Beyond quantum annealing: optimal control solutions to maxcut problems

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 pa-
rameters, 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 dy-
namics. 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.