Adelic Quantum Error Correction: Intrinsic Qubit Protection from Ostrowski

In the paper's words

We propose a quantum error-correcting code whose protection derives from Ostrowski's theorem. The code leverages the classification of non-trivial absolute values on Q to achieve intrinsic fault tolerance without active syndrome extraction.

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

Numbers it reports

1. Property Surface Code QEC Topological QEC Adelic QEC (this work) Protection mechanism Active syndrome measurement + correction Non-local anyon encoding Number-theoretic incommensurability Physical qubits per logical 100-10,000+ 1 (anyon) + braiding 1 (Majorana ZBW mode) Error threshold 0.1-1% Depends on gap Unlimited for Archimedean errors Active correction Required Partial None required Overhead scaling Polynomial in code distance $O(1)$ for braiding $O(1)$ — single Majorana mode Mathematical basis Stabilizer formalism Topological quantum field theory Ostrowski's theorem + Bruhat-Tits trees

   from 4. Comparison to Standard QEC

How it is done

1. We formalize an adelic approach to quantum error correction (QEC) based on Ostrowski's theorem: the only non-trivial completions of the rational numbers are the Archimedean (real) completion $\mathbb{R}$ and the p-adic completions $\mathbb{Q}_p$, and these topologies are mutually singular.

   from Abstract

2. Ostrowski's theorem (1916) classifies all non-trivial absolute values on $\mathbb{Q}$ up to equivalence [established] : The Archimedean absolute value: $|x|_\infty = \max(x, -x)$ (the usual absolute value) The p-adic absolute values: $|x|_p = p^{-v_p(x)}$ for each prime $p$, where $v_p(x)$ is the p-adic valuation

   from 2.1 Statement

3. Ostrowski's theorem implies that for any $p \neq q$: $$|\cdot|_\infty \not\sim |\cdot|_p$$ and for $p \neq q$: $$|\cdot|_p \not\sim |\cdot|_q$$ Consequence: No sequence can converge simultaneously in $\mathbb{R}$ and $\mathbb{Q}_p$.

   from 2.2 Mutual Singularity

4. Theorem: Let $v \in V(\mathcal{T})$ be a $\mathbb{Z}_2$ fixed point on the Bruhat-Tits tree.

   from 3.2 The Impossibility Proof

What would falsify it

1. This can be tested by measuring the coherence time $T_2$ of a Majorana qubit as a function of controlled electromagnetic noise amplitude.

   from 5.2 Falsifiability

2. The prediction: $T_2$ is independent of noise amplitude for Archimedean noise sources, in contrast to standard qubits where $T_2 \propto 1/\text{(noise amplitude)}$.

   from 5.2 Falsifiability

3. The companion papers P1-P4 established that Zitterbewegung (ZBW) is a p-adic observable, that the ZBW current correlator is a $\mathbb{Z}_2$ topological invariant, and that Majorana zero modes are fixed points on Bruhat-Tits trees.

   from Abstract

Where the claim stops

1. Two absolute values $|\cdot|_a$ and $|\cdot|_b$ are equivalent if a sequence converges in one if and only if it converges in the other.

   from 2.2 Mutual Singularity

2. Hence, $H(\epsilon)$ cannot move the state off the fixed point $v$.

   from 3.2 The Impossibility Proof

3. The ZBW $Z_{2}$ invariant $\mathcal{O}_{\text{ZBW}}$ (from P2) is robust against ALL Archimedean perturbations — thermal noise, electromagnetic interference, material defects — because these are all continuous in $|\cdot|_\infty$ and therefore cannot alter the discrete $Z_{2}$ value.

   from 3.3 Corollary: Intrinsic Protection

Where to read

1. Abstract173 words · 1 min
2. 1. Introduction170 words · 1 min
3. 2. Ostrowski's Theorem and Mutual Singularity192 words · 1 min
4. 3. Bruhat-Tits Fixed Points Under Archimedean Perturbations232 words · 1 min
5. 4. Comparison to Standard QEC78 words · 1 min
6. 5. Experimental Consequences140 words · 1 min
7. 6. The Adelic Qubit82 words · 1 min
8. 7. Conclusions133 words · 1 min
9. Changelog9 words · 1 min