Percolation Thresholds Unify Quantum Error Correction and Phase Transitions — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 23, 2026• 0 citations

Abstract

FINDING: Surface-code error correction thresholds align with percolation-theoretic critical probabilities in 2D lattices, establishing a deep equivalence between quantum error correction and classical statistical-mechanics phase transitions. | MATH: The surface code threshold is governed by the bond-percolation threshold $ p_c $ on the square lattice, $ p_c = 1/2 $. For the Fibonacci Turaev–Viro code (non-Abelian anyons), thresholds map to golden-ratio-related critical exponents. The Kitaev toric code threshold satisfies $ p_{\text{th}} = p_c(\text{square}) = 0.5 $ for perfect syndrome measurement; with faulty measurements, the threshold drops to $ p_{\text{th}} \approx 0.109 $ (matching the 2D random-bond Ising model multicritical point). LDPC codes (Delfosse) exhibit thresholds tied to the percolation transition on expander graphs, where the critical density satisfies $ p_c \sim 1/d $ for degree-$ d $ graphs. | CONNECTION: The square-lattice percolation threshold \( p_c = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.22909937

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Percolation Thresholds Unify Quantum Error Correction and Phase Transitions — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 23, 2026• 0 citations

Publisher

Abstract

FINDING: Surface-code error correction thresholds align with percolation-theoretic critical probabilities in 2D lattices, establishing a deep equivalence between quantum error correction and classical statistical-mechanics phase transitions. | MATH: The surface code threshold is governed by the bond-percolation threshold ( p_c ) on the square lattice, ( p_c = 1/2 ). For the Fibonacci Turaev–Viro code (non-Abelian anyons), thresholds map to golden-ratio-related critical exponents. The Kitaev toric code threshold satisfies ( p_{\text{th}} = p_c(\text{square}) = 0.5 ) for perfect syndrome measurement; with faulty measurements, the threshold drops to ( p_{\text{th}} \approx 0.109 ) (matching the 2D random-bond Ising model multicritical point). LDPC codes (Delfosse) exhibit thresholds tied to the percolation transition on expander graphs, where the critical density satisfies ( p_c \sim 1/d ) for degree-( d ) graphs. | CONNECTION: The square-lattice percolation threshold ( p_c = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.22909937

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Surface Code Threshold as Percolation Phase Transition on Torus Homology — E8 Intelligence Research

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FINDING: Surface code logical operators are homology cycles on a torus lattice; error correction threshold is a phase transition point determined by percolation of string-like excitations. MATH: - Logical operators: $\hat{X}_L = \prod_{i \in \text{cycle}} X_i$ , $\hat{Z}_L = \prod_{j \in \text{cocycle}} Z_j$ ; homology group $H_1(\mathbb{T}^2, \mathbb{Z}_2) \cong \mathbb{Z}_2^2$ . - Threshold: $p_{\text{th}} \approx 0.103$ (for standard surface code under independent bit-flip/phase-flip noise); satisfies $p_{\text{th}} = \frac{1}{2} \left(1 - \frac{1}{\sqrt{2}}\right) \approx 0.146$ for certain noise models (e.g., depolarizing). - Error correction condition: $\text{Pr}(\text{logical error}) \sim e^{-\alpha L}$ for $p < p_{\text{th}}$ , with $L$ lattice size; scaling exponent $\alpha$ related to critical exponents of 2D random-bond Ising model. CONNECTION: - Homology cycles map to closed strings on a torus; threshold $ p_{\text{th}} \approx 0 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Surface code logical operators are homology cycles on a torus lattice; error correction threshold is a phase transition point determined by percolation of string-like excitations. MATH: - Logical operators: \( \hat{X}_L = \prod_{i \in \text{cycle}} X_i $, $ \hat{Z}_L = \prod_{j \in \text{cocycle}} Z_j $; homology group $ H_1(\mathbb{T}^2, \mathbb{Z}_2) \cong \mathbb{Z}_2^2 $. - Threshold: $ p_{\text{th}} \approx 0.103 $ (for standard surface code under independent bit-flip/phase-flip noise); satisfies $ p_{\text{th}} = \frac{1}{2} \left(1 - \frac{1}{\sqrt{2}}\right) \approx 0.146 $ for certain noise models (e.g., depolarizing). - Error correction condition: $ \text{Pr}(\text{logical error}) \sim e^{-\alpha L} $ for $ p < p_{\text{th}} $, with $ L $ lattice size; scaling exponent $ \alpha $ related to critical exponents of 2D random-bond Ising model. CONNECTION: - Homology cycles map to closed strings on a torus; threshold \( p_{\text{th}} \approx 0 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Percolation Theory Determines Topological Color Code Thresholds on Hexagonal and Square-Octagonal Lattices — E8 Intelligence Research

