Unconditional Quantum Advantage for Sampling with Shallow Circuits

Recent work by Bravyi, Gosset, and Koenig showed that there exists a search problem that a constant-depth quantum circuit can solve, but that any constant-depth classical circuit with bounded fan-in cannot. They also pose the question: can we achieve a similar proof of separation for an input-independent sampling task? In this paper, we show that the answer to this question is yes. We introduce a distribution $D_{n}$ and give a constant-depth, $n$ qubit, quantum circuit that samples from a distribution close to $D_{n}$ in total variation distance. For any $\delta < 1$ we also prove, unconditionally, that any classical circuit with bounded fan-in gates that takes as input $n + n^\delta$ uniformly random bits and produces output close to $D_{n}$ in total variation distance has depth $\Omega(\log \log n)$. This gives an unconditional proof that constant-depth quantum circuits can sample from distributions which can't be reproduced by constant-depth bounded fan-in classical circuits, even up to additive error. The distribution $D_n$ and classical circuit lower bounds are based on work of Viola, in which he shows a different (but related) distribution cannot be sampled from approximately by constant-depth bounded fan-in classical circuits.