Quantum Circuits for Matrix-Product Unitaries

Abstract

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 $T=O(N^{\alpha })$ realizing the MPU, where the constant $\alpha$ 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 $C^{\ast }$-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 $O(N^{\beta }\mathrm{poly}D)$ where $\beta \le 1+{\log }_2\sqrt{D}/s_{\min }$, with $D$ being the bond dimension and $s_{\min }$ the smallest nonzero Schmidt value of the normalized Choi state corresponding to the MPU.

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