Generalized Code Distance through Rotated Logical States in Quantum Error Correction

TL;DR

This paper introduces rotated logical states in quantum error correction, analyzing how rotations affect code distance and error rates, and demonstrating improved noise resilience through extended logical bases.

Contribution

It presents a novel method of constructing rotated logical states that extend the logical basis and enhance error suppression in quantum codes.

Findings

Rotation operators modify the effective code distance $d_R$ and error correction performance.

Logical error rates decay exponentially with $d_R$, especially under SI noise.

Rotated codes show improved resilience and threshold error rates compared to traditional stabilizer codes.

Abstract

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.