Perturbative approach to continuous-time quantum error correction

We present a discussion of the continuous-time quantum error correction introduced by J. P. Paz and W. H. Zurek [Proc. R. Soc. A 454, 355 (1998)]. We study the general Lindbladian which describes the effects of both noise and error correction in the weak-noise (or strong-correction) regime through a perturbative expansion. We use this tool to derive quantitative aspects of the continuous-time dynamics both in general and through two illustrative examples: the three-qubit and five-qubit stabilizer codes, which can be independently solved by analytical and numerical methods and then used as benchmarks for the perturbative approach. The perturbatively accessible time frame features a short initial transient in which error correction is ineffective, followed by a slow decay of the information content consistent with the known facts about discrete-time error correction in the limit of fast operations. This behavior is explained in the two case studies through a geometric description of the continuous transformation of the state space induced by the combined action of noise and error correction.

Figure 1

Normalized operator distance between the first-order approximation of ${\mathcal{D}}_t^{\left(\eta \right)}\circ \mathcal{R}$ [Eq. (24)] and $\mathcal{R}$ as a function of time (solid line) and the same for ${\mathcal{N}}_t\circ \mathcal{R}$ [Eq. (22)] (dashed line). The distances are normalized by $\parallel {\mathcal{L}}_{\text{n}}\circ \mathcal{R}\parallel$, so that the result does not depend on the choice of operator norm. Here we have set $\gamma =50\eta$.

Figure 2

Normalized operator distance between the second-order approximation of $\mathcal{R}\circ {\mathcal{D}}_t^{\left(\eta \right)}\circ \mathcal{R}$ [Eq. (29)] and $\mathcal{R}$ as a function of time (solid line) and the same for $\mathcal{R}\circ {\mathcal{N}}_t\circ \mathcal{R}$ [Eq. (28)] (dashed line). The distances are normalized by $\parallel \mathcal{R}\circ {\mathcal{L}}_{\text{n}}^2\circ \mathcal{R}\parallel$, so that the result does not depend on the choice of operator norm. Here we have set $\gamma =50\eta$.

Figure 3

Plots of the functions $f_k\left(x\right)$ defined in Eq. (31), for $k=1,\cdots ,5$. Both the initial $\sim \frac{x^{k+1}}{(k+1)!}$ and the asymptotic $\sim x-k$ behaviors are visible.

Figure 4

Time dependence of the coefficients in (38) obtained from the solution of the master equation presented in Appendix pp3 for the following values of the ratio $\gamma /\kappa$: 0, 1, 10, and 100.

Figure 5

Decay of the average recovery fidelities $F_t^{\mathcal{I}}$ and $F_t^{\mathcal{R}}$ from Eq. (8) for the three-qubit code exposed to bit-flip noise of strength $\kappa$ and continuous QEC of strength $\gamma$. (a) No additional recovery operation is applied at readout. (b) A final round of QEC is applied at readout.

Figure 6

Comparison between the average recovery fidelities $F_t^{\mathcal{R}}$ (dashed line) and $F_t^{\mathcal{I}}$ (solid line) for the three-qubit code exposed to bit-flip noise of strength $\kappa$ and continuous QEC of strength $\gamma =32\kappa$. The final round of QEC compensates for the fidelity loss in the initial $t\lesssim \frac{1}{\gamma } \left(\kappa t\lesssim 0.03\right)$ transient but not for the subsequent slow decay. The inset shows the quadratic behavior of $F_t^{\mathcal{R}}$ at short times.

Figure 7

The logical Bloch sphere embedded in the larger state space. The logical operator ${\sigma }_i^L$ is, in general, not parallel to the stable or slow-decaying operator ${\tilde{\sigma }}_i$. The angle goes to zero in the limit of strong QEC (or weak noise).

Figure 8

Relative error of the approximation to the fidelity loss $1-F_t^{\mathcal{R}}$ for the three-qubit code discussed in Sec. 4d4, $\epsilon ={(1-F_t^{\mathcal{R}})}_{\text{pert.}}/{(1-F_t^{\mathcal{R}})}_{\text{exact}}-1$. The values are fitted very accurately by $\epsilon =1.63\frac{\kappa }{\gamma }(\kappa t+2.34)$.

Figure 9

Decay of the average fidelity $F_t^{\mathcal{R}}$ for the five-qubit perfect code subject to depolarizing noise of strength $\kappa$ and continuous error correction of strength $\gamma$, computed numerically for several values of the $\gamma /\kappa$ ratio.

Figure 10

Eigenvalues of $\gamma {\mathcal{L}}_{\text{e.c.}}+\frac{\kappa }{4}{\mathcal{L}}_{\text{n}}$, the Lindbladian superoperator for the five-qubit code subject to depolarizing noise of strength $\kappa$ and continuous error correction of strength $\gamma$. The 1024 eigenvalues cluster into eight distinct values (the nondegenerate zero eigenvalue is not shown). As $\gamma$ is increased, one of the eigenvalues drops like ${\kappa }^2{\gamma }^{-1}$ [blue (dark gray)], while the other 1020 increase like ${\gamma }^{+1}$ [(red (light gray)].

Figure 11

Relative error $\epsilon$ of the approximation to the fidelity loss $1-F_t^{\mathcal{R}}$ for the five-qubit code discussed in Sec. 5c: $\epsilon ={(1-F_t^{\mathcal{R}})}_{\text{pert.}}/{(1-F_t^{\mathcal{R}})}_{\text{num.}}-1$. We computed the relative error for several values of $\kappa /\gamma$ and $t$. The values are fitted very well by $\epsilon =7.9\frac{\kappa }{\gamma }(\kappa t+0.94)$.