Quantum Walks for Chemical Reaction Networks

Abstract

Near a detailed-balance equilibrium, the perturbed mass-action dynamics of a chemical reaction network (CRN) map exactly onto an electrical-flow problem on the bipartite species-reaction graph: chemical potentials become electrical potentials, Onsager coefficients become conductances, and the instantaneous Gibbs free-energy consumption equals the dissipated electrical energy. We exploit this map to design quantum walk algorithms that decide species reachability, sample reachable species, approximate any individual steady-state reaction flux, and estimate the total Gibbs dissipation. The first three follow from standard electrical-flow quantum walks; the last is non-trivial because the chemical flow is not the minimum-energy electrical flow on the same graph. We resolve this via a new use of alternative neighbourhoods in multidimensional quantum walks, which forces the walker onto the mass-action flow whenever the network is $\ddot{I}ƒ-M$ rigid. In an adjacency-matrix QRAM access model the algorithms achieve up to a quadratic speedup over classical methods -- for example $\hat{I}\text{c◯}(n^{3/2})$ vs $\hat{I}\text{c◯}(n^2)$ for reachability -- and dissipation-aware bounds tighten this further when the perturbation is concentrated.