  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.
