Quantum Circuits for Matrix-Product Unitaries

TL;DR

This paper demonstrates how a broad class of matrix-product unitaries, which are important in quantum many-body physics, can be efficiently implemented as polynomial-depth quantum circuits, enabling practical realization of complex quantum operations.

Contribution

The authors explicitly construct polynomial-depth quantum circuits for a large class of MPUs, including nontrivial unitaries with long-range entanglement, extending the understanding of their implementability.

Findings

Polynomial-depth circuits for MPUs with open boundary conditions.

Inclusion of unitaries from representations of C*-weak Hopf algebras.

Extension to nonuniform, translationally-varying MPUs with controlled circuit depth.

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 $C^*$-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.