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  • Open Access

Quantum Circuits for Matrix-Product Unitaries

Georgios Styliaris*, Rahul Trivedi*, and J. Ignacio Cirac

  • *These authors contributed equally to this work.
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Phys. Rev. Lett. 135, 260602 – Published 22 December, 2025

DOI: https://doi.org/10.1103/yshb-hmml

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(Nα) 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(NβpolyD) where β≤1+log2D/smin, with D being the bond dimension and smin the smallest nonzero Schmidt value of the normalized Choi state corresponding to the MPU.

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Phys. Rev. Lett. 135, 260602– Published 22 December, 2025
  • Received 27 August 2025
  • Revised 27 October 2025
  • Accepted 3 December 2025
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