Quantum Circuits for Matrix-Product Unitaries
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911351750787072 |
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| author | Styliaris, Georgios Trivedi, Rahul Cirac, J. Ignacio |
| author_facet | Styliaris, Georgios Trivedi, Rahul Cirac, J. Ignacio |
| contents | 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 $O(N^β \, \mathrm{poly}\, D)$ where $β\le 1 + \log_2 \sqrt{D}/ s_{\min}$, with $D$ being the bond dimension and $s_{\min}$ the smallest nonzero Schmidt value of the normalized Choi state corresponding to the MPU. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_08160 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum Circuits for Matrix-Product Unitaries Styliaris, Georgios Trivedi, Rahul Cirac, J. Ignacio Quantum Physics Strongly Correlated Electrons Mathematical Physics 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 $O(N^β \, \mathrm{poly}\, D)$ where $β\le 1 + \log_2 \sqrt{D}/ s_{\min}$, with $D$ being the bond dimension and $s_{\min}$ the smallest nonzero Schmidt value of the normalized Choi state corresponding to the MPU. |
| title | Quantum Circuits for Matrix-Product Unitaries |
| topic | Quantum Physics Strongly Correlated Electrons Mathematical Physics |
| url | https://arxiv.org/abs/2508.08160 |