Efficient feedback controllers for continuous-time quantum error correction

We present an efficient approach to continuous-time quantum error correction that extends the low-dimensional quantum filtering methodology developed by van Handel and Mabuchi [e-print arXiv:quant-ph/0511221] to include error recovery operations in the form of real-time quantum feedback. We expect this paradigm to be useful for systems in which error recovery operations cannot be applied instantaneously. While we could not find an exact low-dimensional filter that combined both continuous syndrome measurement and a feedback Hamiltonian appropriate for error recovery, we developed an approximate reduced-dimensional model to do so. Simulations of the five-qubit code subjected to the symmetric depolarizing channel suggests that error correction based on our approximate filter performs essentially identically to correction based on an exact quantum dynamical model.

Figure 1

(Color online) Nonzero matrix elements of (a) untruncated and (b) truncated filter. Squares correspond to decoherence terms, crosses correspond to measurement terms, and dots correspond to feedback terms. Note the difference in dimension of the matrices.

Figure 2

(Color online) On the left, a schematic diagram of truncating the filter to only syndrome space and first level feedback blocks. On the right, just a few of the 1024 feedback coefficients of the five-qubit code representing the different feedback block levels.

Figure 3

(Color online) Numerical simulations of the five-qubit code to assess the average code space fidelity. Plot (a) compares the codespace fidelity (averaged over ten trajectories) for filters with different levels of truncation: the full (1024-dimensional) and first-level truncated (136-dimensional) filters are essentially identical. Plot (b) shows the codespace fidelity averaged over 2000 trajectories using the truncated 136-dimensional filter for error correction. (Simulation parameters: ${\lambda }_{\text{max}}=200\gamma$ and $\kappa =100\gamma$.)

Figure 4

(Color online) Comparison between continuous-time and discrete-time error correction for the five-qubit code. For the continuous-time error correction simulations, the codeword fidelity was averaged over 2000 trajectories with $\kappa =100\gamma$ and ${\lambda }_{\text{max}}=200\gamma$.