Quantum error precompensation for quantum noisy channels

Most previous efforts on quantum error correction focused on either extending classical error-correction schemes to the quantum regime by performing a perfect correction on a subset of errors or seeking a recovery operation to maximize the fidelity between an input state and its corresponding output state of a noisy channel. There are few results concerning quantum error precompensation. Here we design an error-precompensated input state for an arbitrary quantum noisy channel and a given target output state. By following a procedure, the required input state, if it exists, can be analytically obtained in single-partite systems. Furthermore, we also present semidefinite programs to numerically obtain the error-precompensated input states with maximal fidelities between the target state and the output state. The numerical results coincide with the analytical results.

Figure 1

(a) Comparison of the QECC and QER methods [19] (usually ${\varrho }_{\mathrm{in}}={\varrho }_{\mathrm{t}}$), with (b) the proposed quantum error-pre-compensation model (usually ${\varrho }_{\mathrm{in}}\ne {\varrho }_{\mathrm{t}}$).

Figure 2

The procedure for analytically designing input state ${\varrho }_{\mathrm{in}}$ with a given quantum channel $\mathcal{E}$ and a target state ${\varrho }_{\mathrm{t}}$. If the whole system is just a single-partite system with a quantum channel, as in Eq. (3), we can obtain $M:={\sum }_i{K}_i\otimes K_i^{*}$, $|{\varrho }_{\mathrm{t}}\rangle :={\varrho }_{\mathrm{t}}\otimes \mathbb{1}{\sum }_i|ii\rangle$, $\delta :={\mathrm{Tr}}_B\left(M^{-1}\right|{\varrho }_{\mathrm{t}}\rangle {\sum }_i\langle ii\left|\right)$, and ${\delta }^{\mathrm{\Psi }}:={\mathrm{Tr}}_B\left(\right|{\varrho }_{\mathrm{in}}^{\mathrm{\Psi }}\rangle {\sum }_i\langle ii\left|\right)$.

Figure 3

(a) Comparison of the fidelity $F_d^{\prime }$ using the QEPC scheme and the fidelity $F_d$ using the Shor code. The solid red line denotes the fidelity $F_d^{\prime }$ using the QEPC scheme, and the dashed line denotes the fidelity $F_d$ using the Shor code. (b) Details of (a) when $0.1\ge p\ge 0$.