Fidelity Budgets for Discrete Regularizations of Gauge Theories: Exact Single-Link Computations, Scaling Laws, and the Limits of Heuristic Bounds

In the paper's words

Discrete regularizations of gauge theories — finite-dimensional link Hilbert spaces in quantum link models, truncated electric bases on quantum registers, and tensor networks on Bruhat–Tits trees — all face the same structural question: when does the discrete object faithfully encode its continuum or infinite-dimensional target? We develop a unified fidelity-budget framework that decomposes a target accuracy ε = 10^-4 into per-component truncation errors with explicit scaling laws. As a fully worked case study, we compute exactly, with all arithmetic shown, the truncation infidelity and energy bias of a single thermal U(1) link at dimensionless inverse temperature β g² = 0.5: the infidelity falls from 1.150 × 10^-2 at n_max=3 to 7.242 × 10^-5 at n_max=5 and 6.44 × 10^-8 at n_max=7, while the register cost grows only from 3 to 4 qubits.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. As a fully worked case study, we compute exactly, with all arithmetic shown, the truncation infidelity and energy bias of a single thermal $\mathrm{U}(1)$ link at dimensionless inverse temperature $\beta g^2 = 0.5$: the infidelity falls from $1.150 \times 10^{-2}$ at $n_{\max}=3$ to $7.242 \times 10^{-5}$ at $n_{\max}=5$ and $6.44 \times 10^{-8}$ at $n_{\max}=7$, while the register cost grows only from 3 to 4 qubits.

   from Abstract

2. We develop a unified fidelity-budget framework that decomposes a target accuracy $\epsilon = 10^{-4}$ into per-component truncation errors with explicit scaling laws.

   from Abstract

3. In a different corner of the literature, holographic quantum error correction has been organized as renormalization-group (RG) flow on the Bruhat–Tits tree $\mathcal{T}_p$ — the $p$-adic analogue of anti-de Sitter space — with a conditional threshold analysis at accuracy $10^{-4}$ [9], building on the identification of the tree as a holographic geometry [11].

   from 1. Introduction

4. Scaling laws and a tree application (Sections 4.3–4.5): the logarithmic cost of multi-link propagation, the Gaussian-versus-Chebyshev gap, and Bruhat–Tits tree geometry at $p=2$, $L=10$.

   from 1. Introduction

5. Our fidelity budget supplies the quantitative instrument such analyses need — a per-link error tolerance derived from a total budget, so that "reaching the limit" becomes a checkable inequality rather than a qualitative comparison. [2] Observation of genuine $2+1$D string dynamics in a U(1) lattice gauge theory with a tunable plaquette term on a trapped-ion quantum computer (arXiv:2604.07436v1).

   from 2. Background and Related Work

6. Our budget in Section 4.4 treats the 4-link plaquette as the coherence multiplier for magnetic-sector errors. [3] Truncation uncertainties for accurate quantum simulations of lattice gauge theories (arXiv:2508.00061v4).

   from 2. Background and Related Work

How it is done

1. As a fully worked case study, we compute exactly, with all arithmetic shown, the truncation infidelity and energy bias of a single thermal $\mathrm{U}(1)$ link at dimensionless inverse temperature $\beta g^2 = 0.5$: the infidelity falls from $1.150 \times 10^{-2}$ at $n_{\max}=3$ to $7.242 \times 10^{-5}$ at $n_{\max}=5$ and $6.44 \times 10^{-8}$ at $n_{\max}=7$, while the register cost grows only from 3 to 4 qubits.

   from Abstract

2. A gauge theory is not directly what an experiment builds.

   from 1. Introduction

3. This experiment demonstrates string dynamics relevant to hadronization in $2+1$ dimensions, where a plaquette term endows the gauge field with dynamics and enables photon-like propagation.

   from 2. Background and Related Work

4. We define the fidelity budget as the constraint $$ \Delta_R(T;\mathcal{O}) \;=\; \sup_{O \in \mathcal{O}} \big| \langle O \rangle_R - \langle O \rangle_T \big| \;\le\; \epsilon, $$ where $\epsilon$ is the target accuracy.

   from 3.1 The fidelity budget

What would falsify it

1. We adopt the same target accuracy and, in Section 4.5, compute the tree-geometry quantities (boundary vertex counts, edge counts, qubit-equivalent costs) that such a threshold analysis requires as inputs. [10] The Trapped-Ion Ultrametric Testbed: A Falsifiability Register for Testing $p$-Adic Structure in Quantum Dynamics (https://papers.qnfo.org/papers/trapped-ion-ultrametric-synthesis/).

   from 2. Background and Related Work

2. It supplies the experimental side of our framework: the fidelity budget is exactly the kind of pre-registered, auditable numerical structure that a falsifiability register needs, and the platform overlap with [2] makes a combined test realistic. [11] Holographic QEC as AdS/CFT RG Flow on Bruhat–Tits Trees (DOI pending).

   from 2. Background and Related Work

3. QNFO: The Trapped-Ion Ultrametric Testbed: A Falsifiability Register for Testing p-Adic Structure in Quantum Dynamics. [11] DOI pending.

   from References

Where the claim stops

1. We document, in an explicit divergence appendix, why a competing heuristic bound proposed in an independent draft fails algebraic scrutiny and was rejected.

   from Abstract

2. A formalism for estimating truncation uncertainties against the Kogut–Susskind limit, leveraging Hilbert-space fragmentation to bound the excitation of large electric fields, has been developed precisely to make these errors quantitative [3].

   from 1. Introduction

3. The three drafts agreed on the framework's structure but diverged on the central quantitative object: one draft proposed a heuristic closed-form bound $\epsilon_{S,a} \le \frac{g^2 a^2}{2S+1}$ for spin-$S$ quantum link truncations; the other two performed exact or model-based computations whose assumptions are fully stated.

   from 1. Introduction

Where to read

1. Abstract155 words · 1 min
2. 1. Introduction529 words · 2 min
3. 2. Background and Related Work1,040 words · 5 min
4. 3. Methods568 words · 2 min
5. 4. Analysis274 words · 1 min