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Quantum Advantage from Sampling Shallow Circuits: Beyond Hardness of Marginals

Authors Daniel Grier , Daniel M. Kane , Jackson Morris , Anthony Ostuni , Kewen Wu

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LIPIcs.ITCS.2026.73.pdf
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ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Complexity classes
Keywords
  • Shallow circuits
  • sampling
  • quantum circuits

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Abstract

We construct a family of distributions {𝒟_n}_n with 𝒟_n over {0, 1}ⁿ and a family of depth-7 quantum circuits {C_n}_n such that 𝒟_n is produced exactly by C_n with the all zeros state as input, yet any constant-depth classical circuit with bounded fan-in gates evaluated on any binary product distribution has total variation distance 1 - e^{-Ω(n)} from 𝒟_n. Moreover, the quantum circuits we construct are geometrically local and use a relatively standard gate set: Hadamard, controlled-phase, CNOT, and Toffoli gates. All previous separations of this type suffer from some undesirable constraint on the classical circuit model or the quantum circuits witnessing the separation.
Our family of distributions is inspired by the Parity Halving Problem of Watts, Kothari, Schaeffer, and Tal (STOC, 2019), which built on the work of Bravyi, Gosset, and König (Science, 2018) to separate shallow quantum and classical circuits for relational problems.

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Daniel Grier, Daniel M. Kane, Jackson Morris, Anthony Ostuni, and Kewen Wu. Quantum Advantage from Sampling Shallow Circuits: Beyond Hardness of Marginals. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 73:1-73:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.73

Author Details

Daniel Grier
  • UC San Diego, CA, USA
Daniel M. Kane
  • UC San Diego, CA, USA
Jackson Morris
  • UC San Diego, CA, USA
Anthony Ostuni
  • UC San Diego, CA, USA
Kewen Wu
  • Institute for Advanced Study, Princeton, NJ, USA

Funding

  • Kane, Daniel M.: Supported in part by NSF Medium Award CCF-2107079.
  • Wu, Kewen: Supported by the National Science Foundation under Grant No. DMS-2424441, and by the IAS School of Mathematics.

Acknowledgements

AO thanks Farzan Byramji for suggestions improving the presentation. KW thanks David Gosset for helpful discussions motivating the problem.

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