Preparation of matrix product states with log-depth quantum circuits

Abstract

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 MPS of $N$ sites requires a circuit depth $T=\Omega (\log N)$. We then introduce an algorithm based on the renormalization-group transformation to prepare normal MPS with an error $ϵ$ in depth $T=O(\log (N/ϵ))$, which is optimal. 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.