How Much Entanglement Do Quantum Optimization Algorithms Require?

Abstract

Many classical optimization problems can be mapped to finding the ground states of diagonal Ising Hamiltonians, for which variational quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) provide heuristic methods.Because the solutions of such classical optimization problems are necessarily product states, it is unclear how entanglement affects their performance.An Adaptive Derivative-Assembled Problem-Tailored (ADAPT) variation of QAOA improves the convergence rate by allowing entangling operations in the mixer layers whereas it requires fewer CNOT gates in the entire circuit. In this work, we study the entanglement generated during the execution of ADAPT-QAOA. Through simulations of the weighted Max-Cut problem, we show that ADAPT-QAOA exhibits substantial flexibility in entangling and disentangling qubits. By incrementally restricting this flexibility, we find that a larger amount of entanglement entropy at earlier stages coincides with faster convergence at later stages. In contrast, while the standard QAOA quickly generates entanglement within a few layers, it cannot remove excess entanglement efficiently. Our results offer implications for favorable features of quantum optimization algorithms.

Information scrambling and entanglement in quantum approximate optimization algorithm circuits

“…The role of entanglement for VQAs has been extensively studied from different perspectives [29][30][31][32][33]. In recent papers, bipartite entanglement entropy in p-layer QAOA circuits was investigated.…”

“…In recent papers, bipartite entanglement entropy in p-layer QAOA circuits was investigated. Chen, et al compared the entanglement required between ADAPT-QAOA and standard QAOA solving certain problems [31], and Dupont, et al characterize entanglement generated in QAOA circuit with entanglement volume law [32]. However, details about entanglement generation in the p-layer circuit and its connection to complexity of problems still remains an open question.…”

Scaling adaptive quantum simulation algorithms via operator pool tiling

“…The ADAPT-VQE algorithm was first applied to finding the ground state energies of small molecules, where it obtained high accuracy with shallower circuit depths compared to other commonly used ansatze. The approach was subsequently generalized to a number of other applications, including optimization problems [10,11], real and imaginary time evolution [12,13], the preparation of excited states [14], and strongly-correlated lattice models [15]. The adaptive philosophy towards ansatz construction has also found other expressions, as in evolutionary algorithms and related methods [16,17].…”