A Variational Approach to the Quantum Separability Problem

TL;DR

This paper introduces a variational quantum algorithm called VSV for identifying the closest separable state to a given quantum state, potentially enabling practical solutions to the quantum separability problem on NISQ devices.

Contribution

The paper presents a novel variational quantum algorithm (VSV) for the quantum separability problem, demonstrating its effectiveness on GHZ states and multipartite X-states, and discussing its potential on NISQ hardware.

Findings

VSV converges for GHZ states up to seven qubits.

VSV can determine the CSS of multipartite X-states.

Current NISQ devices may address the NP-hard separability problem with VSV.

Abstract

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.