Variational approach to the quantum separability problem

Abstract

We present the variational separability verifier (VSV), which is a variational quantum algorithm that determines the closest separable state (CSS) of an arbitrary quantum state with respect to the Hilbert-Schmidt distance (HSD). We first assess the performance of the VSV by investigating the convergence of the optimization procedure for Greenberger-Horne-Zeilinger states of up to seven qubits, using both state-vector and shot-based simulations. We also numerically determine the CSS of maximally entangled mixed X states, and subsequently use the results of the algorithm to surmise the analytical form of the aforementioned CSS. Our results indicate that current noisy intermediate-scale quantum devices may be useful in addressing the NP-hard full separability problem using the VSV, due to the shallow quantum circuit imposed by employing the destructive swap test to evaluate the HSD. The VSV may also possibly lead to the characterization of multipartite quantum states, once the algorithm is adapted and improved to obtain the closest k-separable state of a multipartite entangled state.