Generalized Code Distance through Rotated Logical States in Quantum Error Correction

We construct rotated logical states by applying rotation operators to stabilizer states, extending the logical basis and modifying stabilizer generators. Rotation operators affect the effective code distance $d_R$, which decays exponentially with rotation angles $(\theta, \phi)$, influencing error correction performance. We quantify the scaling behavior of logical error rates under circuit-level noise, comparing standard depolarizing (SD) and superconducting-inspired (SI) noise models with small and large rotations. Our findings show that the rotated code scales as $0.68d_R (0.65d_R)$ for SD and $0.81d_R (0.77d_R)$ for SI, with small rotation angles leading to a steeper decay of logical error rates. At a physical error rate $p_{phy}$ of $10^{-4}$, logical errors decrease exponentially with $d_R$, particularly under SI noise, which exhibits stronger suppression. The threshold error rates for rotated logical states are compared with previous results, demonstrating improved resilience against noise. By extending the logical state basis, rotation-based encoding increases error suppression beyond traditional stabilizer codes, offering a promising approach to advancing quantum error correction.