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Variational quantum algorithms constitute one of the most widespread methodsfor using current noisy quantum computers. However, it is unknown if theseheuristic algorithms provide any quantum-computational speedup, although wecannot simulate them classically for intermediate sizes. Since entanglementlies at the core of quantum computing power, we investigate its role in theseheuristic methods for solving optimization problems. In particular, we usematrix product states to simulate the quantum approximate optimizationalgorithm with reduced bond dimensions $D$, a parameter bounding the systementanglement. Moreover, we restrict the simulation further by deterministicallysampling solutions. We conclude that entanglement plays a minor role in theMaxCut and Exact Cover 3 problems studied here since the simulated algorithmanalysis, with up to $60$ qubits and $p=100$ algorithm layers, shows that itprovides solutions for bond dimension $D \approx 10$ and depth $p \approx 30$.Additionally, we study the classical optimization loop in the approximatedalgorithm simulation with $12$ qubits and depth up to $p=4$ and show that theapproximated optimal parameters with low entanglement approach the exact ones.

The Quantum Approximate Optimization Algorithm performance with low entanglement and high circuit depth