Topological Error Suppression: Exponential Scaling in Surface Code Quantum Correction — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 2, 2026• 0 citations

Abstract

FINDING: Surface code logical error rate scales exponentially with code distance (lattice size), enabling practical quantum error correction via topological order. | MATH: Logical error rate $P_L \propto (\Lambda)^{-d}$, where $d$ = code distance (linear lattice dimension), $\Lambda = P_{th}/P_{phys}$ (error suppression factor per round). Threshold $P_{th} \approx 0.01$ (1% per qubit per cycle). Physical-to-logical overhead scales as $O(d^2)$ qubits per logical qubit. | CONNECTION: The surface code is a **Z₂ toric code** — a lattice gauge theory on a square lattice with **crystallographic symmetry (p4m plane group)**. The logical operators are **non-contractible loops** on the torus (genus-1 surface), whose weight scales as $d$ — directly tied to the **topological winding number** and the **fundamental group $\pi_1(T^2) = \mathbb{Z} \times \mathbb{Z}$**. The threshold ~1% is not a golden ratio, but the **error suppression per distance** $\Lambda$ often approaches values Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23095565

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Exponential Error Suppression in Surface Codes: Scaling with Code Distance — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Surface code logical qubit error suppression scales with code distance $d$, with logical error rate $P_L \propto (\Lambda)^{-(d+1)/2}$ where $\Lambda$ is the error-limit parameter; Google's 2022 experiment demonstrated exponential suppression across $d=3,5,7$. MATH: - Logical error rate: $P_L \propto \Lambda^{-(d+1)/2}$ (exponential in code distance) - Threshold condition: physical error rate $p < p_{th} \approx 0.01$ (for standard surface code with depolarizing noise) - Stabilizer weight: $w = 4$ (X and Z plaquette operators on square lattice) - Logical operator weight: $w_L = d$ (minimum weight path across lattice) - Lattice: square grid, $d \times d$ qubit array, $d^2$ data qubits + $(d^2-1)$ ancilla qubits - Key ratio: $P_L / P_{phys} \sim (\Lambda)^{-(d+1)/2}$ where $\Lambda \approx p_{th}/p$ CONNECTION: - Square lattice = $Z^2$ root system (crystallographic, order 4 symmetry) — stabilizer generators are 4-body plaquette opera Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 1, 20260 citations

Google AI Achieves Sub-Threshold Quantum Error Correction for Logical Qubits — E8 Intelligence Research

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Surface Code Error Scaling and Rotated Qubit Efficiency — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Surface code logical error rate scales as $p_L \propto (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$ for distance-$d$ codes, with threshold $p_{th} \approx 0.01$ (circuit-level) and rotated code halving qubit overhead at equal distance. | MATH: Logical error rate $p_L = A (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$; for odd $d$, exponent $= (d+1)/2$; for even $d$, exponent $= d/2$. Distance $d$ = minimum weight of logical operator (e.g., $d=5$ for 49-qubit rotated code). Qubit count: rotated $= d^2$, unrotated $= 2d^2 - 2d + 1$. Threshold $p_{th} \approx 0.01$ (circuit-level noise), $\approx 0.11$ (code capacity). | CONNECTION: The exponent $\lfloor (d+1)/2 \rfloor$ is the **minimum weight of a logical $Z$ or $X$ operator** — this is the **Manhattan distance** on the square lattice. The rotated surface code is a **$d \times d$ square patch** of the square lattice — a **$D_4$ root system** projection (checkerboard lattice). The ratio of qubit counts (rot Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Topological Error Suppression: Exponential Scaling in Surface Code Quantum Correction — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 2, 2026• 0 citations

Publisher

Abstract

FINDING: Surface code logical error rate scales exponentially with code distance (lattice size), enabling practical quantum error correction via topological order. | MATH: Logical error rate (P_L \propto (\Lambda)^{-d}), where (d) = code distance (linear lattice dimension), (\Lambda = P_{th}/P_{phys}) (error suppression factor per round). Threshold (P_{th} \approx 0.01) (1% per qubit per cycle). Physical-to-logical overhead scales as (O(d^2)) qubits per logical qubit. | CONNECTION: The surface code is a Z₂ toric code — a lattice gauge theory on a square lattice with crystallographic symmetry (p4m plane group). The logical operators are non-contractible loops on the torus (genus-1 surface), whose weight scales as (d) — directly tied to the topological winding number and the fundamental group (\pi_1(T^2) = \mathbb{Z} \times \mathbb{Z}). The threshold ~1% is not a golden ratio, but the error suppression per distance (\Lambda) often approaches values Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23095565

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Google AI Achieves Fault-Tolerant Quantum Computation by Scaling Logical Qubit Fidelity Below Error Threshold — E8 Intelligence Research

