Optimal Protocols in Quantum Annealing and Quantum Approximate Optimization Algorithm Problems

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.

Automated summary

Quantum annealing (QA) and quantum approximate optimization algorithm (QAOA) are two special cases for controlling quantum state energy minimization. Analytical application of optimal control theories showed that optimal procedures have a mix of pulsed (QAOA-like) structure at beginning and end, and smooth annealing structure in middle. Simulation results on Ising models support the theory, suggesting bang-anneal-bang protocols are more common than previously thought and provide guidelines for experimental implementations.