Derandomization of quantum algorithm for triangle finding

TL;DR

This paper presents a method to derandomize quantum algorithms for the triangle sum problem, achieving deterministic solutions with query complexities matching the best randomized algorithms, thus enhancing reliability without sacrificing efficiency.

Contribution

It introduces a deterministic quantum algorithm for the triangle sum problem that matches the query complexity of existing randomized algorithms, using advanced quantum techniques.

Findings

Deterministic quantum algorithm with O(n^{9/7}) queries for triangle sum problem.

Achieves derandomization without increasing query complexity.

Utilizes nested quantum walks, deterministic quantum search, and dimensional reduction techniques.

Abstract

Derandomization is the process of taking a randomized algorithm and turning it into a deterministic algorithm, which has attracted great attention in classical computing. In quantum computing, it is challenging and intriguing to derandomize quantum algorithms, due to the inherent randomness of quantum mechanics. The significance of derandomizing quantum algorithms lies not only in theoretically proving that the success probability can essentially be 1 without sacrificing quantum speedups, but also in experimentally improving the success rate when the algorithm is implemented on a real quantum computer. In this paper, we focus on derandomizing quanmtum algorithms for the triangle sum problem (including the famous triangle finding problem as a special case), which asks to find a triangle in an edge-weighted graph with $n$ vertices, such that its edges sum up to a given weight.We show that when the graph is promised to contain at most one target triangle, there exists a deterministic quantum algorithm that either finds the triangle if it exists or outputs ``no triangle'' if none exists. It makes $O(n^{9/7})$ queries to the edge weight matrix oracle, and thus has the same complexity with the state-of-art bounded-error quantum algorithm. To achieve this derandomization, we make full use several techniques:nested quantum walks with quantum data structure, deterministic quantum search with adjustable parameters, and dimensional reduction of quantum walk search on Johnson graph.