Strong Selective and List-Decoding Direct Product Theorems for Quantum Query Complexity

2026-09-22

AI Summary

本文解决了量子查询复杂度中强选择性和列表解码直接积定理问题，通过新的乘性对手方法证明了适用于所有函数的量子强选择性直接积定理及量子列表解码直接积定理。

Abstract

Quantum strong direct-product theorems for specific functions have been known for nearly two decades. These have been extended to general results for function computation and state generation. The proofs of these results use a version of the multiplicative adversary method that does not naturally extend to relations. Standard strong direct-product theorems apply when algorithms must correctly answer every given question. Prior work extended them to equivalent threshold direct-product theorems, which require answers to all questions but only require that most answers are correct. We focus on two further generalizations. Strong selective direct-products apply to algorithms that adaptively choose, based on what they learn from queries, which questions from a large list to answer. This generalization is relational and useful for proving time-space tradeoffs. We prove a quantum strong selective direct-product theorem for all functions using a new multiplicative adversary formulation for relations that satisfies a strong selective direct product property while being strong enough to capture any query lower bound for functions proven by negative-weights adversaries. This was not previously known even without selectivity. The second generalization is list-decoding direct product problems introduced by Ben-David and Blais for classical randomized query complexity. These allow an algorithm to produce a large list of possible output vectors such that one of them is fully correct. They proved that such theorems hold for classical randomized complexity of all Boolean functions. We prove a quantum analogue of this theorem for all partial Boolean functions. We show that strong list-decoding direct-product theorems are implied by a special case of multiplicative adversaries which we show, via a new reduction, can be obtained from negative-weights adversaries for any Boolean-valued function.

Problem

• quantum query complexity
• strong selective direct-product theorems
• list-decoding direct-product theorems
• multiplicative adversary method

Innovation

• strong selective direct-product theorem
• quantum query complexity
• list-decoding direct-product problems
• multiplicative adversary method