Efficient quantum walks over exponentially large sets of combinatorial objects for optimisation

We present a highly efficient quantum circuit for performing continuous time quantum walks (CTQWs) over an exponentially large set of combinatorial objects, provided that they can be ranked and unranked efficiently. CTQWs form the core mixing operation of a generalised version of the Quantum Approximate Optimisation Algorithm, which works by 'steering' the quantum amplitude into high-quality solutions. The efficient quantum circuit holds the promise of finding high-quality solutions to certain classes of NP-hard combinatorial problems such as the Travelling Salesman Problem, maximum set splitting, graph partitioning and lattice path optimisation.

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