1}{\mathrm{poly}(n)}$, outputting a sample $z$ such that $|\langle z|C|0^n\rangle|^2 \ge \frac{2 + \varepsilon}{2^n}$ on average requires at least $\Omega\left(\frac{2^{n/4}}{\mathrm{poly}(n)}\right)$ queries to $C$, but not more than $O\left(2^{n/3}\right)$ 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 $|\langle z|C|0^n\rangle|^2$ on average.

Recent quantum supremacy experiments were verified using a statistical test called the "Linear Cross-Entropy Benchmark" (or Linear XEB). This benchmark was chosen because of complexity-theoretic evidence that an efficient quantum algorithm can achieve a higher Linear XEB score than any possible efficient classical algorithm.

We argue that this upper bound on the power of classical algorithms for the Linear XEB is analogous to the Bell inequality in nonlocal correlations: both capture inherent limits on the power of classical information and computation that may be violated in the quantum setting. Motivated by this connection, we ask: what is the quantum supremacy analogue of the Tsirelson inequality? That is, what is the highest Linear XEB score achievable by an efficient quantum algorithm? We give evidence that the naive quantum algorithm for passing the benchmark is essentially optimal in this regard.

► BibTeX data

► References

[1] Scott Aaronson. Random circuit sampling: Thoughts and open problems. The Quantum Wave in Computing, 2020. URL https:/​/​​talks/​tbd-124.

[2] Scott Aaronson and Lijie Chen. Complexity-Theoretic Foundations of Quantum Supremacy Experiments. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference (CCC 2017), volume 79 of Leibniz International Proceedings in Informatics (LIPIcs), pages 22:1–22:67, Dagstuhl, Germany, 2017. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. ISBN 978-3-95977-040-8. 10.4230/​LIPIcs.CCC.2017.22. URL http:/​/​​opus/​volltexte/​2017/​7527.

[3] Scott Aaronson and Sam Gunn. On the classical hardness of spoofing linear cross-entropy benchmarking. Theory of Computing, 16 (11): 1–8, 2020. 10.4086/​toc.2020.v016a011. URL http:/​/​​articles/​v016a011.

[4] 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. ISBN 978-3-95977-156-6. 10.4230/​LIPIcs.CCC.2020.7. URL https:/​/​​opus/​volltexte/​2020/​12559.

[5] 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. ISBN 0897919629. 10.1145/​276698.276708. URL https:/​/​​10.1145/​276698.276708.

[6] Andris Ambainis. Understanding quantum algorithms via query complexity. In Proceedings of the 2018 International Congress of Mathematicians, volume 3, pages 3249–3270, 2018. 10.1142/​9789813272880_0181.

[7] Andris Ambainis, Loïck Magnin, Martin Roetteler, and Jeremie Roland. Symmetry-assisted adversaries for quantum state generation. In Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity, CCC '11, page 167–177, USA, 2011. IEEE Computer Society. ISBN 9780769544113. 10.1109/​CCC.2011.24. URL https:/​/​​10.1109/​CCC.2011.24.

[8] 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. ISBN 9781479965175. 10.1109/​FOCS.2014.57. URL https:/​/​​10.1109/​FOCS.2014.57.

[9] 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. ISBN 978-3-95977-146-7. 10.4230/​LIPIcs.TQC.2020.10. URL https:/​/​​opus/​volltexte/​2020/​12069.

[10] Frank Arute, Kunal Arya, Ryan Babbush, et al. Quantum supremacy using a programmable superconducting processor. Nature, 574 (7779): 505–510, 2019. 10.1038/​s41586-019-1666-5. URL https:/​/​​10.1038/​s41586-019-1666-5.

[11] 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. ISSN 0004-5411. 10.1145/​502090.502097. URL https:/​/​​10.1145/​502090.502097.

[12] John Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1: 195–200, Nov 1964. 10.1103/​PhysicsPhysiqueFizika.1.195. URL https:/​/​​doi/​10.1103/​PhysicsPhysiqueFizika.1.195.

[13] Aleksandrs Belovs. Variations on quantum adversary, 2015.

