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 $1 - e^{-Ω(n)}$ 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.

In 2005, Høyer and Špalek showed that constant-depth quantum circuits augmented with multi-qubit Fanout gates are quite powerful, able to compute a wide variety of Boolean functions as well as the quantum Fourier transform. They also asked what other multi-qubit gates could rival Fanout in terms of computational power, and suggested that the quantum Threshold gate might be one such candidate. Threshold is the gate that indicates if the Hamming weight of a classical basis state input is greater than some target value. We prove that Threshold is indeed powerful - there are polynomial-size constant-depth quantum circuits with Threshold gates that compute Fanout to high fidelity. Our proof is a generalization of a proof by Rosenthal that exponential-size constant-depth circuits with generalized Toffoli gates can compute Fanout. Our construction reveals that other quantum gates able to "weakly approximate" Parity can also be used as substitutes for Fanout.