  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.
