A Constant Measurement Quantum Algorithm for Graph Connectivity

TL;DR

This paper presents a new quantum algorithm that efficiently determines graph connectivity with a constant number of measurements, leveraging non-unitary gates from ZX calculus, applicable to any undirected graph.

Contribution

The paper introduces a quantum algorithm for graph connectedness that uses a constant number of measurements and extends to find connected components with linear measurements.

Findings

Algorithm determines graph connectedness with constant measurements.

Numerical simulations validate the algorithm's effectiveness.

Handles graphs with repeated edges and self-loops.

Abstract

We introduce a novel quantum algorithm for determining graph connectedness using a constant number of measurements. The algorithm can be extended to find connected components with a linear number of measurements. It relies on non-unitary abelian gates taken from ZX calculus. Due to the fusion rule, the two-qubit gates correspond to a large single action on the qubits. The algorithm is general and can handle any undirected graph, including those with repeated edges and self-loops. The depth of the algorithm is variable, depending on the graph, and we derive upper and lower bounds. The algorithm exhibits a state decay that can be remedied with ancilla qubits. We provide a numerical simulation of the algorithm.