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Test of Quantumness with Small-Depth Quantum Circuits

Authors Shuichi Hirahara, François Le Gall

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ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum computing
  • small-depth circuits
  • quantum cryptography

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Abstract

Recently Brakerski, Christiano, Mahadev, Vazirani and Vidick (FOCS 2018) have shown how to construct a test of quantumness based on the learning with errors (LWE) assumption: a test that can be solved efficiently by a quantum computer but cannot be solved by a classical polynomial-time computer under the LWE assumption. This test has lead to several cryptographic applications. In particular, it has been applied to producing certifiable randomness from a single untrusted quantum device, self-testing a single quantum device and device-independent quantum key distribution. 
In this paper, we show that this test of quantumness, and essentially all the above applications, can actually be implemented by a very weak class of quantum circuits: constant-depth quantum circuits combined with logarithmic-depth classical computation. This reveals novel complexity-theoretic properties of this fundamental test of quantumness and gives new concrete evidence of the superiority of small-depth quantum circuits over classical computation.

Cite As Get BibTex

Shuichi Hirahara and François Le Gall. Test of Quantumness with Small-Depth Quantum Circuits. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.MFCS.2021.59

Author Details

Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan
François Le Gall
  • Graduate School of Mathematics, Nagoya University, Japan

Funding

JSPS KAKENHI grants Nos. JP19H04066, JP20H05966, JP20H00579, JP20H04139, JP21H04879 and MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) grants No. JPMXS0118067394 and JPMXS0120319794.

Acknowledgements

The authors are grateful to Ryo Hiromasa, Tomoyuki Morimae, Yasuhiko Takahashi and Seiichiro Tani for helpful discussions.

