An entanglement perspective on the quantum approximate optimization algorithm

TL;DR

This paper investigates how entanglement evolves in the QAOA algorithm for Max-Cut problems, revealing a volume-law entanglement barrier and implications for simulation efficiency.

Contribution

It provides a detailed analysis of entanglement growth in QAOA, connecting it with random matrix theory and comparing it to quantum annealing.

Findings

Entanglement exhibits a volume-law barrier during QAOA execution.

Entanglement spectrum aligns with predictions from random matrix theory.

High entanglement impacts the efficiency of tensor network simulations.

Abstract

Many quantum algorithms seek to output a specific bitstring solving the problem of interest--or a few if the solution is degenerate. It is the case for the quantum approximate optimization algorithm (QAOA) in the limit of large circuit depth, which aims to solve quadratic unconstrained binary optimization problems. Hence, the expected final state for these algorithms is either a product state or a low-entangled superposition involving a few bitstrings. What happens in between the initial $N$-qubit product state $\vert 0\rangle^{\otimes N}$ and the final one regarding entanglement? Here, we consider the QAOA algorithm for solving the paradigmatic Max-Cut problem on different types of graphs. We study the entanglement growth and spread resulting from randomized and optimized QAOA circuits and find that there is a volume-law entanglement barrier between the initial and final states. We also investigate the entanglement spectrum in connection with random matrix theory. In addition, we compare the entanglement production with a quantum annealing protocol aiming to solve the same Max-Cut problems. Finally, we discuss the implications of our results for the simulation of QAOA circuits with tensor network-based methods relying on low-entanglement for efficiency, such as matrix product states.