  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.
