Quantum Speed-ups for Semidefinite Programming

We give a quantum algorithm for solving semidefinite programs (SDPs). It has worst-case running time $n^{\frac{1}{2}} m^{\frac{1}{2}} s^2 $\text{poly}$(\log(n), \log(m), R, r, 1/\delta)$,with$n$and$s$ the dimension and row-sparsity of the input matrices, respectively, $m$ the number of constraints, $\delta$ 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 $\mathrm{\Omega }(n^{\frac{1}{2}}+m^{\frac{1}{2}})$ quantum lower bound for solving semidefinite programs with constant $s,R,r$ and $\delta$. 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.