Quantum Advantage from Sampling Shallow Circuits: Beyond Hardness of Marginals

We construct a family of distributions $\{{\mathcal{D}}_n{\}}_n$ with ${\mathcal{D}}_n$ over $\{0,1{\}}^n$ and a family of depth-$7$ quantum circuits $\{C_n{\}}_n$ such that ${\mathcal{D}}_n$ is produced exactly by $C_n$ with the all zeros state as input, yet any constant-depth classical circuit with bounded fan-in gates evaluated on any binary product distribution has total variation distance $1-e^{-\mathrm{\Omega }(n)}$ from ${\mathcal{D}}_n$. Moreover, the quantum circuits we construct are geometrically local and use a relatively standard gate set: Hadamard, controlled-phase, CNOT, and Toffoli gates. All previous separations of this type suffer from some undesirable constraint on the classical circuit model or the quantum circuits witnessing the separation. Our family of distributions is inspired by the Parity Halving Problem of Watts, Kothari, Schaeffer, and Tal (STOC, 2019), which built on the work of Bravyi, Gosset, and König (Science, 2018) to separate shallow quantum and classical circuits for relational problems.