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Unconditional Quantum Advantage for Sampling with Shallow Circuits

Authors Adam Bene Watts , Natalie Parham

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LIPIcs.ITCS.2026.17.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Circuit complexity
Keywords
  • Circuit Complexity
  • Sampling Separation
  • Shallow Quantum Circuits
  • Unconditional Separations
  • Complexity of Distributions

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Abstract

Recent work by Bravyi, Gosset, and Koenig showed that there exists a search problem that a constant-depth quantum circuit can solve, but that any constant-depth classical circuit with bounded fan-in cannot. They also pose the question: Can we achieve a similar proof of separation for an input-independent sampling task? In this paper, we show that the answer to this question is yes when the number of random input bits given to the classical circuit is bounded. 
We introduce a distribution D_{n} over {0,1}ⁿ and construct a constant-depth uniform quantum circuit family {C_n}_n such that C_n samples from a distribution close to D_{n} in total variation distance. For any δ < 1 we also prove, unconditionally, that any classical circuit with bounded fan-in gates that takes as input kn + n^δ i.i.d. Bernouli random variables with entropy 1/k and produces output close to D_{n} in total variation distance has depth Ω(log log n). This gives an unconditional proof that constant-depth quantum circuits can sample from distributions that can't be reproduced by constant-depth bounded fan-in classical circuits, even up to additive error. We also show a similar separation between constant-depth quantum circuits with advice and classical circuits with bounded fan-in and fan-out, but access to an unbounded number of i.i.d random inputs.
The distribution D_n and classical circuit lower bounds are inspired by work of Viola, in which he shows a different (but related) distribution cannot be sampled from approximately by constant-depth bounded fan-in classical circuits.

Cite As Get BibTex

Adam Bene Watts and Natalie Parham. Unconditional Quantum Advantage for Sampling with Shallow Circuits. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 17:1-17:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.17

Author Details

Adam Bene Watts
  • University of Calgary, Canada
Natalie Parham
  • Columbia University, New York, NY, USA

Acknowledgements

The authors would like to thank David Gosset for helpful discussions, and Ansis Rosmanis for sharing an insightful note. They thank Angus Lowe for insightful discussions which motivating the study of classical circuits with biased input in Appendix C of the full version of this paper. They also thank Michael Oliveira for insights regarding compilation of the U_{m, θ} unitary, which are discussed in Appendix A of the full version of the paper.

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