Adelic Quantum Error Correction: Code Constructions Beyond the Stabilizer Formalism

In the paper's words

Adelic generalization of Heydeman holographic QEC. v0.1.1 adds professionally rendered PDF.

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

Numbers it reports

No sentence of this paper reports a figure with a unit or a quantity word, so none is shown.

How it is done

1. We define adelic quantum error-correcting codes as codes constructed on the adelic product of Bruhat-Tits trees $\prod_p \mathrm{BT}_p$, and we ask: can such codes exhibit properties NOT accessible through the standard CSS/stabilizer formalism?

   from Abstract

2. The ring of adeles $\mathbb{A}_{\mathbb{Q}}$ is the restricted product of all $p$-adic completions $\mathbb{Q}_p$ together with the real completion $\mathbb{R}$: $$\mathbb{A}_{\mathbb{Q}} = \left\{(x_p) \in \prod_p \mathbb{Q}_p : x_p \in \mathbb{Z}_p \text{ for all but finitely many } p\right\}$$ The adelic approach has been powerful in number theory — it is the natural setting for class field theory, automorphic forms, and the Langlands program [established].

   from 1.2 The Adelic Idea

3. Section 5 identifies proof obligations.

   from 1.4 Structure of This Paper

4. The burden of proof is to show that the adelic constraint breaks the stabilizer structure.

   from 4.2 Mechanism 2: Adelic Logical Operators Spanning Multiple Geometries

What would falsify it

1. We do not claim to have constructed a provably non-stabilizer adelic QEC code; rather, we define the problem, establish the mathematical framework, and identify the specific constructions whose non-stabilizer status remains to be proven or refuted.

   from Abstract

2. Can adelic QEC produce code constructions not accessible through the standard CSS/stabilizer formalism? [my conjecture — to be verified or refuted by the construction program outlined here] The CSS (Calderbank-Shor-Steane) construction and its generalization to stabilizer codes [2, 3] encompass all known quantum error-correcting codes used in practice, including surface codes, color codes, and their generalizations.

   from 1.3 The Central Question

3. This suggests that adelic QEC codes might encode arithmetic information — logical qubits could represent Galois-invariant quantities [my conjecture, not yet falsifiable].

   from 6.1 Class Field Theory

Where the claim stops

1. This paper does NOT claim to have constructed a provably non-CSS adelic quantum error-correcting code.

   from 7. Conclusion: A Research Program, Not a Result

2. The central question — “Can adelic QEC produce codes beyond CSS/stabilizer?” — remains open.

   from 7. Conclusion: A Research Program, Not a Result

3. We do not claim to have constructed a provably non-stabilizer adelic QEC code; rather, we define the problem, establish the mathematical framework, and identify the specific constructions whose non-stabilizer status remains to be proven or refuted.

   from Abstract

Where to read

1. Abstract208 words · 1 min
2. 1. Introduction316 words · 1 min
3. 2. Single-$p$ QEC on Bruhat-Tits Trees (Review)226 words · 1 min
4. 3. The Adelic Generalization374 words · 2 min
5. 4. Candidate Mechanisms for Non-Stabilizer Codes405 words · 2 min
6. 5. Proof Obligations303 words · 1 min
7. 6. Connections to Number Theory151 words · 1 min
8. 7. Conclusion: A Research Program, Not a Result160 words · 1 min