  The computational power of contemporary quantum processors is limited by
hardware errors that cause computations to fail. In principle, each quantum
processor's computational capabilities can be described with a capability
function that quantifies how well a processor can run each possible quantum
circuit (i.e., program), as a map from circuits to the processor's success
rates on those circuits. However, capability functions are typically unknown
and challenging to model, as the particular errors afflicting a specific
quantum processor are a priori unknown and difficult to completely
characterize. In this work, we investigate using artificial neural networks to
learn an approximation to a processor's capability function. We explore how to
define the capability function, and we explain how data for training neural
networks can be efficiently obtained for a capability function defined using
process fidelity. We then investigate using convolutional neural networks to
model a quantum computer's capability. Using simulations, we show that
convolutional neural networks can accurately model a processor's capability
when that processor experiences gate-dependent, time-dependent, and
context-dependent stochastic errors. We then discuss some challenges to
creating useful neural network capability models for experimental processors,
such as generalizing beyond training distributions and modelling the effects of
coherent errors. Lastly, we apply our neural networks to model the capabilities
of cloud-access quantum computing systems, obtaining moderate prediction
accuracy (average absolute error around 2-5%).
