From Bit-Parallelism to Quantum String Matching for Labelled Graphs

TL;DR

This paper explores converting classical bit-parallel algorithms for string matching in labeled graphs into quantum algorithms, achieving subquadratic time complexity where classical methods are limited by quadratic lower bounds.

Contribution

It demonstrates that a simple bit-parallel algorithm for string matching in level DAGs can be transformed into a quantum algorithm with better than logarithmic speed-up, surpassing classical bounds.

Findings

Quantum algorithm achieves O(|E|√|P|) time complexity.

Classical bit-parallel algorithm can be effectively converted into a quantum algorithm.

Subquadratic speed-up is possible for string matching in restricted graph classes.

Abstract

Many problems that can be solved in quadratic time have bit-parallel speed-ups with factor $w$, where $w$ is the computer word size. A classic example is computing the edit distance of two strings of length $n$, which can be solved in $O(n^2/w)$ time. In a reasonable classical model of computation, one can assume $w=\Theta(\log n)$, and obtaining significantly better speed-ups is unlikely in the light of conditional lower bounds obtained for such problems. In this paper, we study the connection of bit-parallelism to quantum computation, aiming to see if a bit-parallel algorithm could be converted to a quantum algorithm with better than logarithmic speed-up. We focus on string matching in labeled graphs, the problem of finding an exact occurrence of a string as the label of a path in a graph. This problem admits a quadratic conditional lower bound under a very restricted class of graphs (Equi et al. ICALP 2019), stating that no algorithm in the classical model of computation can solve the problem in time $O(|P||E|^{1-\epsilon})$ or $O(|P|^{1-\epsilon}|E|)$. We show that a simple bit-parallel algorithm on such restricted family of graphs (level DAGs) can indeed be converted into a realistic quantum algorithm that attains subquadratic time complexity $O(|E|\sqrt{|P|})$.