Quantum Circuit Complexity of Matrix-Product Unitaries

• URL: http://arxiv.org/abs/2508.08160v1
• Date: Mon, 11 Aug 2025 16:37:14 GMT
• Title: Quantum Circuit Complexity of Matrix-Product Unitaries
• Authors: Georgios Styliaris, Rahul Trivedi, J. Ignacio Cirac,
• Abstract summary: Matrix-product unitaries (MPUs) are unitary operators that preserve entanglement area law in 1D systems.<n>We show that a large class of MPUs can be implemented with a quantum circuit.
• Score: 0.3277163122167433
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 paper, 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 $s_{\min}$ is the smallest nonzero Schmidt value of the normalized Choi state corresponding to the MPU.

Related papers

• Matrix-product entanglement characterizing the optimality of state-preparation quantum circuits [1.9443029989289298]
  We present a class of multipartite entanglement measures that incorporate the matrix product state representation.<n>These measures are dubbed as $chi$-specified matrix product entanglement ($chi$-MPE)<n>Our results establish tensor networks as a powerful and general tool for developing parametrized measures of multipartite entanglement.
  arXiv  Detail & Related papers  (2025-07-08T13:49:53Z)
• Scalable projected entangled-pair state representation of random quantum circuit states [0.0]
  We show the update of projected entangled-pair states (PEPSs) in the Vidal gauge that represent random quantum circuit states.<n>We find the universal scaling behaviors of the state fidelity by treating large-scale circuits ofn leq 104$, using $chi leq 128$ on a conventional CPU.
  arXiv  Detail & Related papers  (2025-04-07T06:47:48Z)
• Matrix-product unitaries: Beyond quantum cellular automata [0.5499796332553706]
  Matrix-product unitaries (MPU) are 1D tensor networks describing time evolution and unitary symmetries of quantum systems.<n>We make the first steps towards a theory of MPU with uniform bulk but arbitrary boundary.
  arXiv  Detail & Related papers  (2024-06-14T17:29:04Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• The Power of Unentangled Quantum Proofs with Non-negative Amplitudes [55.90795112399611]
  We study the power of unentangled quantum proofs with non-negative amplitudes, a class which we denote $textQMA+(2)$. In particular, we design global protocols for small set expansion, unique games, and PCP verification. We show that QMA(2) is equal to $textQMA+(2)$ provided the gap of the latter is a sufficiently large constant.
  arXiv  Detail & Related papers  (2024-02-29T01:35:46Z)
• On sampling determinantal and Pfaffian point processes on a quantum computer [49.1574468325115]
  DPPs were introduced by Macchi as a model in quantum optics the 1970s. Most applications require sampling from a DPP, and given their quantum origin, it is natural to wonder whether sampling a DPP on a classical computer is easier than on a classical one. Vanilla sampling consists in two steps, of respective costs $mathcalO(N3)$ and $mathcalO(Nr2)$ operations on a classical computer, where $r$ is the rank of the kernel matrix.
  arXiv  Detail & Related papers  (2023-05-25T08:43:11Z)
• Quantum and classical low-degree learning via a dimension-free Remez inequality [52.12931955662553]
  We show a new way to relate functions on the hypergrid to their harmonic extensions over the polytorus. We show the supremum of a function $f$ over products of the cyclic group $exp(2pi i k/K)_k=1K$. We extend to new spaces a recent line of work citeEI22, CHP, VZ22 that gave similarly efficient methods for learning low-degrees on hypercubes and observables on qubits.
  arXiv  Detail & Related papers  (2023-01-04T04:15:40Z)
• Average-Case Complexity of Tensor Decomposition for Low-Degree Polynomials [93.59919600451487]
  "Statistical-computational gaps" occur in many statistical inference tasks. We consider a model for random order-3 decomposition where one component is slightly larger in norm than the rest. We show that tensor entries can accurately estimate the largest component when $ll n3/2$ but fail to do so when $rgg n3/2$.
  arXiv  Detail & Related papers  (2022-11-10T00:40:37Z)
• Density Matrix Renormalization Group with Tensor Processing Units [0.0]
  Google's Processing Units (TPUs) are integrated circuits specifically built to accelerate and scale up machine learning workloads. In this work we demonstrate the use of TPUs for accelerating and scaling up the density matrix renormalization group (DMRG), a powerful numerical approach to compute the ground state of a local quantum many-body Hamiltonian.
  arXiv  Detail & Related papers  (2022-04-12T10:40:14Z)
• Small Covers for Near-Zero Sets of Polynomials and Learning Latent Variable Models [56.98280399449707]
  We show that there exists an $epsilon$-cover for $S$ of cardinality $M = (k/epsilon)O_d(k1/d)$. Building on our structural result, we obtain significantly improved learning algorithms for several fundamental high-dimensional probabilistic models hidden variables.
  arXiv  Detail & Related papers  (2020-12-14T18:14:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.