Characterization of exact one-query quantum algorithms

TL;DR

This paper characterizes the exact computational power of one-query quantum algorithms, showing they can only compute specific simple Boolean functions, without relying on polynomial degree methods.

Contribution

It provides a novel characterization of exactly which Boolean functions are computable by one-query quantum algorithms, independent of polynomial degree analysis.

Findings

One-query quantum algorithms can exactly compute only specific simple functions.

The characterization includes functions like individual bits and XOR of two bits.

The proof does not depend on polynomial degree, differing from previous approaches.

Abstract

The quantum query models is one of the most important models in quantum computing. Several well-known quantum algorithms are captured by this model, including the Deutsch-Jozsa algorithm, the Simon algorithm, the Grover algorithm and others. In this paper, we characterize the computational power of exact one-query quantum algorithms. It is proved that a total Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ can be exactly computed by a one-query quantum algorithm if and only if $f(x)=x_{i_1}$ or ${x_{i_1} \oplus x_{i_2} }$ (up to isomorphism). Note that unlike most work in the literature based on the polynomial method, our proof does not resort to any knowledge about the polynomial degree of $f$.