Beyond Quantum Annealing: Optimal control solutions to MaxCut problems

TL;DR

This paper develops and compares optimal control techniques for quantum algorithms solving MaxCut problems, demonstrating improved schedules over traditional linear annealing and enabling adaptation to analog quantum devices.

Contribution

Introduces Fourier and Chebyshev-based optimal control schedules for QA and QAOA, enhancing performance on hard MaxCut instances and bridging digital and analog quantum implementations.

Findings

Optimal schedules improve success probability on hard MaxCut instances.

Smooth protocols exhibit shortcuts to adiabaticity, enhancing efficiency.

Transferability of solutions enables adaptation to analog quantum devices.

Abstract

Quantum Annealing (QA) relies on mixing two Hamiltonian terms, a simple driver and a complex problem Hamiltonian, in a linear combination. The time-dependent schedule for this mixing is often taken to be linear in time: improving on this linear choice is known to be essential and has proven to be difficult. Here, we present different techniques for improving on the linear-schedule QA along two directions, conceptually distinct but leading to similar outcomes: 1) the first approach consists of constructing a Trotter-digitized QA (dQA) with schedules parameterized in terms of Fourier modes or Chebyshev polynomials, inspired by the Chopped Random Basis algorithm (CRAB) for optimal control in continuous time; 2) the second approach is technically a Quantum Approximate Optimization Algorithm (QAOA), whose solutions are found iteratively using linear interpolation or expansion in Fourier modes. Both approaches emphasize finding smooth optimal schedule parameters, ultimately leading to hybrid quantum-classical variational algorithms of the alternating Hamiltonian Ansatz type. We apply these techniques to MaxCut problems on weighted 3-regular graphs with N = 14 sites, focusing on hard instances that exhibit a small spectral gap, for which a standard linear-schedule QA performs poorly. We characterize the physics behind the optimal protocols for both the dQA and QAOA approaches, discovering shortcuts to adiabaticity-like dynamics. Furthermore, we study the transferability of such smooth solutions among hard instances of MaxCut at different circuit depths. Finally, we show that the smoothness pattern of these protocols obtained in a digital setting enables us to adapt them to continuous-time evolution, contrarily to generic non-smooth solutions. This procedure results in an optimized quantum annealing schedule that is implementable on analog devices.