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On Certified Randomness from Fourier Sampling or Random Circuit Sampling

Roozbeh Bassirian, Adam Bouland, Bill Fefferman, Sam Gunn, Avishay Tal

TL;DR

The paper tackles public certified randomness by grounding a Fourier Sampling-based scheme in the quantum random oracle model (QROM), achieving publicly verifiable randomness without computational assumptions. It provides a black-box protocol whose security rests on min-entropy guarantees derived from Fourier-HOG tests and shows strong evidence that LLQSV lies outside $ extsf{BQP}$ and $ extsf{PH}$ via black-box reductions to Squared Forrelation. The work positions these results as a black-box analogue to Aaronson's RCS protocol and contrasts them with the later Yamakawa–Zhandry approach, highlighting unconditional security at the cost of exponential-time verification. It also develops a sophisticated set of proof techniques, including BBBV-style hybrids, double-counting arguments, and derandomization, to connect FHOG performance with high min-entropy and to establish foundational oracle separations. The findings advance the theoretical basis for near-term quantum devices as practical sources of certifiable randomness and outline key open directions toward efficient, provably secure white-box and QROM-free constructions.

Abstract

Certified randomness has a long history in quantum information, with many potential applications. Recently Aaronson (2018, 2020) proposed a novel public certified randomness protocol based on existing random circuit sampling (RCS) experiments. The security of his protocol, however, relies on non-standard complexity-theoretic conjectures which were not previously studied in the literature. Inspired by Aaronson's work, we study certified randomness in the quantum random oracle model (QROM). We show that quantum Fourier Sampling can be used to define a publicly verifiable certified randomness protocol with black-box security without any computational assumptions. In addition to giving a certified randomness protocol in the QROM, our work can also be seen as supporting Aaronson's conjectures for RCS-based randomness generation, as our protocol is in some sense the "black-box version" of Aaronson's protocol. In further support of Aaronson's proposal, we prove a Fourier Sampling version of Aaronson's conjecture by extending Raz and Tal's separation of BQP vs PH. Our work complements the subsequent certified randomness protocol of Yamakawa and Zhandry (2022) in the QROM. Whereas the security of that protocol relied on the Aaronson-Ambainis conjecture, ours does not rely on any computational assumption - at the expense of requiring exponential-time classical verification. Our protocol also has a simple heuristic implementation.

On Certified Randomness from Fourier Sampling or Random Circuit Sampling

TL;DR

The paper tackles public certified randomness by grounding a Fourier Sampling-based scheme in the quantum random oracle model (QROM), achieving publicly verifiable randomness without computational assumptions. It provides a black-box protocol whose security rests on min-entropy guarantees derived from Fourier-HOG tests and shows strong evidence that LLQSV lies outside and via black-box reductions to Squared Forrelation. The work positions these results as a black-box analogue to Aaronson's RCS protocol and contrasts them with the later Yamakawa–Zhandry approach, highlighting unconditional security at the cost of exponential-time verification. It also develops a sophisticated set of proof techniques, including BBBV-style hybrids, double-counting arguments, and derandomization, to connect FHOG performance with high min-entropy and to establish foundational oracle separations. The findings advance the theoretical basis for near-term quantum devices as practical sources of certifiable randomness and outline key open directions toward efficient, provably secure white-box and QROM-free constructions.

Abstract

Certified randomness has a long history in quantum information, with many potential applications. Recently Aaronson (2018, 2020) proposed a novel public certified randomness protocol based on existing random circuit sampling (RCS) experiments. The security of his protocol, however, relies on non-standard complexity-theoretic conjectures which were not previously studied in the literature. Inspired by Aaronson's work, we study certified randomness in the quantum random oracle model (QROM). We show that quantum Fourier Sampling can be used to define a publicly verifiable certified randomness protocol with black-box security without any computational assumptions. In addition to giving a certified randomness protocol in the QROM, our work can also be seen as supporting Aaronson's conjectures for RCS-based randomness generation, as our protocol is in some sense the "black-box version" of Aaronson's protocol. In further support of Aaronson's proposal, we prove a Fourier Sampling version of Aaronson's conjecture by extending Raz and Tal's separation of BQP vs PH. Our work complements the subsequent certified randomness protocol of Yamakawa and Zhandry (2022) in the QROM. Whereas the security of that protocol relied on the Aaronson-Ambainis conjecture, ours does not rely on any computational assumption - at the expense of requiring exponential-time classical verification. Our protocol also has a simple heuristic implementation.
Paper Structure (29 sections, 25 theorems, 95 equations, 1 figure, 1 table)

This paper contains 29 sections, 25 theorems, 95 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Our results in context
  3. Connecting Fourier Sampling to Aaronson's protocol
  4. Our Results
  5. Open Problems
  6. Proof Outline
  7. Min-Entropy Generation
  8. Worst-case, constant min-entropy bound.
  9. Average-case, constant min-entropy bound.
  10. Average-case, linear min-entropy bound.
  11. The Squared Forrelation Problem and Black box llqsv
  12. Lower Bound for $\text{Black-Box }\mathsf{LLQSV}$ via $\mathsf{Squared Forrelation}$
  13. Low-Entropy sampling bounds against
  14. Worst-case, constant min-entropy: A simple BBBV argument
  15. Average-case, constant min-entropy: Vertex degree in a bipartite graph
  16. ...and 14 more sections

Key Result

Theorem 1

(informal) There is no sub-exponential-query quantum algorithm which, given black-box access to $F$, both (i) passes $\mathsf{FHOG}$ with non-negligible probability, and (ii) conditioned on passing $\mathsf{FHOG}$, generates only $o(n)$ bits of min-entropy, for a non-negligible fraction of random fu

Figures (1)

  • Figure 1: The algorithm $A'$ takes enough samples from $A$ to estimate the output distribution, then uses $r$ as a seed to rejection sample from that distribution. For random $r$, the output distributions of $A$ and $A'$ are identical. But for fixed $r$, the output of $A'$ is nearly-deterministic.

Theorems & Definitions (55)

  • Theorem 1
  • Definition 2
  • Remark 3
  • Theorem 4
  • Lemma 8
  • proof
  • Theorem 9
  • Corollary 10
  • proof : Proof of \ref{['thm:entropy']}
  • Theorem 11
  • ...and 45 more