  We consider the quantum-control task of obtaining a target unitary operation
via control fields that couple to the quantum system and are chosen to best
mitigate errors resulting from time-dependent noise. We allow for two sources
of noise: fluctuations in the control fields and those arising from the
environment. We address the issue of error mitigation by means of a formulation
rooted in the Martin-Siggia-Rose (MSR) approach to noisy, classical
statistical-mechanical systems. We express the noisy control problem in terms
of a path integral, and integrate out the noise to arrive at an effective,
noise-free description. We characterize the degree of success in error
mitigation via a fidelity, which characterizes the proximity of the
sought-after evolution to ones achievable in the presence of noise. Error
mitigation is then accomplished by applying the optimal control fields, i.e.,
those that maximize the fidelity subject to any constraints obeyed by the
control fields. To make connection with MSR, we reformulate the fidelity in
terms of a Schwinger-Keldysh (SK) path integral, with the added twist that the
`forward' and `backward' branches of the time-contour are inequivalent with
respect to the noise. The present approach naturally allows the incorporation
of constraints on the control fields; a useful feature in practice, given that
they feature in real experiments. We illustrate this MSR-SK approach by
considering a system consisting of a single spin $s$ freedom (with $s$
arbitrary), focusing on the case of $1/f$ noise. We discover that optimal
error-mitigation is accomplished via a universal control field protocol that is
valid for all $s$, from the qubit (i.e., $s=1/2$) case to the classical (i.e.,
$s \to \infty$) limit. In principle, this MSR-SK approach provides a framework
for addressing quantum control in the presence of noise for systems of
arbitrary complexity.
