Average-Case Quantum Advantage with Shallow Circuits

Author François Le Gall

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François Le Gall
  • Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan


The author is very grateful to Keisuke Fujii, Tomoyuki Morimae, Harumichi Nishimura, Ansis Rosmanis and Yasuhiro Takahashi for helpful discussions. The author also thanks Jalex Stark and Thomas Vidick for comments about the manuscript. This work was partially supported by the JSPS KAKENHI grants No. 15H01677, No. 16H01705 and No. 16H05853.

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François Le Gall. Average-Case Quantum Advantage with Shallow Circuits. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Recently Bravyi, Gosset and König (Science 2018) proved an unconditional separation between the computational powers of small-depth quantum and classical circuits for a relation. In this paper we show a similar separation in the average-case setting that gives stronger evidence of the superiority of small-depth quantum computation: we construct a computational task that can be solved on all inputs by a quantum circuit of constant depth with bounded-fanin gates (a "shallow" quantum circuit) and show that any classical circuit with bounded-fanin gates solving this problem on a non-negligible fraction of the inputs must have logarithmic depth. Our results are obtained by introducing a technique to create quantum states exhibiting global quantum correlations from any graph, via a construction that we call the extended graph. Similar results have been very recently (and independently) obtained by Coudron, Stark and Vidick (arXiv:1810.04233}), and Bene Watts, Kothari, Schaeffer and Tal (STOC 2019).

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Quantum computing
  • circuit complexity
  • constant-depth circuits


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