Adelic Quantum Error Correction: A Synthesis Across All Primes v1.0.0

In the paper's words

The p-adic quantum error correction program has developed three independent but connected strands: a mathematical theory of p-adic stabilizer codes (Phase A), an experimental protocol to test for ultrametric error structure (Phase B), and a hardware co-design proposal for tree-structured qubit layouts (Phase C). These strands share a common origin in the ultrametric geometry of the p-adic numbers Q_p, but they treat each prime p in isolation. Number theory provides a deeper unification: the adele ring A, which stitches together all completions of Q — the real numbers R (Archimedean) and all p-adic fields Q_p (non-Archimedean) — into a single locally compact topological ring.

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. Phase B [Quni-Gudzinas, 2026b] designed an experimental protocol to measure the ultrametricity index $\mathcal{U}_p$ on quantum hardware.

   from 1. Introduction

2. We define Hilbert spaces at each completion: Archimedean Hilbert space: $\mathcal{H}_\infty = L^2(\mathbb{R}^n, d^n x)$, the standard Hilbert space for $n$ continuous-variable quantum systems, or $\mathcal{H}_\infty = (\mathbb{C}^d)^{\otimes m}$ for $m$ qudits embedded in real spacetime. $p$-adic Hilbert space: $\mathcal{H}_p = L^2(\mathbb{Q}_p^n, d\mu_p)$, where $d\mu_p$ is the Haar measure on $\mathbb{Q}_p^n$, normalized so that $\mathbb{Z}_p^n$ has measure 1.

   from 3.1 Local Hilbert Spaces

3. We define: $$\Delta = \min_{L \in \mathcal{N}(\mathcal{S}_{\mathbb{A}})\setminus\mathcal{S}_{\mathbb{A}}} \prod_{v \leq \infty} \|L_v\|_v$$ where $\mathcal{N}(\cdot)$ is the normalizer, $L_v$ is the component of the logical operator $L$ at completion $v$, and $\|\cdot\|_v$ is the appropriate norm: $v = \infty$: Hamming weight (standard code distance $d$) $v = p$: $p$-adic weight $\|P\|_p = \max(|a|_p, |b|_p)$ as defined in Phase A The product structure reflects the adelic philosophy: protection must be provided at ALL active completions simultaneously.

   from 4.2 Adelic Code Distance

4. In analogy with the product formula, we conjecture an adelic quantum bound : $$\prod_{v \leq \infty} \left(\frac{\dim \mathcal{C}_v}{p_v^{n_v}}\right) \leq \text{constant}$$ where $\dim \mathcal{C}_v$ is the logical dimension at completion $v$, $p_v$ is the local qudit dimension, and $n_v$ is the number of physical registers.

   from 4.3 Adelic Product Bound

What would falsify it

1. Experimental accessibility: The adelic product formula prediction ($\mathcal{U}_p$ vs. $p_{\text{phys}}$) is falsifiable with current hardware.

   from 8.3 Limitations and Open Questions

2. Testing this prediction requires measuring $\mathcal{U}_p$ for multiple primes on the same hardware — an extension of the Phase B protocol.

   from 5.2 Which Primes?

3. This interpretation makes a strong, testable prediction: improving local gate fidelity should increase the ultrametricity index $\mathcal{U}_p$.

   from 7.2 Testable Prediction

Where the claim stops

1. Developing an adelic Schrödinger equation or adelic quantum circuit model is an open problem.

   from 8.3 Limitations and Open Questions

2. If $\mathcal{U}_p \approx 0$, the Archimedean assumptions of conventional QEC are validated, and the adelic framework provides a rigorous bound on how non-Archimedean quantum errors could have been but are not.

   from 9. Conclusion

3. This bound would express a trade-off: you cannot maximally encode at all primes simultaneously; increasing the encoding rate at one prime forces a decrease at another.

   from 4.3 Adelic Product Bound

Where to read

1. 1. Introduction416 words · 2 min
2. 2. The Adele Ring and Its Physical Interpretation277 words · 1 min
3. 3. Adelic Hilbert Spaces277 words · 1 min
4. 4. Adelic Stabilizer Codes270 words · 1 min
5. 5. Adelic Concatenation362 words · 2 min
6. 6. The Feynman-Shor Program in Adelic Perspective211 words · 1 min
7. 7. Adelic Product Formula as an Error Budget261 words · 1 min
8. 8. Discussion341 words · 1 min
9. 9. Conclusion329 words · 1 min