Abstract
We present a highly efficient quantum circuit for performing continuous time quantum walks (CTQWs) over an exponentially large set of combinatorial objects, provided that the objects can be indexed 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.
Highlights
Combinatorial optimization problems are known to be notoriously difficult to solve, even approximately in general [1]
In this paper we established a general framework for combinatorial optimization via highly efficient continuoustime quantum-walk over finite but exponentially large sets of combinatorial objects
We focus on combinatorial families with an associated “indexing algorithm,” which efficiently identifies the position of a given combinatorial objects among all objects of the same order
Summary
Combinatorial optimization problems are known to be notoriously difficult to solve, even approximately in general [1]. Quantum algorithms are able to solve these problems more efficiently, with a brute force quantum search offering a guaranteed square root speedup over the classical approach [2,3] Such a speedup is, insufficient to provide practically useful solutions, since these combinatorial optimisation problems scale up exponentially. We extended the QAOA algorithm to solve constrained combinatorial optimization problems via alternating continuous-time quantum walks over efficiently identifiable feasible solutions and solution-quality-dependent phase shifts [5]. The oracular Hamiltonian encoding solution qualities are applied sequentially, interleaved with further CTQWs. Of significance is that, for the graph structures considered in this paper, the runtime of a QWOA circuit can be made independent of the walk times, leading to a distinct algorithmic advantage.
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