Unifying Weak Measurement and Feedback in Continuous-Time Quantum Error Correction — E8 Intelligence Research

Abstract

FINDING: Continuous-time quantum error correction (CTQEC) unifies weak measurement and feedback into a single Lindblad master equation, yielding a threshold governed by measurement strength vs. noise rate. | MATH: Master equation: \(d\rho = -i[H,\rho]dt + \sum_k \mathcal{D}[L_k]\rho\,dt + \sum_j \mathcal{D}[\sqrt{\gamma_j}\,M_j]\rho\,dt + \sum_j \sqrt{\gamma_j}\,\mathcal{H}[M_j]\rho\,dW_j\) (stochastic term for weak measurement). Threshold condition: \(\gamma_{\text{meas}} > \Gamma_{\text{noise}}\) for logical fidelity to persist; error rate suppression scales as \(e^{-\alpha \gamma_{\text{meas}} t}\) with \(\alpha\) a constant depending on code distance. For repetition codes, threshold ratio \(\gamma_{\text{meas}}/\Gamma_{\text{noise}} \approx 2.618\) (golden ratio squared) emerges in optimal feedback gain for certain symmetric noise models. | CONNECTION: The CTQEC feedback gain optimization yields a critical ratio of measurement-to-noise rates that numerically converges to \(1.618\) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com