Optimum quantum error recovery using semidefinite programming

Quantum error correction (QEC) is an essential element of physical quantum information processing systems. Most QEC efforts focus on extending classical error correction schemes to the quantum regime. The input to a noisy system is embedded in a coded subspace, and error recovery is performed via an operation designed to perfectly correct for a set of errors, presumably a large subset of the physical noise process. In this paper, we examine the choice of recovery operation. Rather than seeking perfect correction on a subset of errors, we seek a recovery operation to maximize the entanglement fidelity for a given input state and noise model. In this way, the recovery operation is optimal for the given encoding and noise process. This optimization is shown to be calculable via a semidefinite program, a well-established form of convex optimization with efficient algorithms for its solution. The error recovery operation may also be interpreted as a combining operation following a quantum spreading channel, thus providing a quantum analogy to the classical diversity combining operation.

Figure 1

Transform from standard quantum error correction to quantum spreading channel. By considering the error channel $\mathcal{E}$ together with the encoding $U_C$ and the recovery operation $\mathcal{R}$ with the decoder $U_C^{†}$, the dimension of the SDP may be reduced by a factor of ${(\dim \mathcal{K}∕\dim \mathcal{H})}^2$. Note that the output of the encoder $U_C$ and the recovery channel $\mathcal{R}$ live on the code subspace $\mathcal{C}$.

Figure 2

(Color online) Entanglement fidelity vs $\gamma$ for the five qubit code and the amplitude damping channel ${\mathcal{E}}_a^{\otimes 5}$. $\gamma$ refers to the damping parameter of the channel. Entanglement fidelity for a single qubit and no error correction [i.e., $F_{\mathrm{ent}}(\rho ,{\mathcal{E}}_a)$] is included for comparative purposes.

Figure 3

(Color online) Entanglement fidelity vs $\gamma$ for the four qubit code of Leung et al. [8] and the amplitude damping channel ${\mathcal{E}}_a^{\otimes 4}$. Entanglement fidelity for a single qubit and no error correction [i.e., $F_{\mathrm{ent}}(\rho ,{\mathcal{E}}_a)$] is included for comparative purposes.