A variational quantum algorithm for approximating convex roofs (pp1081-1109)
          George Androulakis and Ryan McGaha
          doi: https://doi.org/10.26421/QIC22.13-14-1
Abstracts: Many entanglement measures are first defined for pure states of a bipartite Hilbert space, and then extended to mixed states via the convex roof extension. In this article we alter the convex roof extension of an entanglement measure, to produce a sequence of extensions that we call $f$-$d$ extensions, for $d \in \mathbb{N}$, where  $f:[0,1]\to [0, \infty)$ is a fixed continuous function which vanishes only at zero. We prove that for any such function $f$, and any continuous, faithful, non-negative function, (such as an entanglement measure), $\mu$  on the set of pure states of a finite dimensional bipartite Hilbert space, the collection of  $f$-$d$ extensions of  $\mu$ detects entanglement, i.e. a mixed state $\rho$ on a finite dimensional bipartite Hilbert space is separable, if and only if there exists $d \in \mathbb{N}$ such that the $f$-$d$ extension of $\mu$ applied to $\rho$ is equal to zero. We introduce a quantum variational algorithm which aims to approximate the $f$-$d$ extensions of entanglement measures defined on pure states. However, the algorithm does have its drawbacks. We show that this algorithm  exhibits barren plateaus when     used to approximate the family of $f$-$d$ extensions of the Tsallis entanglement entropy for a certain function $f$ and unitary ansatz $U(\theta)$ of sufficient depth. In practice, if additional information about the state is known, then one needs to avoid using the suggested ansatz for long depth of circuits.
Key words: Entangelement measure, barren plateau, variational quantum algorithm