# Quantum SDP-Solvers: Better upper and lower bounds

Joran van Apeldoorn1, András Gilyén1, Sander Gribling1, and Ronald de Wolf2

1QuSoft, CWI, Amsterdam, the Netherlands
2QuSoft, CWI and University of Amsterdam, the Netherlands

### Abstract

Brandão and Svore [14] recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension $n$ of the problem and the number $m$ of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure.

We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimization problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with $mn$ when $m\approx n$, which is the same as classical.

Semidefinite programs (SDPs) are an important tool in convex optimization tasks and approximation algorithms. They allow to optimize a linear function over positive semidefinite matrices, subject to linear constraints on those matrices, and they are solvable in polynomial time on a classical computer. Brandão and Svore recently gave a quantum algorithm for solving semidefinite programs that (in some regimes) is faster than the best-possible classical algorithm. In this paper we improve on their algorithm in several ways, in particular we obtain a 4-th root improvement in the running time with respect to the required precision. We also show strong limits for this particular approach to quantum SDP-solvers, for instance for combinatorial optimization problems that have a lot of symmetry, and we prove some general limitations for quantum SDP-solvers.

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