Dissipative preparation of injective tensor network states

The preparation of tensor network states is a fundamental prerequisite for a wide range of quantum simulation tasks. While many unitary protocols for preparing these states have been investigated, dissipative state preparation provides a powerful alternative since it can be robust to noise and initialization errors. In this paper, we construct both continuous-time and discrete-time geometrically local dissipative processes whose unique steady state is a given injective tensor network state. Our method prepares all injective matrix product states on N sites to an error varepsilon in O\(log (N/varepsilon\)) time, yielding an exponential improvement over previously known dissipative preparation schemes. For two and higher-dimensional tensor network states, we prove that when the tensors of the state are highly injective, the constructed dissipative processes are rapid-mixing i.e., they prepare a state varepsilon-close to the N-site target state in O\(log (N/varepsilon\)) time. For these states, our approach provides a polynomial speedup over known unitary methods for states defined on lattices and an exponential speedup for states on general bounded-degree graphs. We corroborate our theoretical results with numerical studies that indicate that the dissipative protocol can rapidly prepares states outside the high-injectivity assumption.