[14] Aleksandrs Belovs and Ansis Rosmanis. Tight quantum lower bound for approximate counting with quantum states, 2020.

[15] 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. 10.1007/​s00220-016-2706-8. URL https:/​/​​10.1007/​s00220-016-2706-8.

[16] Gilles Brassard, Peter Høyer, and Alain Tapp. Quantum cryptanalysis of hash and claw-free functions. SIGACT News, 28 (2): 14–19, June 1997. ISSN 0163-5700. 10.1145/​261342.261346. URL https:/​/​​10.1145/​261342.261346.

[17] 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. ISBN 9780821821404. 10.1090/​conm/​305.

[18] Mark Bun and Justin Thaler. Dual lower bounds for approximate degree and Markov-Bernstein inequalities. Inf. Comput., 243 (C): 2–25, August 2015. ISSN 0890-5401. 10.1016/​j.ic.2014.12.003. URL https:/​/​​10.1016/​j.ic.2014.12.003.

[19] Boris Cirel'son (Tsirelson). Quantum generalizations of Bell's inequality. Letters in Mathematical Physics, 4 (2): 93–100, 1980. 10.1007/​BF00417500. URL https:/​/​​10.1007/​BF00417500.

[20] 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, Oct 1969. 10.1103/​PhysRevLett.23.880. URL https:/​/​​doi/​10.1103/​PhysRevLett.23.880.

[21] 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. ISBN 0769521207. 10.1109/​CCC.2004.1313847.

[22] Aram Harrow and Saeed Mehraban. Approximate unitary t-designs by short random quantum circuits using nearest-neighbor and long-range gates, 2018.

[23] Norman L. Johnson and Samuel Kotz. Urn models and their application: an approach to modern discrete probability theory. Wiley, 1977. ISBN 9780471446309.

[24] 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. 10.1038/​s41534-017-0013-7. URL https:/​/​​10.1038/​s41534-017-0013-7.

[25] William Kretschmer. The Quantum Supremacy Tsirelson Inequality. In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021), volume 185 of Leibniz International Proceedings in Informatics (LIPIcs), pages 13:1–13:13, Dagstuhl, Germany, 2021. Schloss Dagstuhl–Leibniz-Zentrum für Informatik. ISBN 978-3-95977-177-1. 10.4230/​LIPIcs.ITCS.2021.13. URL https:/​/​​opus/​volltexte/​2021/​13552.

[26] 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. ISBN 9780769545714. 10.1109/​FOCS.2011.75. URL https:/​/​​10.1109/​FOCS.2011.75.

[27] 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. ISBN 978-3-95977-134-4. 10.4230/​LIPIcs.ITCS.2020.59. URL https:/​/​​opus/​volltexte/​2020/​11744.

[28] 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. ISBN 9781595936318. 10.1145/​1250790.1250874. URL https:/​/​​10.1145/​1250790.1250874.

[29] Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, USA, 2014. ISBN 1107038324. 10.1017/​CBO9781139814782.

[30] 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. ISBN 354065142X. 10.1007/​3-540-49543-6_13.

[31] 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. 10.1137/​1.9781611973082.44.

[32] Alfréd Rényi. On the theory of order statistics. Acta Mathematica Academiae Scientiarum Hungarica, 4 (3): 191–231, 1953. 10.1007/​BF02127580. URL https:/​/​​10.1007/​BF02127580.

Cited by

[1] Scott Aaronson and Shih-Han Hung, Proceedings of the 55th Annual ACM Symposium on Theory of Computing 933 (2023) ISBN:9781450399135.

[2] Scott Aaronson, "Open Problems Related to Quantum Query Complexity", arXiv:2109.06917, (2021).

[3] William Kretschmer, "Quantum Pseudorandomness and Classical Complexity", arXiv:2103.09320, (2021).

[4] Nicholas LaRacuente, "Quantum Oracle Separations from Complex but Easily Specified States", arXiv:2104.07247, (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2024-04-15 06:14:10) and SAO/NASA ADS (last updated successfully 2024-04-15 06:14:11). The list may be incomplete as not all publishers provide suitable and complete citation data.