Quantum Speed-ups for Semidefinite Programming

We give a quantum algorithm for solving semidefinite programs (SDPs). It has worst-case running time n^1/2 m^1/2 s^2 poly(log(n),log(m),R,r,1/δ) , with n and s the dimension and row-sparsity of the input matrices, respectively, m the number of constraints, δ the accuracy of the solution, and R,r a upper bounds on the size of the optimal primal and dual solutions. This gives a square-root unconditional speed-up over any classical method for solving SDPs both in n and m . We prove the algorithm cannot be substantially improved (in terms of n and m ) giving a Ω(n^1/2+m^1/2) quantum lower bound for solving semidefinite programs with constant s,R,r and δ . The quantum algorithm is constructed by a combination of quantum Gibbs sampling and the multiplicative weight method. In particular it is based on a classical algorithm of Arora and Kale for approximately solving SDPs. We present a modification of their algorithm to eliminate the need for solving an inner linear program which may be of independent interest.