Zhi-Yuan Wei

We consider the preparation of matrix product states (MPS) on quantum devices via quantum circuits of local gates. We first prove that faithfully preparing translation-invariant normal (i.e., short-range correlated) MPS of N sites requires a circuit depth T = Ω(log N). We then introduce an algorithm based on the renormalization-group transformation to prepare normal MPS with an error ε in depth T = O[log(N/ε)]. The algorithm has thus the fastest scaling possible for this class of states. We also show that measurement and feedback leads to an exponential speedup of the algorithm to T = O[log log(N/ε)]. Measurements also allow one to prepare arbitrary translation-invariant MPS, including long-range non-normal ones, in the same depth. Finally, the algorithm naturally extends to inhomogeneous MPS.