On the Quantum Complexity of Closest Pair and Related Problems

TL;DR

This paper explores the quantum computational complexity of the closest pair problem, presenting a near-optimal quantum algorithm in fixed dimensions and establishing complexity bounds and hypotheses for higher dimensions.

Contribution

It introduces a quantum algorithm with $ ilde{O}(n^{2/3})$ complexity for constant dimensions and proposes the Quantum Strong Exponential Time Hypothesis (QSETH) for higher dimensions.

Findings

Quantum algorithm achieves $ ilde{O}(n^{2/3})$ time in constant dimensions.

Lower bounds match the algorithm's complexity up to polylog factors.

QSETH suggests quadratic speedup is nearly optimal in higher dimensions.

Abstract

The closest pair problem is a fundamental problem of computational geometry: given a set of $n$ points in a $d$-dimensional space, find a pair with the smallest distance. A classical algorithm taught in introductory courses solves this problem in $O(n\log n)$ time in constant dimensions (i.e., when $d=O(1)$). This paper asks and answers the question of the problem's quantum time complexity. Specifically, we give an $\tilde{O}(n^{2/3})$ algorithm in constant dimensions, which is optimal up to a polylogarithmic factor by the lower bound on the quantum query complexity of element distinctness. The key to our algorithm is an efficient history-independent data structure that supports quantum interference. In $\mathrm{polylog}(n)$ dimensions, no known quantum algorithms perform better than brute force search, with a quadratic speedup provided by Grover's algorithm. To give evidence that the quadratic speedup is nearly optimal, we initiate the study of quantum fine-grained complexity and introduce the Quantum Strong Exponential Time Hypothesis (QSETH), which is based on the assumption that Grover's algorithm is optimal for CNF-SAT when the clause width is large. We show that the na\"{i}ve Grover approach to closest pair in higher dimensions is optimal up to an $n^{o(1)}$ factor unless QSETH is false. We also study the bichromatic closest pair problem and the orthogonal vectors problem, with broadly similar results.