Quantum Circuits for Matrix-Product Unitaries

Matrix-product unitaries (MPUs) are many-body unitary operators that, as a consequence of their tensor-network structure, preserve the entanglement area law in 1D systems. However, it is unknown how to implement an MPU as a quantum circuit since the individual tensors describing the MPU are not unitary. In this Letter, we show that a large class of MPUs can be implemented with a polynomial-depth quantum circuit. For an N-site MPU built from a repeated bulk tensor with open boundary, we explicitly construct a quantum circuit of polynomial depth T=O(N^{α}) realizing the MPU, where the constant α depends only on the bulk and boundary tensor and not the system size N. We show that this class includes nontrivial unitaries that generate long-range entanglement and, in particular, contains a large class of unitaries constructed from representations of C^{*}-weak Hopf algebras. Furthermore, we also adapt our construction to nonuniform translationally varying MPUs and show that they can be implemented by a circuit of depth O(N^{β}polyD) where β≤1+log_{2}sqrt[D]/s_{min}, with D being the bond dimension and s_{min} the smallest nonzero Schmidt value of the normalized Choi state corresponding to the MPU.