QSlack II: Applications of a slack-variable approach for variational quantum semi-definite programming*

In the prequel presentation, “QSlack I: Theory of a slack-variable approach for variational quantum semi-definite programming,” we introduced a methodology for reformulating semidefinite programs (SDPs) as quantum optimization tasks that can be solved by variational quantum algorithms (VQAs). Here, we provide numerical evidence that showcases how our algorithm works in practice for applications such as estimating trace distance, hypothesis testing relative entropy, entanglement negativity, fidelity, etc. We analyze the convergence of the proposed algorithms by evaluating their proximity to the true optimal value. We explore a broad range of choices for ansatz architecture, optimizers, and classical training techniques. More specifically, we deploy various architectures for both purification ansätze and convex combination ansätze to prepare the parametrized quantum states. We also explore both gradient-based optimizers, including gradient descent (utilizing the parameter shift rule or the Hadamard test) and simultaneous perturbation stochastic approximation (SPSA), as well as gradient-free optimizers like Powell and BFGS. Finally, we analyze how changing various hyperparameters, such as the penalty constant, affects convergence.