  Classical-quantum computational complexity separations are an important
motivation for the long-term development of digital quantum computers, but
classical-quantum complexity equivalences are just as important in our present
era of noisy intermediate-scale quantum devices for framing near-term progress
towards quantum supremacy. We establish one such equivalence using a noisy
quantum circuit model that can be simulated efficiently on classical computers.
With respect to its noise model, quantum states have a robust decomposition
into a sequence of operations that each extend the state by one qubit without
spreading errors between qubits. This enables universal quantum sampling of
states with an efficient representation in this robust form and observables
with low quantum weight that can be sampled from general measurements on a few
qubits and computational basis measurements on the remaining qubits. These
robust decompositions are not unique, and we construct two distinct variants,
both of which are compatible with machine-learning methodology. They both
enable efficiently computable lower bounds on von Neumann entropy and thus can
be used as finite-temperature variational quantum Monte Carlo methods.