References

  1. List of quantum processors. URL: https://en.wikipedia.org/wiki/List_of_quantum_processors.
  2. Scott Aaronson and Alex Arkhipov. The computational complexity of linear optics. In Proceedings of the 43rd ACM Symposium on Theory of Computing, pages 333-342, 2011. URL: https://doi.org/10.1145/1993636.1993682.
  3. Scott Aaronson and Alex Arkhipov. BosonSampling is far from uniform. Quantum Information & Computation, 14(15-16):1383-1423, 2014. URL: https://doi.org/10.26421/QIC14.15-16-7.
  4. Scott Aaronson and Lijie Chen. Complexity-theoretic foundations of quantum supremacy experiments. In Proceedings of the 32nd Computational Complexity Conference, pages 22:1-22:67, 2017. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.22.
  5. Adam Bene Watts, Robin Kothari, Luke Schaeffer, and Avishay Tal. Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits. In Proceedings of the 43rd ACM Symposium on Theory of Computing, pages 515-526, 2019. Google Scholar
  6. Adam Bouland, Bill Fefferman, Chinmay Nirkhe, and Umesh Vazirani. "Quantum supremacy" and the complexity of random circuit sampling. In Proceedings of the 10th Innovations in Theoretical Computer Science conference, pages 15:1-15:2, 2019. https://arxiv.org/abs/1803.04402. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.15.
  7. Zvika Brakerski, Paul Christiano, Urmila Mahadev, Umesh V. Vazirani, and Thomas Vidick. A cryptographic test of quantumness and certifiable randomness from a single quantum device. In Proceedings of the 59th IEEE Annual Symposium on Foundations of Computer Science, pages 320-331, 2018. Full version available as https://arxiv.org/abs/1804.00640. URL: https://doi.org/10.1109/FOCS.2018.00038.
  8. Zvika Brakerski, Venkata Koppula, Umesh V. Vazirani, and Thomas Vidick. Simpler proofs of quantumness. In Proceedings of the 15th Conference on the Theory of Quantum Computation, Communication and Cryptography, volume 158 of LIPIcs, pages 8:1-8:14, 2020. URL: https://doi.org/10.4230/LIPIcs.TQC.2020.8.
  9. Zvika Brakerski and Henry Yuen. Quantum garbled circuits, 2020. URL: http://arxiv.org/abs/2006.01085.
  10. Sergey Bravyi, David Gosset, and Robert König. Quantum advantage with shallow circuits. Science, 362(6412):308-311, 2018. URL: https://doi.org/10.1126/science.aar3106.
  11. Sergey Bravyi, David Gosset, Robert König, and Marco Tomamichel. Quantum advantage with noisy shallow circuits in 3d. In Proceedings of the 60th IEEE Annual Symposium on Foundations of Computer Science, pages 995-999, 2019. URL: https://doi.org/10.1109/FOCS.2019.00064.
  12. Michael J. Bremner, Richard Jozsa, and Dan J. Shepherd. Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 467(2126):459-472, 2010. URL: https://doi.org/10.1098/rspa.2010.0301.
  13. Michael J. Bremner, Ashley Montanaro, and Dan J. Shepherd. Average-case complexity versus approximate simulation of commuting quantum computations. Physical Review Letters, 117:080501, 2016. URL: https://doi.org/10.1103/PhysRevLett.117.080501.
  14. Michael J. Bremner, Ashley Montanaro, and Dan J. Shepherd. Achieving quantum supremacy with sparse and noisy commuting quantum circuits. Quantum, 1:8, 2017. URL: https://doi.org/10.22331/q-2017-04-25-8.
  15. Anne Broadbent and Elham Kashefi. Parallelizing quantum circuits. Theoretical Computer Science, 410(26):2489-2510, 2009. URL: https://doi.org/10.1016/j.tcs.2008.12.046.
  16. Dan E. Browne, Elham Kashefi, and Simon Perdrix. Computational depth complexity of measurement-based quantum computation. In Proceedings of the 5th Conference on Theory of Quantum Computation, Communication, and Cryptography, volume 6519 of Lecture Notes in Computer Science, pages 35-46, 2010. URL: https://doi.org/10.1007/978-3-642-18073-6_4.
  17. Matthew Coudron, Jalex Stark, and Thomas Vidick. Trading locality for time: Certifiable randomness from low-depth circuits. Communications of Mathematical Physics, 2021. URL: https://doi.org/10.1007/s00220-021-03963-w.
  18. Edward Farhi and Aram W. Harrow. Quantum supremacy through the quantum approximate optimization algorithm, 2016. URL: http://arxiv.org/abs/1602.07674.
  19. Keisuke Fujii, Hirotada Kobayashi, Tomoyuki Morimae, Harumichi Nishimura, Shuhei Tamate, and Seiichiro Tani. Impossibility of classically simulating one-clean-qubit model with multiplicative error. Physical Review Letters, 120:200502, 2018. URL: https://doi.org/10.1103/PhysRevLett.120.200502.
  20. Keisuke Fujii and Shuhei Tamate. Computational quantum-classical boundary of noisy commuting quantum circuits. Scientific Reports, 6(25598), 2016. URL: https://doi.org/10.1038/srep25598.
  21. François Le Gall. Average-case quantum advantage with shallow circuits. In Proceedings of the 34th Computational Complexity Conference, volume 137 of LIPIcs, pages 21:1-21:20, 2019. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.21.
  22. Daniel Gottesman and Isaac L. Chuang. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature, 402:390-393, 1999. URL: https://doi.org/10.1038/46503.
  23. Daniel Grier and Luke Schaeffer. Interactive shallow Clifford circuits: quantum advantage against NC¹ and beyond. In Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 875-888, 2020. URL: https://doi.org/10.1145/3357713.3384332.
  24. Lov Grover and Terry Rudolph. Creating superpositions that correspond to efficiently integrable probability distributions, 2002. URL: http://arxiv.org/abs/quant-ph/0208112.
  25. Peter Høyer and Robert Spalek. Quantum fan-out is powerful. Theory of Computing, 1(1):81-103, 2005. URL: https://doi.org/10.4086/toc.2005.v001a005.
  26. Richard Jozsa. An introduction to measurement based quantum computation, 2005. URL: http://arxiv.org/abs/quant-ph/0508124.
  27. Gregory D. Kahanamoku-Meyer, Soonwon Choi, Umesh V. Vazirani, and Norman Y. Yao. Classically-verifiable quantum advantage from a computational Bell test, 2021. URL: http://arxiv.org/abs/2104.00687.
  28. Debbie W. Leung. Quantum computation by measurements. International Journal of Quantum Information, 2(1):33-43, 2004. URL: https://doi.org/10.1142/S0219749904000055.
  29. Urmila Mahadev. Classical verification of quantum computations. In Proceedings of the 59th IEEE Annual Symposium on Foundations of Computer Science, pages 259-267, 2018. URL: https://doi.org/10.1109/FOCS.2018.00033.
  30. Tony Metger, Yfke Dulek, Andrea Coladangelo, and Rotem Arnon-Friedman. Device-independent quantum key distribution from computational assumptions. ArXiv:2010.04175 (Presented as a contributed talk at QIP'21), 2020. URL: http://arxiv.org/abs/2010.04175.
  31. Tony Metger and Thomas Vidick. Self-testing of a single quantum device under computational assumptions. In Proceedings of the 12th Innovations in Theoretical Computer Science Conference, volume 185 of LIPIcs, pages 19:1-19:12, 2021. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.19.
  32. Daniele Micciancio and Chris Peikert. Trapdoors for lattices: Simpler, tighter, faster, smaller. In Proceedings of the 31st Annual International Conference on the Theory and Applications of Cryptographic Techniques, volume 7237 of Lecture Notes in Computer Science, pages 700-718. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-29011-4_41.
  33. Tomoyuki Morimae, Keisuke Fujii, and Joseph F. Fitzsimons. Hardness of classically simulating the one-clean-qubit model. Physical Review Letters, 112:130502, 2014. URL: https://doi.org/10.1103/PhysRevLett.112.130502.
  34. Michael A. Nielsen. Quantum computation by measurement and quantum memory. Physical Letters A, 308:96-100, 2003. URL: https://doi.org/10.1016/S0375-9601(02)01803-0.
  35. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000. Google Scholar
  36. Oded Regev. On lattices, learning with errors, random linear codes, and cryptography. Journal of the ACM, 56(6):34:1-34:40, 2009. URL: https://doi.org/10.1145/1568318.1568324.
  37. Kai-Yeung Siu, Jehoshua Bruck, Thomas Kailath, and Thomas Hofmeister. Depth efficient neural networks for division and related problems. IEEE Transactions on Information Theory, 39(3):946-956, 1993. URL: https://doi.org/10.1109/18.256501.
  38. Yasuhiro Takahashi and Seiichiro Tani. Collapse of the hierarchy of constant-depth exact quantum circuits. Computational Complexity, 25(4):849-881, 2016. URL: https://doi.org/10.1007/s00037-016-0140-0.
  39. Barbara M. Terhal and David P. DiVincenzo. Adaptive quantum computation, constant depth quantum circuits and Arthur-Merlin games. Quantum Information & Computation, 4(2):134-145, 2004. URL: https://doi.org/10.26421/QIC4.2-5.
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