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

Andrew Stewart Caldin

Published Oct 3, 2026• 0 citations

Abstract

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

DOI: 10.5281/zenodo.23115341

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

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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

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

Andrew Stewart Caldin

Published Oct 3, 2026• 0 citations

Publisher

Abstract

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

DOI: 10.5281/zenodo.23115341

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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.comFINDING: 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

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

Andrew Stewart Caldin

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.comFINDING: 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

Oct 2, 20260 citations

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

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Scaling Laws and 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 for this scaling. MATH: For a surface code with distance $d$ , logical error rate $p_L \approx C (p/p_{th})^{d/2}$ (or $p_L \propto (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$ depending on boundary conditions), where $p$ is physical error rate and $p_{th}$ is the threshold (~1% for standard surface codes under circuit-level noise). The quasi-threshold $p^*$ satisfies $p_L(p^*) = p^*$ , i.e., the fixed point of the error correction map. Stabilizer formalism: code space is the simultaneous +1 eigenspace of generators $S_i$ in the Pauli group, with $S_i^2 = I$ , $[S_i,S_j]=0$ . Logical operators $X_L, Z_L$ commute with all $S_i$ but are not in the stabilizer group. The number of encoded qubits $k = n - r$ (n physical, r independent stabilizer generat 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 for this scaling. MATH: For a surface code with distance $d$, logical error rate $p_L \approx C (p/p_{th})^{d/2}$ (or $p_L \propto (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$ depending on boundary conditions), where $p$ is physical error rate and $p_{th}$ is the threshold (~1% for standard surface codes under circuit-level noise). The quasi-threshold $p^*$ satisfies $p_L(p^*) = p^*$, i.e., the fixed point of the error correction map. Stabilizer formalism: code space is the simultaneous +1 eigenspace of generators $S_i$ in the Pauli group, with $S_i^2 = I$, $[S_i,S_j]=0$. Logical operators $X_L, Z_L$ commute with all $S_i$ but are not in the stabilizer group. The number of encoded qubits $k = n - r$ (n physical, r independent stabilizer generat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 2, 20260 citations

Google AI Achieves Quantum Error Correction Below Threshold, Enabling Scalable Logical Qubits — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum error correction (QEC) milestone — Google AI demonstrates error suppression below threshold, enabling scalable logical qubits. | MATH: Key threshold constant: error rate per operation $p < p_{\text{th}} \approx 0.57\%$ (surface code). Logical error rate scales as $p_L \propto (p/p_{\text{th}})^{(d+1)/2}$ for code distance $d$ . | CONNECTION: Surface code is a topological lattice (square grid) with $Z_2$ symmetry — related to toric code, Kitaev's anyon model. Lattice structure: 4-fold rotational symmetry, no direct golden ratio. | DEPTH: 8 — foundational for fault-tolerant quantum computing, but no new geometric constant revealed. FINDING: Deltaflow — a full-stack QEC system achieving >10^4 reliable quantum operations (QuOps). | MATH: Logical fidelity $F = 1 - \epsilon_L$ , with $\epsilon_L \approx c \cdot (p/p_{\text{th}})^{d/2}$ . Target: $10^6$ QuOps for useful computation. | CONNECTION: Error suppression follows power law in code distance — Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum error correction (QEC) milestone — Google AI demonstrates error suppression below threshold, enabling scalable logical qubits. | MATH: Key threshold constant: error rate per operation $ p < p_{\text{th}} \approx 0.57\% $ (surface code). Logical error rate scales as $ p_L \propto (p/p_{\text{th}})^{(d+1)/2} $ for code distance $ d $. | CONNECTION: Surface code is a topological lattice (square grid) with $ Z_2 $ symmetry — related to toric code, Kitaev's anyon model. Lattice structure: 4-fold rotational symmetry, no direct golden ratio. | DEPTH: 8 — foundational for fault-tolerant quantum computing, but no new geometric constant revealed. FINDING: Deltaflow — a full-stack QEC system achieving >10^4 reliable quantum operations (QuOps). | MATH: Logical fidelity $ F = 1 - \epsilon_L $, with $ \epsilon_L \approx c \cdot (p/p_{\text{th}})^{d/2} $. Target: $ 10^6 $ QuOps for useful computation. | CONNECTION: Error suppression follows power law in code distance — Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Aug 11, 20260 citations

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

Andrew Stewart Caldin

FINDING: Quantum error correction (QEC) milestone — Google AI demonstrates error suppression below surface code threshold, enabling logical qubit operation with reduced physical error rates. | MATH: Surface code distance $d$ 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.comFINDING: Quantum error correction (QEC) milestone — Google AI demonstrates error suppression below surface code threshold, enabling logical qubit operation with reduced physical error rates. | MATH: Surface code distance $d$ 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

Aug 14, 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

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

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

d

, logical error rate

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

(or

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

depending on boundary conditions), where

p

is physical error rate and

p_{th}

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

p^*

satisfies

p_L(p^*) = p^*

, i.e., the fixed point of the error correction map. Stabilizer formalism: code space is the simultaneous +1 eigenspace of generators

S_i

in the Pauli group, with

S_i^2 = I

[S_i,S_j]=0

. Logical operators

X_L, Z_L

commute with all

S_i

but are not in the stabilizer group. The number of encoded qubits

k = n - r

(n physical, r independent stabilizer generat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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

(surface code). Logical error rate scales as

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

for code distance

d

. | CONNECTION: Surface code is a topological lattice (square grid) with

Z_2

symmetry — related to toric code, Kitaev's anyon model. Lattice structure: 4-fold rotational symmetry, no direct golden ratio. | DEPTH: 8 — foundational for fault-tolerant quantum computing, but no new geometric constant revealed. FINDING: Deltaflow — a full-stack QEC system achieving >10^4 reliable quantum operations (QuOps). | MATH: Logical fidelity

F = 1 - \epsilon_L

, with

\epsilon_L \approx c \cdot (p/p_{\text{th}})^{d/2}

. Target:

10^6

QuOps for useful computation. | CONNECTION: Error suppression follows power law in code distance — 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