# Quantum SDP-Solvers: Better upper and lower bounds

## TL;DR

This paper improves quantum algorithms for semidefinite programming by enhancing their efficiency and establishing fundamental lower bounds, advancing the understanding of quantum optimization methods.

## Contribution

The paper introduces new techniques for quantum algorithms, improves existing SDP-solvers, and establishes lower bounds on their complexity, especially for symmetric problems.

## Key findings

- Enhanced quantum SDP algorithms with better parameter dependence
- Development of methods to implement smooth functions of sparse Hamiltonians
- Proven lower bounds indicating linear scaling of quantum SDP-solvers in worst-case scenarios

## Abstract

Brand\~ao and Svore very 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 optimizations 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.