Andrew Stewart Caldin

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Topological Quantum Error Correction via Toric Codes and Anyon Braiding — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Topological quantum error correction (TQEC) uses lattice geometry—specifically the toric code and surface codes—to encode logical qubits non-locally, with error thresholds tied to lattice percolation and anyon braiding. MATH: - Toric code: Hamiltonian $H = -\sum_v A_v - \sum_p B_p$ , 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.comFINDING: Topological quantum error correction (TQEC) uses lattice geometry—specifically the toric code and surface codes—to encode logical qubits non-locally, with error thresholds tied to lattice percolation and anyon braiding. MATH: - Toric code: Hamiltonian $ 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

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Surface Code Error Scaling and Rotated Qubit Efficiency — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Surface code logical error rate scales as $p_L \propto (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$ for distance- $d$ codes, with threshold $p_{th} \approx 0.01$ (circuit-level) and rotated code halving qubit overhead at equal distance. | MATH: Logical error rate $p_L = A (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$ ; for odd $d$ , exponent $= (d+1)/2$ ; for even $d$ , exponent $= d/2$ . Distance $d$ = minimum weight of logical operator (e.g., $d=5$ for 49-qubit rotated code). Qubit count: rotated $= d^2$ , unrotated $= 2d^2 - 2d + 1$ . Threshold $p_{th} \approx 0.01$ (circuit-level noise), $\approx 0.11$ (code capacity). | CONNECTION: The exponent $\lfloor (d+1)/2 \rfloor$ is the **minimum weight of a logical $Z$ or $X$ operator** — this is the **Manhattan distance** on the square lattice. The rotated surface code is a ** $d \times d$ square patch** of the square lattice — a ** $D_4$ root system** projection (checkerboard lattice). The ratio of qubit counts (rot Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Surface code logical error rate scales as $p_L \propto (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$ for distance-$d$ codes, with threshold $p_{th} \approx 0.01$ (circuit-level) and rotated code halving qubit overhead at equal distance. | MATH: Logical error rate $p_L = A (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$; for odd $d$, exponent $= (d+1)/2$; for even $d$, exponent $= d/2$. Distance $d$ = minimum weight of logical operator (e.g., $d=5$ for 49-qubit rotated code). Qubit count: rotated $= d^2$, unrotated $= 2d^2 - 2d + 1$. Threshold $p_{th} \approx 0.01$ (circuit-level noise), $\approx 0.11$ (code capacity). | CONNECTION: The exponent $\lfloor (d+1)/2 \rfloor$ is the **minimum weight of a logical $Z$ or $X$ operator** — this is the **Manhattan distance** on the square lattice. The rotated surface code is a **$d \times d$ square patch** of the square lattice — a **$D_4$ root system** projection (checkerboard lattice). The ratio of qubit counts (rot Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Scaling Laws and Quasi-Thresholds in Surface Code Error Correction — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Surface code logical error rate scales exponentially with code distance, with a quasi-threshold defining the crossover where error correction becomes beneficial; stabilizer formalism provides the algebraic framework. MATH: For a surface code with distance $d$ , logical error rate $p_L \approx C (p/p_{th})^{(d+1)/2}$ , where $p$ is physical error rate and $p_{th}$ is the threshold (~1% for standard surface codes under circuit-level noise). Quasi-threshold $p_q(d)$ satisfies $p_L(p_q) = p_q$ , giving $p_q \approx p_{th} (1 - \text{const}/d)$ . Stabilizer formalism: code space is joint $+1$ eigenspace of $n-k$ independent Pauli operators $S_i$ , with $S_i^2 = I$ , $[S_i,S_j]=0$ . Logical operators $L$ satisfy $[L,S_i]=0$ , $L \notin \langle S_i \rangle$ . Decoding via minimum-weight perfect matching (MWPM) on the syndrome graph — a planar lattice with $O(d^2)$ vertices. CONNECTION: The surface code is a topological code on a square lattice — a $Z_2$ g Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Surface code logical error rate scales exponentially with code distance, with a quasi-threshold defining the crossover where error correction becomes beneficial; stabilizer formalism provides the algebraic framework. MATH: For a surface code with distance $d$, logical error rate $p_L \approx C (p/p_{th})^{(d+1)/2}$, where $p$ is physical error rate and $p_{th}$ is the threshold (~1% for standard surface codes under circuit-level noise). Quasi-threshold $p_q(d)$ satisfies $p_L(p_q) = p_q$, giving $p_q \approx p_{th} (1 - \text{const}/d)$. Stabilizer formalism: code space is joint $+1$ eigenspace of $n-k$ independent Pauli operators $S_i$, with $S_i^2 = I$, $[S_i,S_j]=0$. Logical operators $L$ satisfy $[L,S_i]=0$, $L \notin \langle S_i \rangle$. Decoding via minimum-weight perfect matching (MWPM) on the syndrome graph — a planar lattice with $O(d^2)$ vertices. CONNECTION: The surface code is a topological code on a square lattice — a $Z_2$ g Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 24, 20260 citations

