[Paper Review] Exact quantum query complexity of weight decision problems.

Xiaoyu He, Xiaoming Sun+2 more|arXiv (Cornell University)|Jan 17, 2018

Quantum Computing Algorithms and Architecture32 references3 citations

TL;DR

This paper determines the exact quantum query complexity for weight decision problems—distinguishing whether a binary string of length $ n $ has Hamming weight $ k $ or $ l $—by linking the problem to Chebyshev polynomials and introducing a novel quantum padding technique. For most $ (k/n, l/n) $ ratios and large $ n $, the upper and lower bounds differ by at most one query, achieving tight characterization of the exact query complexity.

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.

Motivation & Objective

• To determine the exact quantum query complexity of partial Boolean functions that distinguish between Hamming weights $ k $ and $ l $ in an $ n $-bit string.
• To close the gap between known upper and lower bounds for the query complexity of weight decision problems.
• To establish tight bounds for most $ (k/n, l/n) $ ratios when $ n $ is sufficiently large.
• To introduce and apply a new quantum padding technique to generalize results beyond boundary cases.

Proposed method

• Establish a mathematical connection between the weight decision problem and the properties of Chebyshev polynomials.
• Identify and solve all boundary cases of $ (k/n, l/n) $ where upper and lower bounds match exactly.
• Use the quantum padding technique to extend solutions from boundary cases to general $ (k/n, l/n) $ ratios.
• Apply quantum algorithmic techniques to construct exact quantum algorithms that always return the correct answer.
• Analyze query complexity using spectral methods and polynomial degree bounds derived from Chebyshev polynomials.
• Prove tight bounds by combining structural analysis of the problem with asymptotic behavior of orthogonal polynomials.

Experimental results

Research questions

• RQ1 What is the exact number of quantum queries required to decide whether the Hamming weight of an $ n $-bit string is $ k $ or $ l $, with certainty?
• RQ2 For which ratios $ (k/n, l/n) $ can the exact query complexity be determined precisely using existing techniques?
• RQ3 How can the gap between upper and lower bounds for weight decision problems be minimized or closed?
• RQ4 Can a general method be developed to extend exact query complexity results from boundary cases to arbitrary $ (k/n, l/n) $ ratios?
• RQ5 What novel quantum algorithmic techniques can be used to achieve exactness and tight bounds in query complexity?

Key findings

• For most $ (k/n, l/n) $ ratios and sufficiently large $ n $, the gap between the upper and lower bounds on exact quantum query complexity is at most one query.
• The exact query complexity is fully characterized for all boundary cases of $ (k/n, l/n) $, where upper and lower bounds match precisely.
• The quantum padding technique enables generalization of exact results from boundary cases to arbitrary $ (k/n, l/n) $ ratios, preserving tightness.
• The connection between Chebyshev polynomials and the weight decision problem provides a powerful analytical framework for deriving query bounds.
• The results show that the Deutsch-Jozsa and Grover search problems are special instances of the more general weight decision problem framework.
• The study establishes that exact quantum query complexity for weight decision problems is tightly bounded across a wide range of parameter regimes.