Holographic Quantum Error Correction on Bruhat–Tits Trees: A Conditional Threshold and Resource Analysis at $10^{-4}$

In the paper's words

The holographic quantum error‑correction (QEC) architecture that implements encoding as a renormalization‑group (RG) flow on a Bruhat–Tits tree T_p claims a physical error‑rate threshold ε_th≥10^-4. No hardware data accompany the proposal, so we construct a fully explicit conditional model: (i) independent site errors on a regular tree of valency q=p+1, (ii) a concatenation‑style logical‑error law ε_L=A(ε/ε_th)^(d+1)/2 with A=1, (iii) a reliability target P_fail≤0.01 over T=10⁹ correction cycles at a cycle rate f_c=10⁶ mathrms^-1. Solving the model yields (a) zero logical protection at ε=ε_th=10^-4, (b) a distance d≥21 code sufficient at ε=10^-5, requiring 88,573 physical sites per logical qubit and a measurement bandwidth of 8.8573×10¹0 mathrms^-1, (c) a shallow d=11 code viable only if ε≤1.4678×10^-6, and (d) an exponential resource explosion when operating at ε=5×10^-5.

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

Numbers it reports

1. Assuming a unit cost of $ 1 per physical qubit (engineering projection) and a per‑measurement latency of 10 ns (projection), the $d=21$ configuration would cost roughly $ 8.9 × $10^{4}$ per logical qubit and consume about $0.9$ W of readout power ($8.8573\times10^{10}$ events $s^{-1}$ × 10 ns).

   from 7. Practical Implications and Engineering Pathways

2. The holographic quantum error‑correction (QEC) architecture that implements encoding as a renormalization‑group (RG) flow on a Bruhat–Tits tree $\mathcal{T}_p$ claims a physical error‑rate threshold $\epsilon_{th}\ge10^{-4}$.

   from Abstract

3. An alternative quadratic‑map convention shows the threshold equals $1/\kappa$, so the $10^{-4}$ target holds iff $\kappa\le10^{4}$.

   from Abstract

4. The holographic QEC architecture of [9] proposes to encode logical information by an RG flow on a $p$‑adic Bruhat–Tits tree $\mathcal{T}_p$, and it asserts a physical error‑rate threshold $\epsilon_{th}\ge10^{-4}$.

   from 1. Introduction

5. If the claim holds, a modest physical error rate of $10^{-5}$ would suffice to achieve a logical failure probability below $1\%$ over $10^{9}$ cycles, potentially enabling medium‑scale quantum algorithms with far fewer resources than conventional surface‑code stacks.

   from 1. Introduction

6. Exact qubit‑overhead $N_{\text{phys}}=V_{3}(11)=265{,}720$ and measurement bandwidth $B=2.6572\times10^{11}\,\mathrm{s^{-1}}$ for the $d=21$ code (Sections 4.4–4.5).

   from 1. Introduction

How it is done

1. Correction (version 2.0.1, 2026-10-19): statements that described experiments, measurements, reviews or registrations as done were corrected; none took place.

   from Holographic Quantum Error Correction on Bruhat–Tits Trees: A Conditional Threshold and Resource Analysis at $10^{-4}$

2. No hardware data accompany the proposal, so we construct a fully explicit conditional model: (i) independent site errors on a regular tree of valency $q=p+1$, (ii) a concatenation‑style logical‑error law $\epsilon_L=A(\epsilon/\epsilon_{th})^{(d+1)/2}$ with $A=1$, (iii) a reliability target $P_{fail}\le0.01$ over $T=10^{9}$ correction cycles at a cycle rate $f_c=10^{6}\,\mathrm{s^{-1}}$.

   from Abstract

3. Assumptions (A1)–(A7) are listed in the main text and reproduced here for completeness: (A1) Independent site errors with physical error rate $\epsilon$ per correction cycle.

   from 3.1 Convention 1 (primary): distance‑based conditional model

4. Each site is measured once per cycle, giving per‑logical‑qubit event rate $$ B = V_q(R)\,f_c . $$

   from 3.3 Measurement bandwidth

What would falsify it

1. Observation of $\kappa\gt 10^{4}$ would falsify the $10^{-4}$ threshold claim under Convention 2; failure to achieve logical error suppression at $\epsilon=10^{-5}$ for $d=21$ would falsify the distance‑based claim.

   from 6. Discussion

2. An observed $\kappa\gt 10^{4}$ would falsify the $10^{-4}$ threshold claim under Convention 2; a logical failure probability exceeding $0.01$ at $\epsilon=10^{-5}$ would falsify the distance‑based claim (R2).

   from 7. Practical Implications and Engineering Pathways

Where the claim stops

1. The analysis is purely analytical, unreviewed, and based on assumed error models; according to NASA TRL definitions it resides at TRL 2 (concept formulation).

   from 6. Discussion

2. Correlation handling: The model assumes independent site errors (A1); correlated noise would increase effective $A$ and raise required $d$.

   from 6. Discussion

3. Assuming a $1$ USD per physical qubit (a rough engineering projection) and a $10$ ns per measurement event, the $d=21$ configuration would cost roughly $8.9\times10^{4}$ USD per logical qubit and consume $≈0.9$ W of readout power (projected from $B\times10$ ns).

   from 6. Discussion

Where to read

1. Abstract171 words · 1 min
2. 1. Introduction216 words · 1 min
3. 2. Background and Related Work220 words · 1 min
4. 3. Methods140 words · 1 min
5. 4. Analysis147 words · 1 min
6. 5. Results81 words · 1 min
7. 6. Discussion368 words · 2 min
8. 7. Practical Implications and Engineering Pathways500 words · 2 min
9. 9. Conclusion136 words · 1 min
10. Correction Notice302 words · 1 min
11. Provenance and AI Involvement230 words · 1 min
12. Appendix A. Divergence report88 words · 1 min