Beyond Bond Gauge: Exact Tensor-Network Tangent Spaces at Weighted Graph States
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≥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≥2, 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.
Related
- Tensor-network approach to quantum optical state evolution beyond the Fock basis
- Leg-Tied Tensor Network States: Entanglement Beyond Virtual Bonds
- Excitation spectra and rank tomography of finite MPS tangent spaces
- Universal fusion category symmetries on tensor products of infinite-dimensional Hilbert spaces
- Approximating Local Properties by Tensor Network States With Constant Bond Dimension
BibTeX
@misc{cheng2026beyond,
title = {{Beyond Bond Gauge: Exact Tensor-Network Tangent Spaces at Weighted Graph States}},
author = {Song Cheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30066},
eprint = {2609.30066},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30066},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Beyond Bond Gauge: Exact Tensor-Network Tangent Spaces at Weighted Graph States
AU - Song Cheng
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30066
UR - https://arxiv.org/abs/2609.30066
AB - 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≥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≥2, 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.
ER -