On Certified Randomness from Fourier Sampling or Random Circuit Sampling

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 unconditional black-box security. 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, our protocol is unconditionally secure - at the expense of requiring exponential-time classical verification. Our protocol also has a simple heuristic implementation.