Variational determination of multi-qubit geometrical entanglement in NISQ computers

Abstract: Current noise levels in physical realizations of qubits and quantum operations limit the applicability of conventional methods to characterize entanglement. In this adverse scenario, we follow a quantum variational approach to estimate the geometric measure of entanglement of multiqubit pure states. The algorithm requires only single-qubit gates and measurements, so it is well suited for NISQ devices. This is demonstrated by successfully implementing the method on IBM Quantum devices for Greenberger-Horne-Zeilinger states of $3$, $4$, and $5$ qubits. Numerical simulations with random states show the robustness and accuracy of the method. The scalability of the protocol is numerically demonstrated via matrix product states techniques up to $25$ qubits.