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arXiv:2609.30066

Beyond Bond Gauge: Exact Tensor-Network Tangent Spaces at Weighted Graph States

Song Cheng

quant-phcond-mat.str-elSubmitted Sep 24, 2026• Announced Sep 25, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $q\ge2$, 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 $q\ge2$, and for six at $q=2$. Supports outside every neighborhood give the missing directions: at qubit graph states on the eleven Archimedean tilings every nearest-neighbor $XX$ and $YY$ direction is missing, including on honeycomb, where the non-bond kernel is empty. On large honeycomb and square tori, small nonzero coherent $XX$ errors leave the bond-two PEPS set on the same graph. Removing redundant parameters and supplying missing directions are therefore distinct operations.

Comments
84 pages including appendices, 15 figures

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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.
Abstract Projected entangled pair states (PEPS) are very useful in the description of strongly correlated systems, partly because they allow encoding symmetries, either global or local (gauge), naturally. In recent years, PEPS with local symmetries have increasingly been used in the study of non-perturbative regimes of lattice gauge theories, most prominently as a way to construct variational ansatz states depending only on a small number of parameters and yet capturing the relevant physical properties. For the case of one-dimensional PEPS (matrix product states—MPS) a bidirectional connection was established between the internal structure of the tensor network, i.e. the mathematical properties of the constituent tensors, and the symmetry. In higher dimensions this has only been done for global symmetries, where in the local (gauge) case it is known only how to construct gauge-invariant states, but not what the symmetry implies on the internal structure of the PEPS. In the present work we complete this missing piece and study the internal structure of PEPS with a gauge symmetry. The PEPS we consider consist of matter and gauge field tensors placed on the vertices and edges, respectively, of arbitrary graphs.
We propose a single-layer tensor network framework for the variational determination of ground states in two-dimensional quantum lattice models. By combining the nested tensor network method [Phys. Rev. B 96, 045128 (2017)] with the automatic differentiation technique, our approach can reduce the computational cost by three orders of magnitude in bond dimension, and therefore enables highly efficient variational ground-state calculations. We demonstrate the capability of this framework through two quantum spin models: the antiferromagnetic Heisenberg model on a square lattice and the frustrated Shastry-Sutherland model. Even without GPU acceleration or symmetry implementation, we have achieved a bond dimension of nine and obtained accurate ground-state energy and consistent order parameters compared to prior studies. In particular, we confirm the existence of an intermediate empty-plaquette valence bond solid ground state in the Shastry-Sutherland model. We have further discussed the convergence of the algorithm and its potential improvements. Our work provides a promising route for large-scale tensor network calculations of two-dimensional quantum systems.
We introduce a gauge-propagation approach for approximately converting generic tensor-network states into an isometric tensor-network 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 applies to general local input-output partitions for which the input-side dimension is no smaller than the output-side dimension and provides a local truncation criterion for gauge propagation. Benchmarks on random tensors show that the proposed decomposition is effective in the low-term regime and that the residual decreases for larger local clusters. For the loop-gas tensor representation of the Kitaev spin liquid, two structured terms reduce the local residual to numerical precision, and the same cluster refinement further lowers the leading-term truncation error and reduces error accumulation during gauge propagation on a finite honeycomb network. These results identify a propagation-compatible local decomposition as a useful building block for approximate isometrization and as a potential initializer or preconditioner for variational isoTNS algorithms.
Tensor networks have a gauge degree of freedom on the virtual degrees of freedom that are contracted. A canonical form is a choice of fixing this degree of freedom. For matrix product states, choosing a canonical form is a powerful tool, both for theoretical and numerical purposes. On the other hand, for tensor networks in dimension two or greater there is only limited understanding of the gauge symmetry. Here we introduce a new canonical form, the minimal canonical form, which applies to projected entangled pair states (PEPS) in any dimension, and prove a corresponding fundamental theorem. Already for matrix product states this gives a new canonical form, while in higher dimensions it is the first rigorous definition of a canonical form valid for any choice of tensor. We show that two tensors have the same minimal canonical forms if and only if they are gauge equivalent up to taking limits; moreover, this is the case if and only if they give the same quantum state for any geometry. In particular, this implies that the latter problem is decidable – in contrast to the well-known undecidability for equality of PEPS on grids. We also provide rigorous algorithms for computing minimal canonical forms. To achieve this we draw on geometric invariant theory and recent progress in theoretical computer science in non-commutative group optimization.
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.
We study the conditions under which Matrix Product States (MPS) or Matrix Product Operators are exact eigenvectors of an extensive local operator, such as a Hamiltonian. By suitably choosing the local operator, this covers a wide range of settings: Exact eigenstates of Hamiltonians, including scar states, exact MPS trajectories for driven quantum systems, steady states of local Lindbladians, generalized symmetries of either Hamiltonians or density matrices, and many more. Our key result is that a local, fixed-size equation---namely, how a single term in the operator acts on a block of tensors---provides a necessary and sufficient condition for exact solutions. This local characterization allows us to obtain the full space of MPS solutions to all of the aforementioned problems; in particular, we exemplify the power of our results by recovering the quantum group symmetries of the XXZ model. We also discuss applications to numerical algorithms with MPS and the generalization of our results to 2D, i.e., projected entangled pair states (PEPS).