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 $q\ge2$, 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.

Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el)

arXiv:2609.30066 [quant-ph]

(or arXiv:2609.30066v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.30066 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.