Derandomization of quantum algorithm for triangle finding

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 quantum 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 $O\left(n^{9/7}\right)$ queries to the edge weight matrix oracle, and thus has the same complexity as the state-of-the-art bounded-error quantum algorithm. To achieve this derandomization, we make full use of several techniques: nested quantum walk with quantum data structure, deterministic quantum search with adjustable parameters, and dimensional reduction of quantum walk search on Johnson graph.