Calibrating the role of entanglement in variational quantum circuits

Abstract: Entanglement is a key property of quantum computing that separates it from its classical counterpart, however, its exact role in the performance of quantum algorithms, especially variational quantum algorithms, is not well understood. In this work, we utilise tensor network methods to systematically probe the role of entanglement in the working of two variational quantum algorithms, the Quantum Approximate Optimisation Algorithm (QAOA) and Quantum Neural Networks (QNNs), on prototypical problems under controlled entanglement environments. We find that for the MAX-CUT problem solved using QAOA, the fidelity as a function of entanglement is highly dependent on the number of layers, layout of edges in the graph, and edge density, generally exhibiting that a high number of layers indicates a higher resilience to truncation of entanglement. This is in contrast to previous studies based on no more than four QAOA layers which show that the fidelity of QAOA follows a scaling law with respect to the entanglement per qubit of the system. Contrarily, in the case of QNNs, trained circuits with high test accuracies are underpinned by higher entanglement, with any enforced limitation in entanglement resulting in a sharp decline in test accuracy. This is corroborated by the entanglement entropy of these circuits which is consistently high suggesting that, unlike QAOA, QNNs may require quantum devices capable of generating highly entangled states. Overall our work provides a deeper understanding of the role of entanglement in the working of variational quantum algorithms which may help to implement these algorithms on NISQ-era quantum hardware in a way that maximises their accuracies.