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Optimal Gain Balancing in Continuous-Time Quantum Error Correction — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 1, 2026• 0 citations
Publisher

Abstract

FINDING: Continuous-time quantum error correction (CTQEC) via weak measurements and feedback yields a fidelity threshold governed by a stochastic master equation (SME), with optimal gain balancing measurement back-action against correction drive. | MATH: The SME for CTQEC takes the form \( d\rho = -i[H,\rho]dt + \mathcal{D}[\sqrt{\gamma}L]\rho\,dt + \sqrt{\eta\gamma}\,\mathcal{H}[L]\rho\,dW \), where \(\mathcal{D}\) is the Lindblad dissipator, \(\mathcal{H}\) is the measurement back-action superoperator, \(\eta\) is detection efficiency, \(\gamma\) is measurement strength, and \(dW\) is a Wiener increment. The steady-state fidelity \(F_\infty\) obeys \( F_\infty = 1 - \frac{\gamma_{\text{eff}}}{\gamma_{\text{corr}}} + O(\gamma_{\text{eff}}^2/\gamma_{\text{corr}}^2) \), with \(\gamma_{\text{corr}}\) the correction rate. The threshold condition is \(\gamma_{\text{corr}} > \gamma_{\text{eff}} \cdot (1 + 2/\eta)\), and the optimal feedback gain \(g^*\) satisfies \( g^* = \sqrt{\gamma_{\tex Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.23075418

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Continuous-time quantum error correction (CTQEC) is an approach to protecting quantum information from noise in which both the noise and the error correcting operations are treated as processes that are continuous in time. This chapter investigates CTQEC based on continuous weak measurements and feedback from the point of view of the subsystem principle, which states that protected quantum information is contained in a subsystem of the Hilbert space. We study how to approach the problem of constructing CTQEC protocols by looking at the evolution of the state of the system in an encoded basis in which the subsystem containing the protected information is explicit. This point of view allows us to reduce the problem to that of protecting a known state, and to design CTQEC procedures from protocols for the protection of a single qubit. We show how previously studied CTQEC schemes with both direct and indirect feedback can be obtained from strategies for the protection of a single qubit via weak measurements and weak unitary operations. We also review results on the performance of CTQEC with direct feedback in cases of Markovian and non-Markovian decoherence, where we have shown that due to the existence of a Zeno regime in non-Markovian dynamics, the performance of CTQEC can exhibit a quadratic improvement if the time resolution of the weak error-correcting operations is high enough to reveal the non-Markovian character of the noise process.
Continuous quantum error correction (CQEC) replaces discrete syndrome measurements and recovery operations with continuous syndrome extraction and real-time Hamiltonian feedback. Here we investigate the thermodynamic resources required by this process by formulating measurement-based continuous quantum error correction as an information engine. We distinguish the system-side power associated with energy transfer between the feedback field and the protected system from the controller-side power associated with the feedback Hamiltonian. The framework is first developed for the one-qubit Ahn-Doherty-Landahl (ADL) protocol and subsequently extended to the three-qubit repetition code under continuous stabilizer monitoring. Numerical simulations show that increasing the feedback strength improves the steady-state fidelity and reduces the conditional-state entropy, while both energetic contributions continue to increase in magnitude after the fidelity begins to saturate. These results expose a direct tradeoff between logical stabilization and the energetic resources required for continuous feedback, closely related to previously established energy--precision tradeoffs in quantum measurement and quantum-Zeno stabilization.