Test of Quantumness with Small-Depth Quantum Circuits

Authors Shuichi Hirahara, François Le Gall

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


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

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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)


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.

Subject Classification

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


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