FINDING: Surface code logical qubit error suppression scales with code distance

d

, with logical error rate

P_L \propto (\Lambda)^{-(d+1)/2}

where

\Lambda

is the error-limit parameter; Google's 2022 experiment demonstrated exponential suppression across

d=3,5,7

. MATH: - Logical error rate:

P_L \propto \Lambda^{-(d+1)/2}

(exponential in code distance) - Threshold condition: physical error rate

p < p_{th} \approx 0.01

(for standard surface code with depolarizing noise) - Stabilizer weight:

w = 4

(X and Z plaquette operators on square lattice) - Logical operator weight:

w_L = d

(minimum weight path across lattice) - Lattice: square grid,

d \times d

qubit array,

d^2

data qubits +

(d^2-1)

ancilla qubits - Key ratio:

P_L / P_{phys} \sim (\Lambda)^{-(d+1)/2}

where

\Lambda \approx p_{th}/p

CONNECTION: - Square lattice =

Z^2

root system (crystallographic, order 4 symmetry) — stabilizer generators are 4-body plaquette opera Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

d

yields logical error rate

P_L \propto (p/p_{th})^{d/2}

where

p

is physical error rate,

p_{th} \approx 0.57\%

for standard depolarizing noise; threshold condition

p < p_{th}

. Stabilizer formalism uses Pauli group

\mathcal{P}_n

and syndrome extraction via

n

ancilla qubits. | CONNECTION: Surface code lattice is a 2D square grid (crystallographic symmetry

p4m

wallpaper group). Error correction cycles correspond to toric code on a planar boundary — topological order with anyonic excitations. No direct golden ratio or base-60 link. | DEPTH: 8 — milestone demonstrates fault-tolerant quantum memory below threshold, a necessary condition for scalable quantum computing. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

p < p_{\text{th}} \approx 0.57\%

for surface code; logical error rate

p_L \propto (p/p_{\text{th}})^{(d+1)/2}

where

d

is code distance; stabilizer formalism uses Pauli group

\mathcal{P}_n

and syndrome extraction via parity checks. | CONNECTION: Surface code is a topological QEC code based on a 2D square lattice (crystallographic symmetry

p4m

); code distance

d

corresponds to lattice size; logical operators are non-contractible loops on torus (homology). No direct golden ratio or base-60 link. | DEPTH: 8 — QEC is essential for scalable quantum computing; surface code threshold is a precise mathematical condition; topological nature ties to lattice geometry and homology, but no new fundamental constants or ratios emerge. 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

FINDING: Surface code logical error rate scales as

p_L \propto (p/p_{th})^{\lfloor (d+1)/2 \rfloor}

for distance-

d

codes, with threshold

p_{th} \approx 0.01

(circuit-level) and rotated code halving qubit overhead at equal distance. | MATH: Logical error rate

p_L = A (p/p_{th})^{\lfloor (d+1)/2 \rfloor}

; for odd

d

, exponent

= (d+1)/2

; for even

d

, exponent

= d/2

. Distance

d

= minimum weight of logical operator (e.g.,

d=5

for 49-qubit rotated code). Qubit count: rotated

= d^2

, unrotated

= 2d^2 - 2d + 1

. Threshold

p_{th} \approx 0.01

(circuit-level noise),

\approx 0.11

(code capacity). | CONNECTION: The exponent

\lfloor (d+1)/2 \rfloor

is the **minimum weight of a logical

Z

X

operator** — this is the **Manhattan distance** on the square lattice. The rotated surface code is a **

d \times d

square patch** of the square lattice — a **

D_4

root system** projection (checkerboard lattice). The ratio of qubit counts (rot Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

d

, logical error rate

p_L \approx C (p/p_{th})^{(d+1)/2}

, where

p

is physical error rate and

p_{th}

is the threshold (~1% for standard surface codes under circuit-level noise). Quasi-threshold

p_q(d)

satisfies

p_L(p_q) = p_q

, giving

p_q \approx p_{th} (1 - \text{const}/d)

. Stabilizer formalism: code space is joint

+1

eigenspace of

n-k

independent Pauli operators

S_i

, with

S_i^2 = I

[S_i,S_j]=0

. Logical operators

L

satisfy

[L,S_i]=0

L \notin \langle S_i \rangle

. Decoding via minimum-weight perfect matching (MWPM) on the syndrome graph — a planar lattice with

O(d^2)

vertices. CONNECTION: The surface code is a topological code on a square lattice — a

Z_2

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