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arXiv:2606.22816

Isometrization of Tensor Network States via Gauge Propagation

Zhiyu Jiang, Hiroshi Ueda

cond-mat.str-elphysics.comp-phquant-phSubmitted Jun 22, 2026• Announced Jun 23, 2026• 0 citations
v1 · Jun 22, 2026v2 · Aug 20, 2026
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Abstract

We introduce a gauge-propagation approach for approximately converting generic tensor-network states into an isometric tensor-network state form with a prescribed orthogonality center. In one dimension, this propagation is exact because the non-isometric factor produced by a QR or singular-value decomposition is supported on a single virtual bond. In higher-dimensional networks, however, a local step can have several outgoing directions, and the residual factor is generally not separable into independent single-bond contributions. We address this local obstruction by approximating a local tensor, or a contracted local cluster, by structured terms consisting of an isometric factor multiplied by a tensor product of output-leg factors. The isometric factor is retained at the current site or cluster, while the output-leg factors are absorbed into neighboring tensors along the propagation directions. This construction provides a local truncation criterion for gauge propagation and a practical route to refinement by increasing the number of retained terms or enlarging the local cluster. Benchmarks on random tensors and on the loop-gas tensor representation of the Kitaev spin liquid show that this refinement reduces both local residuals and accumulated propagation errors. For the loop-gas tensor, two structured terms reduce the local residual to numerical precision, and enlarging the local object from 2-in-2-out to 4-in-2-out and 6-in-2-out clusters lowers both local truncation errors and accumulated errors in finite honeycomb gauge propagation. These results identify propagation-compatible local decomposition as a useful building block for approximate isometrization and as a potential initializer or preconditioner for variational isoTNS algorithms.

