Abstract
We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. This class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine ideas from quantum Gibbs sampling and matrix exponent updates. A de-quantization of the algorithm also leads to a faster classical solver. For generic instances, our quantum solver gives a nearly quadratic speedup over state-of-the-art algorithms. Such instances include approximating the ground state of spin glasses and MaxCut on Erdös-Rényi graphs. We also provide an efficient randomized rounding procedure that converts approximately optimal SDP solutions into approximations of the original quadratic optimization problem.
Highlights
Quadratic optimization problems with binary constraints are an important class of optimization problems
We present Hamiltonian Updates – a meta-algorithm for solving convex optimization problems over the set of quantum states based on quantum Gibbs sampling – in a more general setting, as we expect it to find applications to other problems
This view is similar in spirit to [43, Lemma 4.6], here we focus on using this approach to construct solutions and to show that this notion of approximate feasibility is good enough for the MaxQP semidefinite programming (SDP)
Summary
Quadratic optimization problems with binary constraints are an important class of optimization problems. We capitalize on this correspondence by devising a meta-algorithm – Hamiltonian Updates (HU) – that is inspired by matrix exponentiated gradient updates [62], see [43, 14, 30] for similar approaches Another key insight is that the diagonal constraints have a clear quantum mechanical interpretation: the feasible states are those that are indistinguishable from the uniform or maximally mixed state when measured in the computational basis. This interpretation points the way to another key component to obtaining speedups for MaxQP SDP: by setting > 0 we further relax the problem and optimize over all states that are approximately indistinguishable from the maximally mixed state when measured in the computational basis. This result on the randomized rounding relies on a detailed analysis of the stability of the rounding procedure w.r.t. to approximate solutions to the problem
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