Quantum Error Correction under Full Noise using a Unified Geometric Equation

We present a unified equation for modeling quantum error correction that connects logical fidelity, error amplitude, and the intrinsic geometry of quantum systems. The total error E is defined as: E = ρ * (1 + ε) + γ * π In this expression: ● ρ (rho) represents the logical fidelity after the correction is applied. ● ε (epsilon) is the magnitude of the introduced error. ● γ (gamma) is a coupling factor for phase-based influence. ● π (pi) reflects the irreducible geometric curvature of the quantum phase space. To test this equation, we simulate a three-qubit repetition code using Cirq, including full depolarizing noise (10%) applied to all qubits. The results demonstrate that this equation remains valid even under realistic noisy conditions. Even when the fidelity is high, the geometric term γ * π reveals a base-level error that cannot be corrected — a limit set by the system’s geometry. This model offers a predictive framework for understanding quantum error, going beyond operational recovery to include structural constraints.