Approximating Entanglement Measures Using VQA

Many common measures of entanglement are defined using the convex roof construction, which requires one to minimize a certain quantity over the set of all pure state ensembles of a given density matrix. This minimization is often difficult to compute directly for a given state and is usually approximated via classical optimization techniques. We supply a quantum algorithm to access the various pure state ensembles of a density matrix using parametrized quantum circuits, made possible by the Schrodinger-HJW theorem. Using standard optimization techniques, one can find an approximate optimal pure state ensemble for a given density matrix and entanglement measure. We also attempted to analyze the landscapes of the cost functions induced by both the Von Neumann entanglement entropy and the Tsallis entanglement entropy for barren plateaus, but were unsuccessful in doing so. We instead supply an entanglement witness derived from the Tsallis entropy, whose landscape exhibits barren plateaus. This entanglement witness illustrates that highly non-linear cost functions can still have barren plateaus even if they involve exponentially many terms in their definition.