From Exponential to Quadratic: Optimal Control for a Frustrated Ising Ring Model

Abstract: Exponentially small spectral gaps are known to be the crucial bottleneck for traditional Quantum Annealing (QA) based on interpolating between two Hamiltonians, a simple driving term and the complex problem to be solved, with a linear schedule in time. One of the simplest models showing exponentially small spectral gaps was introduced by Roberts et al., PRA 101, 042317 (2020): a ferromagnetic Ising ring with a single frustrating antiferromagnetic bond. A previous study of this model (C^ot\'e et al., QST 8, 045033 (2023)) proposed a continuous-time diabatic QA, where optimized non-adiabatic annealing schedules provided good solutions, avoiding exponentially large annealing times. In our work, we move to a digital framework of Variational Quantum Algorithms, and present two main results: 1) we show that the model is digitally controllable with a scaling of resources that grows quadratically with the system size, achieving the exact solution using the Quantum Approximate Optimization Algorithm (QAOA); 2) We combine a technique of quantum control -- the Chopped RAndom Basis (CRAB) method -- and digitized quantum annealing (dQA) to construct smooth digital schedules yielding optimal solutions with very high accuracy.