A semidefinite program (SDP) is a particular kind of convex optimization problem with applications in operations research, combinatorial optimization, quantum information science, and beyond. In this work, we propose variational quantum algorithms for approximately solving SDPs. For one class of SDPs, we provide a rigorous analysis of their convergence to approximate locally optimal solutions, under the assumption that they are weakly constrained (i.e., $N\gg M$, where $N$ is the dimension of the input matrices and $M$ is the number of constraints). We also provide algorithms for a more general class of SDPs that requires fewer assumptions. Finally, we numerically simulate our quantum algorithms for applications such as MaxCut, and the results of these simulations provide evidence that convergence still occurs in noisy settings.

翻译：半定规划（SDP）是一类特殊的凸优化问题，在运筹学、组合优化、量子信息科学等领域具有广泛应用。本文提出用于近似求解半定规划的变分量子算法。针对一类半定规划，我们在弱约束假设（即$N\gg M$，其中$N$为输入矩阵的维度，$M$为约束数量）下，严格分析了其收敛至近似局部最优解的特性。我们还为更一般类别的半定规划设计了所需假设更少的算法。最后，我们针对MaxCut等应用对量子算法进行了数值仿真，仿真结果证明在含噪声环境下算法仍能保持收敛性。