  The quantum approximate optimization algorithm (QAOA) is a hybrid
quantum-classical algorithm that seeks to achieve approximate solutions to
optimization problems by iteratively alternating between intervals of
controlled quantum evolution. Here, we examine the effect of analog precision
errors on QAOA performance both from the perspective of algorithmic training
and canonical state- and observable-dependent QAOA-relevant metrics. Leveraging
cumulant expansions, we recast the faulty QAOA as a control problem in which
precision errors are expressed as multiplicative control noise and derive
bounds on the performance of QAOA. We show using both analytical techniques and
numerical simulations that errors in the analog implementation of QAOA circuits
hinder its performance as an optimization algorithm. In particular, we find
that any fixed precision implementation of QAOA will be subject to an
exponential degradation in performance dependent upon the number of optimal
QAOA layers and magnitude of the precision error. Despite this significant
reduction, we show that it is possible to mitigate precision errors in QAOA via
digitization of the variational parameters, therefore at the cost of increasing
circuit depth. We illustrate our results via numerical simulations and analytic
and empirical error bounds as a comparison. While focused on precision errors,
our approach naturally lends itself to more general noise scenarios and the
calculation of error bounds on QAOA performance and broader classes of
variational quantum algorithms.
