  Using quantum devices supported by classical computational resources is a
promising approach to quantum-enabled computation. One example of such a hybrid
quantum-classical approach is the variational quantum eigensolver (VQE) built
to utilize quantum resources for the solution of eigenvalue problems and
optimizations with minimal coherence time requirements by leveraging classical
computational resources. These algorithms have been placed among the candidates
for first to achieve supremacy over classical computation. Here, we provide
evidence for the conjecture that variational approaches can automatically
suppress even non-systematic decoherence errors by introducing an exactly
solvable channel model of variational state preparation. Moreover, we show how
variational quantum-classical approaches fit in a more general hierarchy of
measurement and classical computation that allows one to obtain increasingly
accurate solutions with additional classical resources. We demonstrate
numerically on a sample electronic system that this method both allows for the
accurate determination of excited electronic states as well as reduces the
impact of decoherence, without using any additional quantum coherence time or
formal error correction codes.
