A physics-informed quantum-computing workflow has resolved low-lying closed-flux excitations in a finite Z2 lattice model and tracked the changing spatial profile of its lightest state. In the study’s normalized radius measure, the profile grew from approximately 1.06 at x = 0.8 to approximately 2.04 at x = 0.4. Here, glueball-like is an operational label for localized closed-flux excitations in an Abelian model, not an identification with the non-Abelian glueballs of quantum chromodynamics.
Building the gauge vacuum
The calculation used a two-stage vacuum preparation. First, a gauge-invariant loop-gas manifold supplied the initial structure; then a Hamiltonian variational ansatz, or HVA, added correlations while keeping the state in the Gauss-law sector.
At x = 0.2, the reported energy and fidelity both improved rapidly as HVA layers were added. In the study, this result serves as a vacuum-preparation benchmark.
Finding excitations in a reduced space
To construct the excitations, the workflow used Wilson-loop quantum subspace expansion, or QSE. Physically interpretable loop operators were applied to the variational vacuum, and the Hamiltonian, symmetry operators and local loop observables were projected into a nonorthogonal excitation manifold.
On the 6 × 4 periodic lattice at x = 0.6, a Wilson-loop basis with loop area no larger than two plaquettes resolved four low-lying states. The reported energies were E0 = -19.2750, E1 = -12.9122, and a closely spaced pair at E2,3 = -12.8997; the corresponding gaps were 6.3628 and 6.3753. Energy variance vanished within the reported numerical precision.
The reported check was vanishing energy variance within numerical precision, and no statistical uncertainty interval was given for the spectrum.
Borrowing information across couplings
Eigenvector continuation, or EC, supplied a second way to build the subspace. Nearby ground states encode parameter-dependent dressing, while explicit Wilson-loop operators provide excitation directions; the two are treated as complementary basis resources.
At target x = 0.4, the combined EC-and-loop basis produced substantially smaller excitation errors than vacuum-only EC in the reported comparison. With an enlarged loop set and larger training sets, the first excitation reached the 10^-3 scale and the second reached a few 10^-3.
No statistical uncertainty intervals or replication estimates were reported for this comparison.
A spatial footprint with a finite-size warning
The spatial result comes with an important qualification. The BS radius is an operator-dependent measure of the spatial profile of a chosen glueball interpolating field, rather than a unique mass radius.
On the 6 × 4 lattice, the exact normalized lightest-state radius was approximately 1.06 at x = 0.8 and 2.04 at x = 0.4. At x = 0.4, loop-only cutoffs labeled 1p, 2p and 3p gave approximately 1.87, 1.94 and 1.99, while EC with C = 3 gave approximately 2.03 without an explicit loop operator.
Toward x = 0.3, the exact radius approached a plateau near 2.11 r0. The paper identifies that behavior as finite-size spatial extension on the 6 × 4 torus, rather than saturation of a thermodynamic radius.
The radius values were reported as approximate and without statistical error bars.
A quench changes the loop pattern
A separate quench calculation started from the electric vacuum at x0 = 1 and compared target couplings x1 = 0.1 and x1 = 0.6. The x1 = 0.1 case showed substantially more loop weight, including larger motifs, while x1 = 0.6 remained electric-field dominated and suppressed extended loops.
Those motif counts are not exact particle numbers or spectral production probabilities, and the quench is not a model of a high-energy hadronic collision.
The boundary of the result
The authors’ claimed polynomial-dimensional regime depends on support only on loops up to qmax = O(1), plus a modest EC training set spanning smooth parameter dependence. The construction can fail near criticality or for highly excited states when the needed loop size grows with the system.
That boundary matters for how the result should be read: it is evidence for an algorithmic workflow in a finite Abelian Z2 model, not a demonstration of non-Abelian QCD glueballs. Larger-lattice validation is needed to separate finite-size behavior from thermodynamic spatial growth.
The document carries the arXiv version-1 identifier arXiv:2608.25696v1. It reports support from the National Natural Science Foundation of China and the Guangdong Provincial Quantum Science Strategic Initiative.
Paper data and sources
Original title: Physics-informed quantum algorithms for glueball-like excitations in a $\mathbb{Z}_2$ lattice gauge theory
Authors: Dan-Bo Zhang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text