A Paturi Theorem for Signed Subcube Representations

2026-08-06

AI Summary

This work investigates the approximate generalized weight and sparsity of Boolean functions under the symbolic subcube representation and establishes a tight connection to quantum query complexity. By introducing the notion of maximum alternation depth $D(F)$ and exponential restriction profiles, the authors extend Paturi’s theorem to this representation framework for the first time, thereby developing a dual-based lower bound technique. The main contributions include deriving tight exponential upper bounds—optimal up to constant factors—on approximate generalized weight and sparsity, providing a complete characterization of approximate generalized sparsity, and uncovering its precise relationship with quantum query complexity.

Abstract

We obtain exponential upper bounds on approximate generalized weight and generalized sparsity in terms of the deepest transition depth D(F), and show that these bounds are optimal up to constant factors in the exponent. We further characterize approximate generalized sparsity, establish a dual lower-bound framework based on exponential restriction profiles, and derive a Paturi-type characterization relating approximate generalized weight to quantum query complexity.

Problem

• approximate generalized weight
• approximate generalized sparsity
• deepest transition depth
• quantum query complexity
• Paturi theorem

Innovation

• Signed Subcube Representations
• Generalized Sparsity
• Quantum Query Complexity
• Paturi Theorem
• Exponential Restriction Profiles