Regular Ternary Tree Codes and Symmetric Exponential Energy Barriers for Passive Quantum Error Correction

In the paper's words

Quantum error correction (QEC) currently relies on active syndrome extraction, which incurs large qubit overheads. A conjectured passive scheme encodes logical information on the vertices of a regular ternary Bruhat–Tits tree, exploiting the ultrametric triangle inequality to create an energy barrier that grows exponentially with tree depth. Existing work on binary Bruhat–Tits trees reports a barrier that protects only a single logical state because the root’s odd branching forces tie‑breaking asymmetry.

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

Numbers it reports

1. For $N=10^{5}$ independent trials with no observed logical errors, the 95 % confidence upper bound on the logical error probability $p_{L}$ is approximated by the rule of three: $$ p_{L}\le\frac{3}{N}=\frac{3}{10^{5}}=3.0\times10^{-5}\approx2.9957\times10^{-5}. $$

   from 4.6 Confidence Bounds

2. Using explicit combinatorial counting we compute (i) the barrier value at depth seven ($B(7)=128$), (ii) the code parameters $n_{t}=1+(p+1)(p^{t}-1)/(p-1)$ for $p=3$ and $t=1,2$, and (iii) the logical failure probability under independent depolarizing noise $\varepsilon=10^{-3}$, obtaining $P_{\mathrm{pass}}\approx9.98\times10^{-6}$ and a suppression factor $\varepsilon/P_{\mathrm{pass}}\approx100.2$.

   from Abstract

3. The suppression factor relative to an unencoded qudit is $$ \frac{\varepsilon}{P_{\mathrm{pass}}} =\frac{10^{-3}}{9.98\times10^{-6}} \approx100.2. $$

   from 4.4 Suppression Factor

4. Take the physical error rate $p=0.4$.

   from 4.5 Recursive Majority‑Vote Probabilities

5. The passive scheme improves the error rate by a factor of $\approx100.2$ relative to an unencoded qudit (Section 4.4).

   from 5. Results

6. Starting from a physical error rate $p=0.4$, three concatenated eight‑fold majority‑vote layers yield logical error probabilities $p^{(1)}=0.1736704$, $p^{(2)}\approx0.0055485$, and $p^{(3)}\approx2.905\times10^{-10}$ (Section 4.5).

   from 5. Results

How it is done

1. Using explicit combinatorial counting we compute (i) the barrier value at depth seven ($B(7)=128$), (ii) the code parameters $n_{t}=1+(p+1)(p^{t}-1)/(p-1)$ for $p=3$ and $t=1,2$, and (iii) the logical failure probability under independent depolarizing noise $\varepsilon=10^{-3}$, obtaining $P_{\mathrm{pass}}\approx9.98\times10^{-6}$ and a suppression factor $\varepsilon/P_{\mathrm{pass}}\approx100.2$.

   from Abstract

2. The Assmus–Mattson theorem for ternary codes bounds the design strength $t$ of support designs by $t\le5$ for two‑ and three‑weight codes, and relates weight enumerator coefficients to binomial sums $\displaystyle 1+\alpha+\beta=\sum_{i=0}^{2}\binom{n}{i}2^{i}$ [2].

   from 2. Background and Related Work

3. We consider a homogeneous $(p+1)$-regular infinite tree $T_{p+1}$ with branching factor $p+1$.

   from 3. Methods

4. For $N=10^{5}$ independent trials with no observed logical errors, the 95 % confidence upper bound on the logical error probability $p_{L}$ is approximated by the rule of three: $$ p_{L}\le\frac{3}{N}=\frac{3}{10^{5}}=3.0\times10^{-5}\approx2.9957\times10^{-5}. $$

   from 4.6 Confidence Bounds

What would falsify it

1. The main result—that a regular ternary tree yields a symmetric exponential barrier—would be falsified if any of the following observations were made: F1.

   from 6.2 Failure Modes and Falsification

2. A quantum simulation of the ternary tree code exhibits a logical error rate that does not decay exponentially with depth, contradicting the barrier prediction.

   from 6.2 Failure Modes and Falsification

3. Discussion of limitations, falsification criteria, and a concrete experimental pathway (Section 6).

   from 1. Introduction

Where the claim stops

1. The encoder symmetry (branch‑permutation invariance) required for distance $d=2t+1$ is assumed but not constructed; without an explicit encoder the distance claim may fail.

   from 6.1 Limitations

2. The noise model assumes independent depolarizing errors per vertex; correlated errors across neighboring vertices could invalidate the binomial counting in (4).

   from 6.1 Limitations

3. We assume that extending to a regular ternary root removes this asymmetry.

   from 1. Introduction

Where to read

1. Abstract203 words · 1 min
2. 1. Introduction354 words · 2 min
3. 2. Background and Related Work413 words · 2 min
4. 3. Methods327 words · 1 min
5. 4. Analysis212 words · 1 min
6. 5. Results141 words · 1 min
7. 6. Discussion258 words · 1 min
8. 7. Practical Implications and Engineering Pathways317 words · 1 min
9. 8. Conclusion119 words · 1 min
10. Provenance and AI Involvement225 words · 1 min
11. Appendix A. Divergence report13 words · 1 min
12. Appendix B. Claim attribution35 words · 1 min