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Ostrowski to Fault Tolerance: A Proof That Adelic Encoding is Necessary for Quantum Error Correction

QNFO Research Collective, Rowan Brad Quni-Gudzinas

Published Jul 11, 2026• 0 citations
Publisher

Abstract

We prove that any fault-tolerant quantum error correction (QEC) scheme must encode logical information in an adelic representation — a representation simultaneously well-defined with respect to both the Archimedean valuation and at least one p-adic valuation on Q. The proof proceeds from Ostrowski's theorem on the classification of absolute values on Q and three key lemmas covering metric classification and insufficiency proofs for single-metric error models. Computational verification across 132 code parameter combinations found no counter-examples. We derive corollaries concerning code optimality, dimension constraints (prime-power qudit codes), and the natural fit of the Bruhat-Tits quantum processor architecture. Part of the QNFO Kepler Program, Phase 1.

DOI
10.5281/zenodo.21304468

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quant-pharXiv:2609.26660•Sep 23, 2026•0 citations
We prove that any fault-tolerant quantum error correction (QEC) scheme must encode logical information in an adelic representation — a representation simultaneously well-defined with respect to both the Archimedean valuation and at least one p-adic valuation on Q. The proof proceeds from Ostrowski's theorem on the classification of absolute values on Q and three key lemmas covering metric classification and insufficiency proofs for single-metric error models. Computational verification across 132 code parameter combinations found no counter-examples. We derive corollaries concerning code optimality, dimension constraints (prime-power qudit codes), and the natural fit of the Bruhat-Tits quantum processor architecture. Part of the QNFO Kepler Program, Phase 1.
The Kepler Program establishes that fault-tolerant quantum computing requires departure from purely Archimedean (Boolean) representations. Through Ostrowski's theorem, we prove that any complete quantum error correction scheme must encode logical information in an adelic representation — simultaneously well-defined in both Archimedean and p-adic metrics. BUNDLE CONTENTS (10 Phases): - Phase 1: OFT Theorem Proof (Ostrowski → Fault Tolerance) - Phase 2: QEC Architecture + Provisional Patent (20 claims) - Phase 3: Multi-Prime Hensel Codec (GPU/SIMD, 73K enc/s) - Phase 4: Quantum Hardware Lab Identification + Collaboration Proposals - Phase 5: Silent Radix Defense + Braided Memory Patent + LoF↔Ultrametric - Phase 6: Planck Derivation (19→3 params) + Informational Universe + Infomatics - Phase 7: 2-adic Signal Confirmation (p<0.001) + Posner Molecule Protocol - Phase 8: Re-Entry + Ultrametric Signatures + Autaxys Falsification + Time-as-Tree-Depth - Phase 9: Infrastructure Decommission + Archive Migration + Discovery Momentum - Phase 10: 7 Synthesis Papers + Kepler Program Capstone CORRECTIVE NOTICE: This publication replaces a previous incorrect DOI claim (10.5281/zenodo.21304674 — which belongs to an unrelated ML paper by Zhang, Jincheng). The Kepler Program bundle was never properly published in Iteration 2. This Iteration 3 publication is the first verified Kepler Program bundle on Zenodo.
Noise is one of the central obstacles to building useful quantum computers, and quantum error correction (QEC) provides the framework for protecting quantum information against it. Unlike classical error correction, QEC must preserve fragile quantum states without copying them, measuring them directly, or destroying the information they encode. Driven by rapid progress in both theory and experiment, this challenge has grown into one of the most active areas of quantum information science. This tutorial gives a guided introduction to modern QEC, developing the core concepts of codes, syndromes, stabilizers, decoding, and fault tolerance before connecting them to major code families and current research directions. We cover both established constructions and newer developments, including topological and subsystem codes, bosonic and qudit codes, dynamical codes, and quantum low-density parity-check (qLDPC) codes. The emphasis is on building operational understanding: explaining not only what the main objects are, but how they are used in code design, error diagnosis, decoding, and fault-tolerant computation. The tutorial is intended for newcomers seeking a first path through QEC, as well as researchers looking for a coherent reference for the concepts, code families, and tools that arise in current work.
We prove a fault-tolerance theorem for quantum computation against adversarial noise. For every quantum circuit on Nˉ\bar{N}Nˉ logical qudits of depth Tˉ\bar{T}Tˉ, we construct a fault-tolerant circuit on N=poly(Nˉ)N=\text{poly}(\bar{N})N=poly(Nˉ) physical qudits of depth Tˉ⋅Nˉo(1)\bar{T}\cdot\bar{N}^{o(1)}Tˉ⋅Nˉo(1), which is robust against an adversary who may arbitrarily choose and corrupt an almost-linear number N1−o(1)N^{1-o(1)}N1−o(1) of physical qudits at each time step. This robustness significantly improves upon prior fault-tolerance theorems, which assumed corruptions were either local and stochastic, or else only act on a polynomially vanishing fraction of qudits. Our fault-tolerance scheme addresses a key bottleneck towards constructing quantum PCPs via the circuit-to-Hamiltonian mapping of Anshu, Breuckmann, and Nguyen (STOC'24). More fundamentally, our result demonstrates that fault-tolerant quantum computation remains possible under noise models that are global, worst-case, and non-Markovian over the full duration of the computation, directly countering concerns that correlated noise could fundamentally undermine quantum fault tolerance. Our construction is based on a new family of subsystem product codes we develop, which have large dimension and distance along with low-weight parity-checks, and which support transversal non-Clifford gates. We show how to perform single-shot fault-tolerant error correction on these codes using a Floquet-like procedure based on the local testability of classical tensor codes. We then obtain a universal fault-tolerance scheme using repeated code switching in a hypercubic qudit architecture. Finally, we recursively compose our scheme with itself to reduce an initially exponential qudit dimension down to a constant.
Bosonic codes offer a promising approach towards hardware-efficient fault tolerance, with many leading examples such as the cat-code and the GKP code organized by an underlying symmetry. We build on a representation-theoretic framework of quantum error correction to formalize the construction and analysis of multimode symmetric bosonic codes, focusing on finite groups. Here, a physical, logical representation and an initial seed state together define the codewords via a covariant encoding. A central observation is that QEC matrices can be viewed as morphisms of group representations, so that Schur's lemma forces entire irreducible representations blocks to vanish. We recover several known single- and two-mode bosonic codes and build and analyze the error protection capabilities of three codes based on finite subgroups of SU(2). We analyze the impact of reducibility of the logical representation and optimize the seed state using the near-optimal fidelity.