Engineering autonomous error correction in stabilizer codes at finite temperature

We present an error-correcting protocol that enhances the lifetime of stabilizer code–based qubits which are susceptible to the creation of pairs of localized defects (due to stringlike error operators) at finite temperature, such as the toric code. The primary tool employed is periodic application of a local, unitary operator, which exchanges defects and thereby translates localized excitations. Crucially, the protocol does not require any measurements of stabilizer operators and therefore can be used to enhance the lifetime of a qubit in the absence of such experimental resources.

Figure 1

A snapshot of a linear stabilizer code with detection and correction operators $O$. The optimal error correction operator has nontrivial system size scaling and in general depends on the particular code being used.

Figure 2

Circuit for performing error suppression for a subregion of the 1D Ising model. Qubits $q_1$ through $q_4$ are lattice sites on an Ising chain. $s_1$ through $s_3$ are ancillas used to read out the syndrome measurements of $ZZ$ on the nearest-neighbor Ising lattice sites. Based on the results of the syndrome measurements, the conditional unitary operator $CU$ corrects the errors present. A table that defines $CU$ is provided in Appendix pp1. This entire circuit represents a possible realization of one such operator $O$ from Fig. 1.

Figure 3

Sequence of $\mathrm{DSWAP}\mathrm{s}$, denoted $C$, for a $\lambda =3 \lambda$-mixing protocol. If a pair of domain walls exist anywhere between sites $q_1$ through $q_7$, then they will necessarily be brought adjacent to each other by this sequence of swaps. Gates are applied sequentially with waiting time $1/\chi$ between each gate.

Figure 4

The same sequence from Fig. 3 but shown acting on domain-wall variables. Here, it is clear that the sequence of $\mathrm{DSWAP}\mathrm{s}$ is designed not to mix domain walls between the two regions of size $\lambda =3$. Site $b_1$ sits between $q_1$ and $q_2, b_2$ between $q_2$ and $q_3$, etc.

Figure 5

One possible snapshot of the error correction process. (a) A system with two domain walls present, each sitting in adjacent $\lambda$-domains. (b) The state of the system after the protocol has been applied—domain walls have been shuttled to the shared boundary. In (c), the bath dissipates the domain walls, and the system returns to the ground state.

Figure 6

Measured lifetime of the Ising model, ${\mathrm{\Gamma }}^{-1}$, expressed in terms of inverse units of ${\mathrm{\Gamma }}_0$ (i.e., Eq. (13)) as a function of the correction rate, $\chi$, rescaled by the translation rate, ${\gamma }_0$, for different values of $\lambda$ and for fixed system size $L=96$, temperature, $T=.07$, and translation rate, ${\gamma }_0=.0007$. Protocols were implemented in parallel on $48/\lambda$ blocks (see text). In the absence of the protocol, the lifetime of the Ising model for these parameters corresponds to approximately ${\mathrm{\Gamma }}_0^{-1}$, i.e., Eq. (13). This is the value which all three protocols converge toward in the limit of $\chi /{\gamma }_0\ll 1$. Note the decrease in lifetime for $\chi /{\gamma }_0\approx 1$, as well as the universal scaling in the lifetime up until the $\lambda$-dependent cutoff (i.e., Eq. (17)), which increases as $\lambda$ decreases. The only remaining mismatch between the curves is due to the residual $\lambda$ dependence in the $1/(L-2-2\lambda )$ term of Eq. (15).

Figure 7

Lifetime of the Ising model as a function of the correction rate, $\chi$, rescaled by the translation rate, ${\gamma }_0$, for different values of $\lambda$, and for fixed system size, $L=96$, temperature, $T=.07$, and translation rate, ${\gamma }_0=.0007$. Protocols were implemented serially (see text). The scaling of lifetime with $\chi$ is characteristically similar to the parallel case; however, the maximal lifetime is correspondingly smaller for the serial implementation. Note that smaller $\lambda$ still yields a larger enhanced lifetime.

Figure 8

Lifetime of the Ising model as a function of the correction rate, $\chi$, rescaled by the translation rate, ${\gamma }_0$, for different values of system size, $L$, and for fixed $\lambda =3, T=.07, {\gamma }_0=.0007$. Protocols were implemented in parallel on $L/2\lambda$ blocks (see text). Note the linear scaling in $\chi$ for small values, as well as the shift in the maximum of the lifetime as a function of $L$.

Figure 9

This figure contains the same data as Fig. 8, but with the $\chi$ axis rescaled to $\chi L$, and the $1/\mathrm{\Gamma }$ axis rescaled by Eq. (14). Hence, the linear scaling in $\chi$, and the slight residual system size dependence have been removed. Note the steep, sudden dropoff in lifetime after $\chi L\sim 55$.

Figure 10

The critical cycling rate, ${\chi }_c$, rescaled by the translation rate, ${\gamma }_0$, as a function of $\frac{1}{L}$ for $\lambda =3, T=0.07, {\gamma }_0=0.0007$. Protocols were implemented in parallel on $L/2\lambda$ blocks. This scaling is consistent with the error model ansatz in Eq. (17). Fit to $1/L$ in red. Errors are dominated by systematic effects, not sampling error.

Figure 11

A single random application of the bath-hopping operator causes a defect to move between adjacent regions of size $\lambda$. Once a pair of defects are separated by this distance, the protocol will not be able to correct them with certainty.

Figure 12

A single random application of the bath-hopping operator causes a defect to move between nonintersecting regions of size $\lambda$, depicted in (a) and (b). (c) A new lattice tiling where no single swap can move defects into two $\lambda$ regions that do not share a boundary.

Figure 13

An example of the result of applying a winning sequence $M\left(G\right)$ to a graph with defect pairs present. No matter where the defects are, the sequence of DSWAPs brings pairs adjacent, whereupon they immediately fuse.