Locality Bounds for Sampling Hamming Slices

TL;DR

This work advances the study of locality versus sampling by proving superconstant locality lower bounds for sampling distributions over binary strings with fixed Hamming properties, notably $\mathcal{D}_k$ and $\mathcal{M}_n$, without input-size restrictions. The authors develop a general framework combining graph-elimination, conditioning, anticoncentration, and local limit theorems to reduce arbitrary $d$-local samplers to structured subproblems with many non-connected outputs, leading to strong distance guarantees. They translate these bounds into concrete applications, including data-structure lower bounds for dictionaries with modular or fixed-weight constraints and input-independent quantum–classical separations, where quantum circuits can sample certain distributions efficiently while classical NC^0 circuits cannot. The paper also provides matching upper bounds, showing explicit constructions of $O(\log n\cdot \log(n/\varepsilon))$- or $O(\log(n/k)+\log^2(k/\varepsilon))$-local samplers for $\mathcal{D}_k$ and $O(q^2\log^2(n/\varepsilon))$-local samplers for $\mathcal{D}_{q,\Lambda}$, highlighting the tightness and trade-offs of locality in sampling tasks. Overall, the results offer a detailed locality–distance landscape for sampling Hamming-slice distributions with broad implications for complexity, data structures, and quantum–classical separations.

Abstract

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. We build upon and make explicit earlier implicit results of Viola to provide superconstant lower bounds on the locality of Boolean functions approximately sampling the uniform distribution over binary strings of particular Hamming weights, both exactly and modulo an integer, answering questions of Viola (Journal of Computing 2012) and Filmus, Leigh, Riazanov, and Sokolov (RANDOM 2023). Applications to data structure lower bounds and quantum-classical separations are discussed.

Figures (2)

• Figure 1: The tree $\mathcal{L}_k$. The vertices in the green box are identified as the left vertices of the bipartite graph $G$, while those outside the box correspond to the right vertices.
• Figure 2: Left: The sub-tree rooted at a node $v \in \mathcal{L}_k$ of depth $i$. The blue dashed edges represent that the $u_{2,j}$'s are leaves in the sub-tree rooted at $v_2$. Right: The corresponding $\mathcal{B}_v$. The blue dashed edges correspond to edges in the bipartite graph $G$.

Theorems & Definitions (96)

• Theorem 1.1: Consequence of \ref{['thm:locality_single_hamming_slice']}
• Theorem 1.2: Consequence of \ref{['thm:locality_single_hamming_slice']}
• Theorem 1.3: Consequence of \ref{['thm:locality_single_non-dyadic']}
• Theorem 1.4: Consequence of \ref{['thm:mod_slice']}
• Definition 1.5: Dictionary Problem
• Claim 1.6: viola2012complexity
• Corollary 1.7: Via \ref{['thm:informal_mod']}
• Corollary 1.8: Via \ref{['thm:informal_mod']}
• Corollary 1.9: Via \ref{['thm:informal_single_1']}
• Theorem 1.10: Consequence of \ref{['thm:locality_biased']}