Quantum algorithms and lower bounds for convex optimization

09/04/2018

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by   Shouvanik Chakrabarti, et al.

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While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a convex function over an n-dimensional convex body using Õ(n) queries to oracles that evaluate the objective function and determine membership in the convex body. This represents a quadratic improvement over the best-known classical algorithm. We also study limitations on the power of quantum computers for general convex optimization, showing that it requires Ω̃(√(n)) evaluation queries and Ω(√(n)) membership queries.