Optimal Protocols in Quantum Annealing and Quantum Approximate Optimization Algorithm Problems

TLDR

Quantum Annealing and QAOA are special cases of a control problem that seeks to minimize the energy of a quantum state, yet it has been unclear which protocol is more effective, contrary to prior suggestions that bang‑bang (QAOA) is optimal. The study analytically applies optimal control theory to demonstrate that, within a fixed time, the optimal protocol combines bang‑bang QAOA at the start and end with a smooth annealing segment in between. The authors support their theory with simulations of transverse‑field Ising models that reveal bang‑anneal‑bang protocols are more prevalent. The analysis and simulations show that bang‑anneal‑bang protocols are common and provide guidance for early experimental implementations of quantum optimization algorithms.

Abstract

Quantum Annealing (QA) and the Quantum Approximate Optimization Algorithm (QAOA) are two special cases of the following control problem: apply a combination of two Hamiltonians to minimize the energy of a quantum state. Which is more effective has remained unclear. Here we analytically apply the framework of optimal control theory to show that generically, given a fixed amount of time, the optimal procedure has the pulsed (or "bang-bang") structure of QAOA at the beginning and end but can have a smooth annealing structure in between. This is in contrast to previous works which have suggested that bang-bang (i.e., QAOA) protocols are ideal. To support this theoretical work, we carry out simulations of various transverse field Ising models, demonstrating that bang-anneal-bang protocols are more common. The general features identified here provide guideposts for the nascent experimental implementations of quantum optimization algorithms.