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。本文提出了用于近似求解半定规划的变分量子算法。针对一类特定的半定规划问题，我们在弱约束假设（即 $N\gg M$，其中 $N$ 为输入矩阵的维度，$M$ 为约束条件数量）下，严格分析了算法收敛至近似局部最优解的过程。我们还为更广泛的半定规划类别提供了所需假设更少的算法。最后，我们对MaxCut等应用场景中的量子算法进行了数值模拟，模拟结果表明算法在含噪声环境中依然保持收敛性。