Query-efficient classical implementation of Bell difference sampling

Develop a query-efficient classical randomized procedure that, given query access to a 1-bounded function f: F_2^n -> C, implements Bell difference sampling by sampling from the convoluted distribution Q_f using few queries to f, ideally with complexity comparable to the six-copy quantum protocol, to enable stabilizer learning and related quadratic Goldreich–Levin applications in the classical query model.

Background

The paper dequantizes techniques from stabilizer learning to build a near-optimal quadratic Goldreich–Levin algorithm. A central tool in quantum stabilizer learning is Bell difference sampling, which samples from the convoluted distribution Q_f and can be performed exactly with only a few copies of a quantum state. Translating this capability to the classical query model is challenging: direct classical sampling from Q_f with few queries is not straightforward, and the authors instead construct an approximate distribution μ_f that suffices for their analysis.

A classical, query-efficient implementation of Bell difference sampling would strengthen dequantization approaches, potentially simplifying algorithms and improving constants, and could have broader impact on classical analogues of quantum property testing and learning tasks.