Comments
14 pages, 4 figures, 1 table

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We introduce a gauge-propagation approach for approximately converting generic tensor-network states into an isometric tensor-network state form with a prescribed orthogonality center. In one dimension, this propagation is exact because the non-isometric factor produced by a QR or singular-value decomposition is supported on a single virtual bond. In higher-dimensional networks, however, a local step can have several outgoing directions, and the residual factor is generally not separable into independent single-bond contributions. We address this local obstruction by approximating a local tensor, or a contracted local cluster, by structured terms consisting of an isometric factor multiplied by a tensor product of output-leg factors. The isometric factor is retained at the current site or cluster, while the output-leg factors are absorbed into neighboring tensors along the propagation directions. This construction provides a local truncation criterion for gauge propagation and a practical route to refinement by increasing the number of retained terms or enlarging the local cluster. Benchmarks on random tensors and on the loop-gas tensor representation of the Kitaev spin liquid show that this refinement reduces both local residuals and accumulated propagation errors. For the loop-gas tensor, two structured terms reduce the local residual to numerical precision, and enlarging the local object from 2-in-2-out to 4-in-2-out and 6-in-2-out clusters lowers both local truncation errors and accumulated errors in finite honeycomb gauge propagation. These results identify propagation-compatible local decomposition as a useful building block for approximate isometrization and as a potential initializer or preconditioner for variational isoTNS algorithms.
Recently, a class of tensor networks called isometric tensor network states (isoTNS) was proposed which generalizes the canonical form of matrix product states to tensor networks in higher dimensions. While this ansatz allows for efficient numerical computations, it remained unclear which phases admit an isoTNS representation. In this work, we show that two-dimensional string-net liquids, which represent a wide variety of topological phases including discrete gauge theories, admit an exact isoTNS representation. We further show that the isometric form can be preserved after applying a finite-depth local quantum circuit. Taken together, these results show that long-range entanglement by itself is not an obstruction to isoTNS representation and suggest that all two-dimensional gapped phases with gappable edges admit an isoTNS representation.
Isometric tensor network states (isoTNS) generalize the isometric form of the one-dimensional matrix product states to tensor networks in two and higher dimensions. Here, we introduce an alternative isometric form for isoTNS by incorporating auxiliary tensors to represent the orthogonality hypersurface. We implement the time evolving block decimation algorithm on this new isometric form and benchmark the method by computing ground states and the real time evolution of the transverse field Ising model in two dimensions on large square lattices of up to 1250 sites. Our results demonstrate that isoTNS can efficiently capture the entanglement structure of two-dimensional area law states. The short-time dynamics is also accurately reproduced even at the critical point. Our isoTNS formulation further allows for a natural extension to different lattice geometries, such as the honeycomb or kagome lattice.
Although tensor networks are powerful tools for simulating low-dimensional quantum physics, tensor network algorithms are very computationally costly in higher spatial dimensions. We introduce quantum gauge networks : a different kind of tensor network ansatz for which the computation cost of simulations does not explicitly increase for larger spatial dimensions. We take inspiration from the gauge picture of quantum dynamics, which consists of a local wavefunction for each patch of space, with neighboring patches related by unitary connections. A quantum gauge network (QGN) has a similar structure, except the Hilbert space dimensions of the local wavefunctions and connections are truncated. We describe how a QGN can be obtained from a generic wavefunction or matrix product state (MPS). All 2k -point correlation functions of any wavefunction for M many operators can be encoded exactly by a QGN with bond dimension O(Mk) . In comparison, for just k=1 , an exponentially larger bond dimension of 2M/6 is generically required for an MPS of qubits. We provide a simple QGN algorithm for approximate simulations of quantum dynamics in any spatial dimension. The approximate dynamics can achieve exact energy conservation for time-independent Hamiltonians, and spatial symmetries can also be maintained exactly. We benchmark the algorithm by simulating the quantum quench of fermionic Hamiltonians in up to three spatial dimensions.
Tensor network methods, most prominently matrix product states (MPS), have become fundamental tools in modern quantum many-body physics. While MPS and extensions like the multiscale entanglement renormalization ansatz (MERA) and tree tensor networks (TTN) efficiently capture area-law entanglement and its logarithmic violations, they inherently struggle to represent highly entangled wavefunctions. Specifically, reaching the volume-law regime typically demands exponential resources within these conventional frameworks. Motivated by this challenge, we propose holographic isometric tensor network states (holographic isoTNS) that simulate quantum lattice models in DDD spatial dimensions via (D+1)(D+1)(D+1)-dimensional networks of tensors. The additional dimension substantially enlarges the representational manifold, while isometric constraints on each tensor ensure efficient contractibility. Using one-dimensional systems as testbeds, we analyze the properties of holographic isoTNS. First, we show that randomly initialized holographic isoTNS typically display volume-law entanglement at fixed bond dimension. Second, through analytic constructions and variational optimization, we demonstrate that holographic isoTNS can faithfully describe a broad class of highly entangled yet low-complexity states. In particular, the ansatz can represent arbitrary fermionic Gaussian states, Clifford states, extensions of rainbow states, and certain short-time-evolved states under local evolution. Third, to exploit this expressivity in broader contexts, we implement a time-evolving block decimation (TEBD) algorithm on holographic isoTNS. While the method remains efficient and scalable, error accumulation over TEBD sweeps suggests the need for further algorithmic improvement. Overall, holographic isoTNS broaden the scope of tensor-network methods, opening new avenues to study physics in the volume-law regime.
Tensor network states are expected to be good representations of a large class of interesting quantum many-body wave functions.In higher dimensions, their utility is however severely limited by the difficulty of contracting the tensor network, an operation needed to calculate quantum expectation values.Here we introduce a method for the time evolution of three-dimensional isometric tensor networks which respects the isometric structure and therefore renders contraction simple through a special canonical form.Our method involves a tetrahedral site splitting which allows one to move the orthogonality center of an embedded tree tensor network in a simple cubic lattice to any position.Using imaginary time evolution to find an isometric tensor network representation of the ground state of the three-dimensional transverse field Ising model across the entire phase diagram, we perform a systematic benchmark study of this method in comparison with exact Lanczos and quantum Monte Carlo results.We show that the obtained energy matches the exact ground-state result accurately deep in the ferromagnetic and polarized phases, while the regime close to the critical point requires larger bond dimensions.This behavior is in close analogy with the two-dimensional case, which we also discuss for comparison.
For normal PEPS, the fundamental theorem settles at the finite level whether virtual-bond gauge exhausts a tensor network's representation freedom. We ask the first-order question at a fixed representation, from both sides of the differential: which tensor variations leave the state unchanged without being bond gauges, and which physical directions can no variation reach? We answer both exactly for weighted graph state (WGS) tensor networks with equal physical and bond dimension q≥2q\ge2q≥2, which include the graph-state resources of measurement-based quantum computation. Dividing a single-tensor variation by the nonvanishing amplitude turns it into an arbitrary function on the closed neighborhood of its vertex. Overlapping neighborhoods give the complete kernel. Its non-bond part is generated on periodic regular-polygon tilings by triangles, chordless squares, and edge-sharing diamonds. Under stated period conditions, generic gauge completeness on all thirty-nine families studied identifies this nullity as a WGS rank drop: for thirty-three families at every q≥2q\ge2q≥2, and for six at q=2q=2q=2. Supports outside every neighborhood give the missing directions: at qubit graph states on the eleven Archimedean tilings every nearest-neighbor XXXXXX and YYYYYY direction is missing, including on honeycomb, where the non-bond kernel is empty. On large honeycomb and square tori, small nonzero coherent XXXXXX errors leave the bond-two PEPS set on the same graph. Removing redundant parameters and supplying missing directions are therefore distinct operations.