Optimizing the Convergence Rate of the Quantum Consensus: A Discrete Time Model

TL;DR

This paper develops a mathematical framework to optimize the convergence rate of discrete quantum consensus algorithms in quantum networks, using spectral graph theory and representation theory to derive analytical solutions.

Contribution

It generalizes Aldous' conjecture to all partitions of N, providing a novel spectral analysis approach for optimizing quantum consensus convergence rates.

Findings

Spectral gap of Laplacian is the same for all partitions except (N).

Analytical solutions for N ≤ d^2 + 1 are derived.

Optimal solutions for complete graph topologies are provided.

Abstract

Motivated by the recent advances in the field of quantum computing, quantum systems are modelled and analyzed as networks of decentralized quantum nodes which employ distributed quantum consensus algorithms for coordination. In the literature, both continuous and discrete time models have been proposed for analyzing these algorithms. This paper aims at optimizing the convergence rate of the discrete time quantum consensus algorithm over a quantum network with $N$ qudits. The induced graphs are categorized in terms of the partitions of integer $N$ by arranging them as the Schreier graphs. It is shown that the original optimization problem reduces to optimizing the Second Largest Eigenvalue Modulus (SLEM) of the weight matrix. Exploiting the Specht module representation of partitions of $N$, the Aldous' conjecture is generalized to all partitions (except ($N$)) in the Hasse diagram of integer $N$. Based on this result, it is shown that the spectral gap of Laplacian of all induced graphs corresponding to partitions (other than ($N$)) of $N$ are the same, while the spectral radius of the Laplacian is obtained from the feasible least dominant partition in the Hasse diagram of integer $N$. The semidefinite programming formulation of the problem is addressed analytically for $N \leq d^2 + 1$ and a wide range of topologies where closed-form expressions for the optimal results are provided. For a quantum network with complete graph topology, solution of the optimization problem based on group association schemes is provided for all values of $N$.