A Fast Exact Quantum Algorithm for Solitude Verification

TL;DR

This paper introduces a quantum algorithm that efficiently verifies solitude and computes symmetric Boolean functions in anonymous distributed networks with minimal assumptions, achieving exact results in linear or near-linear rounds.

Contribution

It presents the first exact quantum algorithms for solitude verification and symmetric Boolean functions in anonymous networks, with optimal round complexity under minimal assumptions.

Findings

Solitude verification can be solved in O(N) rounds with quantum algorithms.

Symmetric Boolean functions can be computed in O(N log(max{k,2})) rounds quantumly.

Algorithms operate with polynomial bit complexity in N.

Abstract

Solitude verification is arguably one of the simplest fundamental problems in distributed computing, where the goal is to verify that there is a unique contender in a network. This paper devises a quantum algorithm that exactly solves the problem on an anonymous network, which is known as a network model with minimal assumptions [Angluin, STOC'80]. The algorithm runs in $O(N)$ rounds if every party initially has the common knowledge of an upper bound $N$ on the number of parties. This implies that all solvable problems can be solved in $O(N)$ rounds on average without error (i.e., with zero-sided error) on the network. As a generalization, a quantum algorithm that works in $O(N\log_2 (\max\{k,2\}))$ rounds is obtained for the problem of exactly computing any symmetric Boolean function, over $n$ distributed input bits, which is constant over all the $n$ bits whose sum is larger than $k$ for $k\in \{0,1,\dots, N-1\}$. All these algorithms work with the bit complexities bounded by a polynomial in $N$.