Sublinear-Time Quantum Computation of the Diameter in CONGEST Networks

TL;DR

This paper presents a quantum distributed algorithm that computes the diameter of a network in sublinear rounds, demonstrating a quantum advantage over classical methods and establishing lower bounds for quantum algorithms in the CONGEST model.

Contribution

It introduces the first sublinear-time quantum algorithm for exact diameter computation in the CONGEST model, showing quantum speedup over classical algorithms.

Findings

Quantum algorithm computes diameter in rac{7}{7}O(7)7(7nD) rounds.

Quantum algorithms outperform classical counterparts in diameter computation.

Lower bounds establish fundamental limits for quantum distributed algorithms.

Abstract

The computation of the diameter is one of the most central problems in distributed computation. In the standard CONGEST model, in which two adjacent nodes can exchange $O(\log n)$ bits per round (here $n$ denotes the number of nodes of the network), it is known that exact computation of the diameter requires $\tilde \Omega(n)$ rounds, even in networks with constant diameter. In this paper we investigate quantum distributed algorithms for this problem in the quantum CONGEST model, where two adjacent nodes can exchange $O(\log n)$ quantum bits per round. Our main result is a $\tilde O(\sqrt{nD})$-round quantum distributed algorithm for exact diameter computation, where $D$ denotes the diameter. This shows a separation between the computational power of quantum and classical algorithms in the CONGEST model. We also show an unconditional lower bound $\tilde \Omega(\sqrt{n})$ on the round complexity of any quantum algorithm computing the diameter, and furthermore show a tight lower bound $\tilde \Omega(\sqrt{nD})$ for any distributed quantum algorithm in which each node can use only $\textrm{poly}(\log n)$ quantum bits of memory.