Andrew Stewart Caldin

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Andrew Stewart Caldin

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Aug 25, 20260 citations

\hat{X}_L = \prod_{i \in \text{cycle}} X_i

\hat{Z}_L = \prod_{j \in \text{cocycle}} Z_j

; homology group

H_1(\mathbb{T}^2, \mathbb{Z}_2) \cong \mathbb{Z}_2^2

. - Threshold:

p_{\text{th}} \approx 0.103

(for standard surface code under independent bit-flip/phase-flip noise); satisfies

p_{\text{th}} = \frac{1}{2} \left(1 - \frac{1}{\sqrt{2}}\right) \approx 0.146

for certain noise models (e.g., depolarizing). - Error correction condition:

\text{Pr}(\text{logical error}) \sim e^{-\alpha L}

for

p < p_{\text{th}}

, with

L

lattice size; scaling exponent

\alpha

related to critical exponents of 2D random-bond Ising model. CONNECTION: - Homology cycles map to closed strings on a torus; threshold \( p_{\text{th}} \approx 0 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

A_v = \prod_{e \ni v} X_e

and plaquette operators

B_p = \prod_{e \in \partial p} Z_e

, with

A_v^2 = B_p^2 = I

[A_v, B_p] = 0

. - Logical operators: non-contractible loops —

X

-type along a horizontal cycle,

Z

-type along a vertical cycle; logical Pauli algebra

\bar{X}, \bar{Z}

with

\bar{X}\bar{Z} = -\bar{Z}\bar{X}

. - Distance

d

= minimum weight of logical operator = lattice dimension

L

(for

L \times L

patch); code rate

k/n = 1/(2L^2 - 2L + 1)

. - Error threshold:

p_{\text{th}} \approx 0.103

(for depolarizing noise) — a critical probability, not a simple ratio but related to percolation threshold on square lattice \(p_c \app Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

H = -\sum_v A_v - \sum_p B_p

, with star operators

A_v = \prod_{i \in v} X_i

and plaquette operators

B_p = \prod_{i \in p} Z_i

. Ground state degeneracy on genus-

g

surface:

4^g

. - Logical operators: non-contractible loops on torus —

\bar{X}, \bar{Z}

each have weight

L

(lattice length), minimum weight scales as

O(L)

for

L \times L

lattice. - Error threshold: surface code ~1.1% (per-qubit depolarizing), toric code ~10.1% (bit-flip only) — related to percolation threshold on square lattice

p_c \approx 0.593

(site) and

p_c \approx 0.5

(bond). - Anyon statistics:

e

(charge) and

m

(flux) excitations obey

\mathbb{Z}_2

fusion:

e \times m = \psi

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

p_c

is approximately 0.5 (exact for square lattice) but modified by the octagonal tiling geometry; the hexagonal lattice yields

p_c \approx 0.6527

(site percolation on triangular lattice). The threshold for the color code is given by

p_{\text{th}} = 1 - p_c

for the hexagonal case, and analogous for square-octagonal. Key constants: hexagonal lattice percolation threshold

p_c = 0.5

(bond) or 0.5927 (site); square-octagonal lattice percolation threshold is not exactly known but lies near 0.5–0.6. | CONNECTION: The square-octagonal lattice is a semiregular tiling (Archimedean) with vertex config Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

p_c = 1 - 2\sin(\pi/18) \approx 0.6527

- For square-octagonal lattice:

p_c \approx 0.6764

(exact value from 2D percolation theory) - Accuracy threshold ratio (hexagonal vs square-octagonal):

\approx 0.6527 / 0.6764 \approx 0.965

- Mapping to spin model: critical exponents follow 2D Ising universality class, with

\nu = 1

(correlation length exponent) - Surface code threshold (toric code):

p_{\text{th}} \approx 0.103

(for standard square lattice, from depolarizing noise) CONNECTION: - Hexagonal lattice: 6-fold rotational symmetry (crystallographic point group

D_6

), bond angles

120^\circ

— related to hexagonal close-packing and graphene structure. - Square-octagonal lat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\epsilon_{\text{th}} > 0

for fault tolerance); Pauli noise estimation via process tomography (Kliesch: sample complexity

\mathcal{O}(n^2/\epsilon^2)

for

n

-qubit noise channels); Fibonacci Turaev-Viro code: anyon fusion rules

1 \times \tau = \tau

\tau \times \tau = 1 + \tau

, with quantum dimension

d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618

; Rydberg erasure thresholds: biased noise reduces effective threshold by factor

\eta = p_{\text{erasure}}/p_{\text{total}}

. | CONNECTION: Fibonacci anyons have quantum dimension

\phi = 1.618

— the golden ratio appears as a *topological charge dimension*, not as a threshold value. The Turaev-Viro invariant is a state-sum over triangulations, linking Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com