Non-variational Quantum Combinatorial Optimisation
arxiv(2024)
Abstract
This paper introduces a non-variational quantum algorithm designed to solve a
wide range of combinatorial optimisation 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.
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