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.

Let G be a finite abelian group and A be a subset of $G \times G$ which is corner-free, meaning that there are no $x, y \in G$ and $d \in G \backslash\{0\}$ such that $(x, y),(x+d, y),(x, y+d) \in A$. We prove that \begin{equation*}|A| \leq|G|^{2} \cdot \exp \left(-(\log |G|)^{\Omega{1}}\right)\end{equation*}As a consequence, we obtain polynomial (in the input length) lower bounds on the non-deterministic communication complexity of Exactly-N in the 3-player Number-on-Forehead model. We also obtain the first “reasonable” lower bounds on the coloring version of the 3 -dimensional corners problem, as well as on the non-deterministic communication complexity of Exactly-N in the 4-player Number-on-Forehead model. This is an extended abstract. The full version of the paper can be found at https://arxiv.org/abs/2504.07006.

We characterize the power of constant-depth Boolean circuits in generating uniform symmetric distributions. Let fλ¶{0,1}m→{0,1}n be a Boolean function where each output bit of f depends only on O(1) input bits. Assume the output distribution of f on uniform input bits is close to a uniform distribution D with a symmetric support. We show that D is essentially one of the following six possibilities: (1) point distribution on 0n, (2) point distribution on 1n, (3) uniform over {0n,1n}, (4) uniform over strings with even Hamming weights, (5) uniform over strings with odd Hamming weights, and (6) uniform over all strings. This confirms a conjecture of Filmus, Leigh, Riazanov, and Sokolov (RANDOM 2023). This is an extended abstract. The full paper can be found at https://arxiv.org/abs/2411.08183v1. An updated version with a stronger result can be found at https://arxiv.org/abs/2411.08183.

The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients. In this note, we refute two conjectures. The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang et al. (FOCS'13), and the second one is due to Mande and Sanyal (FSTTCS'20). These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. We refute these conjectures by constructing two specific boolean functions tailored to each.

Spurred by the influential work of Viola (Journal of Computing 2012), the past decade has witnessed an active line of research into the complexity of (approximately) sampling distributions, in contrast to the traditional focus on the complexity of computing functions.