Quantum error correction under numerically exact open-quantum-system dynamics

The known quantum error-correcting codes are typically built on approximative open-quantum-system models such as Born-Markov master equations. However, it is an open question how such codes perform in actual physical systems that, to some extent, necessarily exhibit phenomena beyond the limits of these models. To this end, we employ numerically exact open-quantum-system dynamics to analyze the performance of a five-qubit error-correction code where each qubit is coupled to its own bath. We first focus on the performance of a single error-correction cycle covering timescales and coupling strengths beyond those of Born-Markov models. We observe distinct power-law behavior of the error-corrected channel infidelity $\propto t^{2a}$: $a\lesssim 2$ in the ultrashort times $t<3/{\omega }_{\mathrm{c}}$ and $a\approx 1/2$ in the short-time range $3/{\omega }_{\mathrm{c}}<t<30/{\omega }_{\mathrm{c}}$, where ${\omega }_{\mathrm{c}}$ is the cutoff angular frequency of the bath. Furthermore, the observed scaling of the performance of error correction is found to be robust against various imperfections in physical qubit systems, such as perturbations in qubit-qubit coupling strengths and parametric disorder. For repeated error correction, we demonstrate the breaking of the five-qubit error-correction code and of the Born-Markov models if the repetition rate of the error-correction cycles exceeds $2\pi /\omega$ or the coupling strength $\kappa \gtrsim \omega /10$, where $\omega$ is the angular frequency of the qubit. Our results provide bounds of validity for the standard quantum error-correction codes and pave the way for applying numerically exact open-quantum-system methods for further studies of error correction beyond simple error models and for other strongly coupled many-body models.

Figure 1

Schematic of (a) the single-cycle and (b) the repeated error-correction processes. Here, $\mathcal{U}$ is the encoding process, $\mathcal{E}$ is the error process, and $\mathcal{R}$ is the recovery process including the decoding operation also.

Figure 2

Channel infidelity $1-F_{{\mathrm{\Phi }}_1}$ of the non-error-corrected channel (dashed lines) and the error-corrected channel (solid, dotted and dash-dotted lines) as a function of the time interval $t$ from the encoding to the recovery, i.e., the duration of the error process, for the qubit-bath coupling strength $\kappa /\omega =0.01$. Here we compare the results calculated by SLED (blue), the Lindblad master equation (ME, red), and the analytic short-time methods by Eq. (6) (ST, green). (a) All qubits are coupled to baths, ${\kappa }_j=\kappa$, without internal qubit-qubit coupling, $J_j/\omega =0$. (b) Only a single qubit is coupled to a bath, ${\kappa }_3=\kappa$, and all qubits are coupled to each other with internal qubit-qubit coupling parameters, $J_j=J$, where $J/\omega =0.01$ (dotted lines) and 0.1 (dash-dotted lines). (c) All qubits are coupled to baths and the internal qubit-qubit coupling parameters are $J_j=J$ such that $J/\omega =0$ (solid lines) and 0.1 (dash-dotted lines). Note that the non-error-corrected channel fidelity is independent of the value of $J$ in (b) and (c). The qubits have identical frequencies, ${\omega }_j=\omega$. The baths are also identical with temperatures ${\beta }_j=2$ and cutoff frequencies ${\omega }_{\mathrm{c}\mathrm{j}}/\omega =20$.

Figure 3

Channel infidelity $1-F_{{\mathrm{\Phi }}_1}$ for the non-error-corrected channel (dashed lines) and the error-corrected channel (solid lines) as a function of the length of the time interval $t$ from the encoding to the recovery for a Drude-type cutoff at the frequency ${\omega }_{\mathrm{c}\mathrm{j}}/\omega =10$ (cyan), 20 (blue), and 60 (dark blue). The Lindblad master equation results are independent of ${\omega }_c$. The qubits and baths are assumed identical with ${\omega }_j=\omega , {\kappa }_j=\kappa =0.01\omega , J/\omega =0$, and ${\beta }_j=\beta =2$.

Figure 4

Channel infidelity of the repeated error correction $1-F_{{\mathrm{\Phi }}_{\mathrm{N}}}$ as a function of the number of the error correction cycles, $N$, within a fixed total time $t_{\max }\kappa =1$ for different coupling strengths $\kappa /\omega =0.001$ (yellow), 0.01 (blue), and 0.1 (magenta). The solid lines show the results computed by the SLED and the black dash-dotted line represents the $\kappa$-independent infidelity arising from the Lindblad master equation. We have ${\omega }_j=\omega , {\kappa }_j=\kappa , J_j=J=0, {\beta }_j=\beta =2$, and ${\omega }_{\mathrm{c}j}={\omega }_{\mathrm{c}}=20\omega$.

Figure 5

Channel infidelity $1-F_{{\mathrm{\Phi }}_1}$ for the non-error-corrected channel (dashed lines) and the error-corrected channel (solid lines) as a function of the length of the time interval $t$ from the encoding to the recovery. Here we compare the results calculated by SLED (blue), the Lindblad master equation (ME, red), and the analytic short-time methods (ST, green). The qubits and baths are assumed identical with ${\omega }_j=\omega , {\kappa }_j=\kappa =0.1\omega , J/\omega =0$, and ${\beta }_j=\beta =2$.

Figure 6

Channel infidelity $1-F_{{\mathrm{\Phi }}_1}$ for the non-error-corrected channel (dashed lines) and the error-corrected channel (solid lines) as a function of the length of the time interval $t$ from the encoding to the recovery. We compare the results calculated by SLED (blue) and the Lindblad master equation (ME, red). The qubits and baths are assumed to be nonidentical with an average qubit angular frequency ${\omega }_{\mathrm{avg}}=\omega$ and $20\%$ disorder, an average ${\kappa }_{\mathrm{avg}}=\kappa =0.01\omega$ and $20\%$ disorder, and ${\beta }_j=\beta =2$.