Why Measure Error Correlations Before Choosing a Code: An Industry Brief and Pre-Registered Benchmark Protocol

Why Measure Error Correlations Before Choosing a Code

A one-page brief for quantum-hardware teams and funders.

Source record: *The Trapped-Ion Ultrametric Testbed: A Falsifiability Register for

Testing p-Adic Structure in Quantum Dynamics*, v1.4 — DOI 10.5281/zenodo.22025544

(concept 10.5281/zenodo.22013263). All numbers cited below are published in the

record and its source archive.

The problem: threshold theorems are theorems about noise models, not hardware

Every error-correction threshold — the surface code's ~$1.1\times10^{-2}$ included — is proven

under an error model. That model is overwhelmingly independent, Markovian, additive

noise: the Archimedean geometry of the error channel. Real devices violate it:

cosmic-ray multi-qubit events, crosstalk, two-level-system defects, leakage,

non-Markovian baths, and thermal correlations grow with scale. If correlated

structure dominates at the scales where fault tolerance must actually run, the

threshold that matters is not the textbook one — it is the threshold under the

measured noise, and that number is not yet on anyone's datasheet.

The honest disclosure first (the number that hurts)

Under independent errors, the ultrametric qudit code path shows a threshold near

$2.0\times10^{-4}$ — approximately 55× worse than surface codes (published in the

program's own QEC record, 10.5281/zenodo.21046993; recomputed from its own numbers).

We publish this number because it is the crux: **if independent errors dominate at

scale, the ultrametric path is dead, and the industry is right to ignore it.** The

same discipline published the program's own nulls: no CMB log-periodic signal

(10.5281/zenodo.21902891), an anti-ultrametric biological counterexample

(10.5281/zenodo.21651892), and a QEC-Darwinism no-go theorem whose proof chain

assumes Archimedean geometry (Maity et al., arXiv:2608.03944).

The one bet everything rests on

> At scale, error structure is not independent.

That is a falsifiable statement about hardware, not a philosophical preference. The

case for taking it seriously: (1) the industry's own correlated-error pain

(cosmic-ray events, crosstalk, TLS defects); (2) the QEC-Darwinism theorem — QEC and

emergent classical objectivity cannot coexist above F_L > 0.874 *under the

Archimedean assumption*, which is exactly the assumption correlated/hierarchical

noise violates; (3) the thermal/energy argument — active correction pays an energy

tax that grows with distance, while structure-suppressing (passive) designs do not.

The ask: a rounding error of R&D budget, spent on measurement

We are not asking anyone to abandon surface codes. We are asking teams to spend a

small fraction of their error-characterization budget on **measuring the correlation

structure of errors at scale* — because that measurement improves any* code choice,

ultrametric or not:

1. R1 — Page–Wootters ultrametricity probe (8 weeks on existing trapped-ion

apparatus; predicted UVR split 0% diagonal vs 29–35% nondiagonal; kill-condition

pre-registered).

1. R3 — laptop benchmark (effective transient dimension on p-regular trees;

code deposited with the source record).

1. Artifact 4 — energy-audit template (joules-per-solution methodology, applied

to any platform).

1. Artifact 5 — QEC-Darwinism constraint checker (audit any candidate

architecture against the F_L > 0.874 no-go; check which Archimedean assumptions

the proof uses — if an advantage survives only by violating a named assumption,

the geometry is doing real work).

The correlated-error benchmark spec (companion document) turns the bet into a

pre-registered protocol: pairwise correlation functions, spatiotemporal clustering,

non-Markovian memory, event-burst statistics, and decision rules that settle the

question either way — **including the rule that kills the ultrametric path if

independent errors dominate.**

What we promise in return

If the measurement says independent errors dominate: the ultrametric path is dead,

and we will have published the result — the program's ledger already includes its own

nulls. If it says correlated structure dominates: the 55× comparison was the wrong

comparison, the geometry of the error structure is the code, and the teams that

measured first own the transition.

**One measurement, pre-registered, cheap, valuable either way. That is the entire

ask.**

*Companion: correlated-error-benchmark-spec-2026-08-20.md (pre-registered protocol

and decision rules). Full register: 10.5281/zenodo.22025544. This brief is an

engagement artifact, not a claim of hardware advantage.*