We construct protocols for classically verifiable quantum advantage and classical verification of
BQP computations using \emph{quantum indistinguishability obfuscation} (qiO). Specifically, given qiO and assuming a slightly stronger version of
BQP=BPP, we construct a two-message quantum-advantage protocol that is efficiently and publicly verifiable. Our result can be viewed as a rigorous cryptographic foundation for the heuristic quantum advantage proposals based on \emph{peaked random circuit sampling} of Aaronson and Zhang (arXiv:2404.14493).
We also construct two simple protocols for classically verifying arbitrary
BQP computations. The first protocol is privately verifiable and assumes only the existence of qiO. This gives a rare example of a nontrivial cryptographic application of (quantum) iO that does not make additional computational hardness assumptions. The second protocol additionally assumes post-quantum one-way functions and is \emph{publicly verifiable}. To our knowledge, this is the first publicly verifiable protocol for classical verification of
BQP computations under computational assumptions in the standard model.
As qiO is the basis for our results, we propose an approach to constructing qiO by adapting the local-mixing framework of Canetti--Chamon--Mucciolo--Ruckenstein.
In recent years, achieving verifiable quantum advantage on a NISQ device has emerged as an important open problem in quantum information. The sampling-based quantum advantages are not known to have efficient verification methods. This article investigates the verification of quantum advantage from a cryptographic perspective. We establish a strong connection between the verifiability of quantum advantage and cryptographic and complexity primitives, including efficiently samplable, statistically far but computationally indistinguishable pairs of (mixed) quantum states ( EFI ), pseudorandom states ( PRS ), and variants of minimum circuit size problems ( MCSP ). Specifically, we prove that a) a sampling-based quantum advantage is either verifiable or can be used to build EFI and even PRS and b) polynomial-time algorithms for a variant of MCSP would imply efficient verification of quantum advantages. Our work shows that the quest for verifiable quantum advantages may lead to applications of quantum cryptography, and the construction of quantum primitives can provide new insights into the verifiability of quantum advantages.
Demonstrating quantum advantage on current quantum hardware is a central goal of quantum computing, and random quantum circuits underpin many leading proposals. Yet sampling-based demonstrations are often difficult to verify, while observable-based approaches face a different challenge: concentration can suppress differences between circuit instances. Recent experiments have put forward the estimation of out-of-time-order correlators (OTOCs) in random circuits as a promising task for verifiable quantum advantage, yet whether their circuit-to-circuit fluctuations survive concentration as system size grows has remained open. Here we show that they do. For broad classes of local random circuits in any fixed spatial dimension, we prove inverse-polynomial fluctuations of fixed-order OTOCs at system-scale depths. In one-dimensional Haar-random brickwork circuits, we further show that macroscopically many gates contribute to these fluctuations, yet in each layer they remain confined to a sublinear-width region. As a byproduct, we develop a classical algorithm exhibiting the first rigorous improvement over brute-force classical simulation of OTOCs in 1D. Although classical hardness remains open, our results rule out strong concentration as an obstruction to OTOC-based proposals for verifiable quantum advantage.