On verifiable quantum advantage with peaked circuit sampling

TL;DR

This paper proposes a new class of quantum circuits called peaked circuits, which have high output concentration on specific states, potentially enabling verifiable quantum advantage without exponential classical verification.

Contribution

It introduces and analyzes peaked circuits, demonstrating their potential for verifiable quantum advantage and providing bounds on their peakedness properties.

Findings

Peakedness requires .19 power of ( au_r/n) for (1/poly(n)) peakedness.

Numerical evidence shows nontrivial peakedness more than Haar-random states.

Peaked circuits could enable future verifiable quantum advantage experiments.

Abstract

Over a decade after its proposal, the idea of using quantum computers to sample hard distributions has remained a key path to demonstrating quantum advantage. Yet a severe drawback remains: verification seems to require classical computation exponential in the system size, $n$. As an attempt to overcome this difficulty, we propose a new candidate for quantum advantage experiments with otherwise random "peaked circuits", i.e., quantum circuits whose outputs have high concentrations on a computational basis state. Naturally, the heavy output string can be used for classical verification. In this work, we analytically and numerically study an explicit model of peaked circuits, in which $\tau_r$ layers of uniformly random gates are augmented by $\tau_p$ layers of gates that are optimized to maximize peakedness. We show that getting $1/\text{poly}(n)$ peakedness from such circuits requires $\tau_{p} = \Omega((\tau_r/n)^{0.19})$ with overwhelming probability. However, we also give numerical evidence that nontrivial peakedness is possible in this model -- decaying exponentially with the number of qubits, but more than can be explained by any approximation where the output of a random quantum circuit is treated as a Haar-random state. This suggests that these peaked circuits have the potential for future verifiable quantum advantage experiments. Our work raises numerous open questions about random peaked circuits, including how to generate them efficiently, and whether they can be distinguished from fully random circuits in classical polynomial time.