Exact quantum query complexity of weight decision problems via Chebyshev polynomials

TL;DR

This paper precisely determines the exact quantum query complexity for weight decision problems using Chebyshev polynomials, establishing tight bounds and introducing a novel quantum padding technique.

Contribution

It provides tight bounds for the quantum query complexity of weight decision problems and introduces a new quantum padding method for algorithm design.

Findings

Upper and lower bounds differ by at most one for most cases.

Chebyshev polynomials are key to analyzing quantum query complexity.

Quantum padding is a versatile technique for quantum algorithms.

Abstract

The weight decision problem, which requires to determine the Hamming weight of a given binary string, is a natural and important problem, with applications in cryptanalysis, coding theory, fault-tolerant circuit design and so on. In particular, both Deutsch-Jozsa problem and Grover search problem can be interpreted as special cases of weight decision problems. In this work, we investigate the exact quantum query complexity of weight decision problems, where the quantum algorithm must always output the correct answer. More specifically we consider a partial Boolean function which distinguishes whether the Hamming weight of the length-$n$ input is $k$ or it is $l$. Our contribution includes both upper bounds and lower bounds for the precise number of queries. Furthermore, for most choices of $(\frac{k}{n},\frac{l}{n})$ and sufficiently large $n$, the gap between our upper and lower bounds is no more than one. To get the results, we first build the connection between Chebyshev polynomials and our problem, then determine all the boundary cases of $(\frac{k}{n},\frac{l}{n})$ with matching upper and lower bounds, and finally we generalize to other cases via a new \emph{quantum padding} technique. This quantum padding technique can be of independent interest in designing other quantum algorithms.