Polynomial representation of general partial Boolean functions with a single quantum query

TL;DR

This paper explores the polynomial degree of partial Boolean functions computable with a single quantum query, establishing new equivalences, classifications for small functions, and methods to identify functions with quantum advantages.

Contribution

It introduces a new equivalence transforming partial Boolean functions to simple forms, classifies small functions with single quantum queries, and provides a constructive method to find functions with quantum advantages.

Findings

Any partial Boolean function with one quantum query can be transformed to a degree-one polynomial function.

There are only 10 non-trivial partial Boolean functions with up to four bits that have a single quantum query.

A constructive method exists to identify all partial Boolean functions computable exactly by a given quantum 1-query algorithm.

Abstract

Early in 1992, Deutsch-Jozsa algorithm computed a symmetric partial Boolean function with a single quantum query, and thus achieved the best separation between classical deterministic and exact quantum query complexity. Until recent years, it was clarified that all symmetric partial Boolean functions with a single quantum query can be computed exactly by Deutsch-Jozsa algorithm. For the general partial Boolean functions with a single quantum query, the latest characterizations is complex and not very satisfactory. Based on this, this paper proves and discovers three new results: (1) Establishing a new equivalence, each partial Boolean function with a single quantum query can be transformed to a simple partial Boolean function whose polynomial degree is just one; (2) For partial Boolean functions up to four bits, there are only 10 non-trivial partial Boolean functions with a single quantum query; (3) For each quantum 1-query algorithm with undefined measurement, there exists a constructive method for finding out all partial Boolean functions that can be computed exactly by the algorithm. Essentially, the first discovery represent a step forward for a fundamental conclusion that the polynomial degree of partial Boolean functions with a single quantum query is one or two, and the last two results contribute a way for searching more nontrival partial Boolean functions that have quantum advantages.