Quantifying the efficiency of state preparation via quantum variational eigensolvers

TL;DR

This paper investigates how the success of quantum variational eigensolvers, specifically QAOA, in preparing complex quantum states relates to the interaction distance, linking state structure to algorithm efficiency across various models.

Contribution

It introduces the interaction distance as a metric to quantify QAOA's efficiency in state preparation and demonstrates its correlation with entanglement spectrum structure in multiple models.

Findings

Interaction distance correlates with QAOA success.

QAOA finds solutions near the closest free-fermion state.

Entanglement spectrum structure indicates phase information.

Abstract

Recently, there has been much interest in the efficient preparation of complex quantum states using low-depth quantum circuits, such as Quantum Approximate Optimization Algorithm (QAOA). While it has been numerically shown that such algorithms prepare certain correlated states of quantum spins with surprising accuracy, a systematic way of quantifying the efficiency of QAOA in general classes of models has been lacking. Here, we propose that the success of QAOA in preparing ordered states is related to the interaction distance of the target state, which measures how close that state is to the manifold of all Gaussian states in an arbitrary basis of single-particle modes. We numerically verify this for several examples of non-integrable quantum models, including Ising models with two- and three-spin interactions and the cluster model in an external field. Our results suggest that the structure of the entanglement spectrum, as witnessed by the interaction distance, correlates with the success of QAOA state preparation, and that this correlation also contains information about different phases present in the model. We conclude that QAOA typically finds a solution that perturbs around the closest free-fermion state.