  Analog quantum algorithms are formulated in terms of Hamiltonians rather than
unitary gates and include quantum adiabatic computing, quantum annealing, and
the quantum approximate optimization algorithm (QAOA). These algorithms are
promising candidates for near-term quantum applications, but they often require
fine tuning via the annealing schedule or variational parameters. In this work,
we explore connections between these analog algorithms, as well as limits in
which they become approximations of the optimal procedure.Notably, we explore
how the optimal procedure approaches a smooth adiabatic procedure but with a
superposed oscillatory pattern that can be explained in terms of the
interactions between the ground state and first excited state that effect the
coherent error cancellation of diabatic transitions. Furthermore, we provide
numeric and analytic evidence that QAOA emulates this optimal procedure with
the length of each QAOA layer equal to the period of the oscillatory pattern.
Additionally, the ratios of the QAOA bangs are determined by the smooth,
non-oscillatory part of the optimal procedure. We provide arguments for these
phenomena in terms of the product formula expansion of the optimal procedure.
With these arguments, we conclude that different analog algorithms can emulate
the optimal protocol under different limits and approximations. Finally, we
present a new algorithm for better approximating the optimal protocol using the
analytic and numeric insights from the rest of the paper. In practice,
numerically, we find that this algorithm outperforms standard QAOA and naive
quantum annealing procedures.
