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

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