Quantum walk informed variational algorithm design

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 $\mathcal{O}(e^{n \log m})$.