Quantum walk informed variational algorithm design

TL;DR

This paper introduces a theoretical framework for designing efficient quantum variational algorithms based on continuous-time quantum walks and graph automorphisms, leading to improved algorithms for combinatorial optimization.

Contribution

The paper develops a novel framework linking quantum walk theory with variational algorithm design, providing heuristics and metrics for problem-specific optimization.

Findings

New algorithms for unconstrained and constrained optimization with improved convergence.

Simulation results showing significant improvements over existing QVAs.

Derived bounds and metrics for evaluating graph structures in quantum algorithms.

Abstract

We present a theoretical framework for the analysis of amplitude transfer in Quantum Variational Algorithms (QVAs) for combinatorial optimisation with mixing unitaries defined by vertex-transitive graphs, based on their continuous-time quantum walk (CTQW) representation and the theory of graph automorphism groups. This framework leads to a heuristic for designing efficient problem-specific QVAs. Using this heuristic, we develop novel algorithms for unconstrained and constrained optimisation. We outline their implementation with polynomial gate complexity and simulate their application to the parallel machine scheduling and portfolio rebalancing combinatorial optimisation problems, showing significantly improved convergence over preexisting QVAs. Based on our analysis, we derive metrics for evaluating the suitability of graph structures for specific problem instances, and for establishing bounds on the convergence supported by different graph structures. For mixing unitaries characterised by a CTQW over a Hamming graph on $m$-tuples of length $n$, our results indicate that the amplification upper bound increases with problem size like $\mathcal{O}(e^{n \log m})$.