Quantum algorithms and lower bounds for convex optimization
Shouvanik Chakrabarti, Andrew M. Childs, Tongyang Li, Xiaodi Wu · Quantum · 2020
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 O ~ ( 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 Ω ~ ( n ) evaluation queries and Ω ( n ) membership queries.