On verifiable quantum advantage with peaked circuit sampling

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.
Comment: Update: We analytically show that peaked circuits are rare in completely random RQCs in Section II. We show that even when peakedness delta~1/poly(n) instead of being a constant, the lower bound for generating peaked circuits still holds. We added a plot in Fig. 2