Iterative Optimization of Quantum Error Correcting Codes

We introduce a convergent iterative algorithm for finding the optimal coding and decoding operations for an arbitrary noisy quantum channel. This algorithm does not require any error syndrome to be corrected completely, and hence also finds codes outside the usual Knill-Laflamme definition of error correcting codes. The iteration is shown to improve the figure of merit “channel fidelity” in every step.

Figure 1

Fidelity of the five-bit code applied to fivefold tensor product of the depolarizing qubit channel (solid line) compared to the fidelity of the depolarizing channel (dashed line).

Figure 2

Comparison of the channel fidelity of no error correction (dotted line), five-bit code (dashed lines), and the iteration (solid line) applied to the fivefold tensor product of depolarizing channel with parameter $p$. For $p>1$ also the fidelity for five-bit encoder combined with optimized decoder is shown.