Efficient conversion from MPO/MPS descriptions to shallow channel circuits

Develop an efficient algorithm that, given either (i) a one-dimensional mixed quantum state specified by a matrix product operator (MPO) representation with constant bond dimension or (ii) a one-dimensional quantum state with constant fidelity to a matrix product state (MPS) whose bond dimension scales polynomially with system size, constructs an explicit shallow local channel circuit that generates the state.

Background

In one dimension, mixed quantum states that admit matrix product operator (MPO) representations with constant bond dimension can be efficiently reconstructed from local measurements, and there are algorithms to find MPS approximations when a state has constant fidelity with an MPS of polynomial bond dimension.

However, having an MPO/MPS description does not directly yield a constructive, shallow local channel circuit for state generation. The paper points out that bridging this gap—turning such compact tensor-network or approximation descriptions into explicit shallow channel circuits—remains unresolved.