Sublinear-Time Quantum Computation of the Diameter in CONGEST Networks

TL;DR

The paper proves that in the quantum CONGEST model, the diameter can be computed exactly in $\tilde O(\sqrt{nD})$ rounds, a sublinear advance over the classical $\tilde\Omega(n)$ bound and a demonstrated quantum-classical separation. It builds a distributed quantum optimization framework (Initialization, Setup, Evaluation) and refines it with DFS-based candidate sets to boost success probability, achieving the $\tilde O(\sqrt{nD})$ bound with polylogarithmic per-node memory. Additionally, it provides a $3/2$-approximation algorithm in $\tilde O(\sqrt[3]{nD}+D)$ rounds and two tight lower bounds: $\tilde \Omega(\sqrt{n})$ for small diameter and $\tilde \Omega(\sqrt{nD/s})$ under memory constraints, by reductions to disjointness in quantum communication complexity. The results illustrate a clear quantum advantage in distributed graph problems and establish fundamental limits under memory bounds, with potential practical impact for scalable quantum distributed computing.

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 $\tilde Ω(n)$ 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 $\tilde Ω(\sqrt{n})$ on the round complexity of any quantum algorithm computing the diameter, and furthermore show a tight lower bound $\tilde Ω(\sqrt{nD})$ for any distributed quantum algorithm in which each node can use only $\textrm{poly}(\log n)$ quantum bits of memory.

Figures (8)

• Figure 1: Construction of $\mathsf{BFS}(\mathsf{leader})$ in Proposition \ref{['prop:init']}.
• Figure 2: The procedure $\mathsf{Evaluation}$.
• Figure 3: Our quantum algorithm computing a $3/2$-approximation of the diameter.
• Figure 4: The graph $G_n$ used in the proof of Theorem \ref{['th:construction1']}, for $n=2$.
• Figure 5: The graph $\mathcal{G}_d$.

Theorems & Definitions (37)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5: BGK+15
• Theorem 6: Amplitude amplification Brassard+ICALP98
• Corollary 1: Quantum optimization
• proof
• Theorem 7: Distributed quantum optimization
• proof