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-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.

arXiv:1311.2485Cambridge University Press eBooksSep 5, 20135 citations

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

Andrew Stewart Caldin

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.comFINDING: 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

Aug 24, 20260 citations

Foundations of quantum error correction

Wen-Ching Li

AMS/IP studies in advanced mathematicsJul 20, 20074 citations

Realization of quantum error correction

John Chiaverini, Dietrich G. Leibfried, Tobias Schaetz +9 more

NatureDec 1, 2004517 citations

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