  We present the variational separability verifier (VSV), which is a novel
variational quantum algorithm (VQA) 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 (GHZ)
states of up to seven qubits, using both statevector and shot-based
simulations. We also numerically determine the (CSS) of maximally entangled
multipartite $X$-states ($X$-MEMS), 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 (NISQ) 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 ($k$-CSS) of a
multipartite entangled state.
