Robust decompositions of quantum states

Abstract: 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.