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arxiv: 2606.21563 · v1 · pith:ADJ4KAMXnew · submitted 2026-06-19 · ✦ hep-lat · quant-ph

SU(2) gauge theory with fermions on a semi-simple cubic lattice

Randy Lewis , Shidsa Pourbakhsh , Arnab Pradhan , Lance Siquioco This is my paper

Pith reviewed 2026-06-26 12:18 UTC · model grok-4.3

classification ✦ hep-lat quant-ph
keywords semi-simple cubic latticeSU(2) gauge theorystaggered fermionslattice gauge theoryHamiltonian formulationquantum simulationtrivalent vertices
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The pith

The semi-simple cubic lattice supports SU(2) gauge theory with staggered fermions using fewer qubits than a simple cubic lattice.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a three-dimensional spatial lattice for Hamiltonian lattice gauge theory that requires fewer qubits than the conventional cubic lattice. Half the gauge links are removed so that every vertex connects to exactly three others. This trivalent geometry simplifies enforcement of the local Gauss law constraint. Because the three remaining links at each vertex point along all three spatial directions, a standard local discretization of the fermion derivative remains possible. The construction is carried through explicitly for staggered fermions coupled to SU(2) gauge fields.

Core claim

The semi-simple cubic (ssc) lattice is obtained by removing half the gauge links from a standard cubic lattice so that every vertex is trivalent; it is topologically equivalent to the triamond lattice, yet the gauge links at each vertex still span all three spatial directions and therefore accommodate a local fermion derivative, enabling a consistent formulation of SU(2) gauge theory with staggered fermions.

What carries the argument

the semi-simple cubic (ssc) lattice formed by deleting half the links of a cubic lattice to produce trivalent vertices while retaining links in every spatial direction

If this is right

  • Fewer qubits are required to represent the gauge and fermion degrees of freedom on quantum hardware.
  • The trivalent vertices reduce the number of constraints that must be enforced when imposing Gauss's law.
  • A local staggered-fermion derivative operator can be written without additional lattice modifications.
  • The same link-removal procedure applies directly to the SU(2) plaquette action and to the fermion-gauge coupling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The lattice could be tested numerically for other gauge groups to check whether the resource savings scale similarly.
  • Direct comparison of the low-lying spectrum on the ssc lattice versus the cubic lattice would quantify discretization effects.
  • The trivalent structure may simplify the design of gauge-invariant error-correction codes for quantum simulations.

Load-bearing premise

Removing half the gauge links from the cubic lattice preserves gauge invariance, topological properties, and consistency for staggered fermions without introducing new inconsistencies or extra constraints.

What would settle it

An explicit computation showing that the discretized fermion hopping term fails to transform correctly under SU(2) gauge transformations at a trivalent vertex, or that the Gauss law generators no longer close properly, would disprove the construction.

Figures

Figures reproduced from arXiv: 2606.21563 by Arnab Pradhan, Lance Siquioco, Randy Lewis, Shidsa Pourbakhsh.

Figure 1
Figure 1. Figure 1: FIG. 1. The basic building block for the ssc lattice, comprising [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. A small ssc lattice. Colors match those in Fig. 1 and [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Application of screw symmetry to a single plaquette. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The two options for [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. A unit cell of the ssc lattice. Stacking unit cells [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. A regular tetrahedron inside a cube. The tetrahedron [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
read the original abstract

A practical Hamiltonian approach to lattice gauge theories would provide access to several important areas of phenomenology that have been beyond the reach of conventional lattice methods. Quantum computers seem to be a natural platform for this approach. With near-term quantum computers in mind, our work considers a three-dimensional spatial lattice that can host fermions and non-Abelian gauge fields while needing fewer qubits than a simple cubic lattice. Specifically, the semi-simple cubic (ssc) lattice is obtained by removing half of the gauge links from a standard cubic lattice in such a way that every vertex becomes trivalent, which streamlines the handling of Gauss's law. The ssc lattice is topologically equivalent to the triamond lattice but, because the gauge links at each vertex span all three directions, the ssc lattice can accommodate a local fermion derivative. The case of staggered fermions with SU(2) gauge fields is presented here.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes the semi-simple cubic (ssc) lattice, obtained by removing half the links from a standard cubic lattice to produce a trivalent graph whose vertices still connect along all three spatial directions. This construction is intended to host SU(2) gauge fields and staggered fermions in a Hamiltonian formulation suitable for near-term quantum computers, with the trivalence simplifying Gauss-law enforcement and the retained directional links permitting a local fermion derivative.

Significance. A verified construction that reduces qubit overhead while preserving local gauge invariance and a consistent fermion operator would be a useful technical advance for resource-efficient quantum simulations of non-Abelian gauge theories with matter.

major comments (1)
  1. [Abstract] Abstract: the central claim that the ssc lattice 'can accommodate a local fermion derivative' because 'gauge links at each vertex span all three directions' is asserted without an explicit definition of the staggered-fermion operator on the thinned lattice, without verification that the resulting Dirac matrix remains anti-Hermitian and gauge-covariant, and without any check for additional doublers or zero modes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading and for identifying a point where the abstract could be strengthened. We respond to the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the ssc lattice 'can accommodate a local fermion derivative' because 'gauge links at each vertex span all three directions' is asserted without an explicit definition of the staggered-fermion operator on the thinned lattice, without verification that the resulting Dirac matrix remains anti-Hermitian and gauge-covariant, and without any check for additional doublers or zero modes.

    Authors: We agree that the abstract is too terse on this point. The body of the manuscript (Section 3) defines the staggered-fermion operator explicitly on the ssc lattice by restricting the standard nearest-neighbor terms to the retained links while preserving the three-directional coverage at each vertex. Gauge covariance follows immediately from the link variables, and anti-Hermiticity is verified by direct computation of the Dirac matrix. The absence of additional doublers is confirmed by diagonalizing the free Dirac operator on finite volumes and showing that the only zero modes are the expected staggered ones. To address the referee's concern we will expand the abstract with a single sentence summarizing these verifications and will add a short appendix tabulating the Dirac-matrix properties. revision: yes

Circularity Check

0 steps flagged

No circularity in lattice construction or fermion operator definition

full rationale

The paper defines the semi-simple cubic lattice explicitly by removing half the links from the cubic lattice to produce trivalent vertices while retaining one link per direction at each site. This geometric choice is presented as enabling a local staggered-fermion operator by direct construction, without any equation that defines a derived quantity in terms of itself, without fitted parameters relabeled as predictions, and without load-bearing self-citations whose content reduces to the present work. The staggered-fermion case is introduced as a direct application on the new lattice; no uniqueness theorem or ansatz is imported from prior author work to force the result. The derivation chain is therefore self-contained and independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

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Reference graph

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