Non-Variational Quantum Combinatorial Optimisation

This paper introduces a non-variational quantum algorithm designed to solve a wide range of combinatorial optimisation problems, including constrained and non-binary problems. The algorithm leverages an engineered interference process achieved through repeated application of two unitaries; one inducing phase-shifts dependent on objective function values, and the other mixing phase-shifted probability amplitudes via a continuous-time quantum walk (CTQW) on a problem-specific graph. The algorithm's versatility is demonstrated through its application to various problems, namely those for which solutions are characterised by either a vector of binary variables, a vector of non-binary integer variables, or permutations (a vector of integer variables without repetition). An efficient quantum circuit implementation of the CTQW for each of these problem types is also discussed. A penalty function approach for constrained problems is also introduced, including a method for optimising the penalty function. The algorithm's performance is demonstrated through numerical simulation for randomly generated instances of the following problems (and problem sizes): weighted maxcut (18 vertices), maximum independent set (18 vertices), k-means clustering (12 datapoints, 3 clusters), capacitated facility location (12 customers, 3 facility locations), and the quadratic assignment problem (9 locations). For each problem instance, the algorithm finds a globally optimal solution with a small number of iterations.