Self-correcting quantum memory in a thermal environment

The ability to store information is of fundamental importance to any computer, be it classical or quantum. To identify systems for quantum memories, which rely, analogously to classical memories, on passive error protection (“self-correction”), is of greatest interest in quantum information science. While systems with topological ground states have been considered to be promising candidates, a large class of them was recently proven unstable against thermal fluctuations. Here, we propose two-dimensional (2D) spin models unaffected by this result. Specifically, we introduce repulsive long-range interactions in the toric code and establish a memory lifetime polynomially increasing with the system size. This remarkable stability is shown to originate directly from the repulsive long-range nature of the interactions. We study the time dynamics of the quantum memory in terms of diffusing anyons and support our analytical results with extensive numerical simulations. Our findings demonstrate that self-correcting quantum memories can exist in 2D at finite temperatures.

Figure 1

Quantum memory based on the toric code. Illustrated is an $8\times 8$ lattice (periodic boundary conditions) with a total of 128 spins-$\frac{1}{2}$ [gray (smaller) circles] on its edges. The four-body plaquette and star operators are indicated in the background. A particular choice for all logical operators $X_1,Z_1,X_2$, and $Z_2$ is shown, although we will focus only on the decay of $Z_1\equiv Z$ (see main text). A number of spins is affected by ${\sigma }_x$ errors (solid dots), which leads to excited plaquettes, or “plaquette anyons” (striped plaquettes). Measurement of the plaquette operators yields the positions of the excited plaquettes, but reveals no information about how they were originally paired or which path (indicated by the framed plaquettes) they took. A minimum-weight error-correction procedure (see Sec. 3b) applies ${\sigma }_x$ operators to the spins marked by the larger orange circles. While the vertically striped green anyons are annihilated properly (with a trivial loop of errors remaining from the top pair and no error from the bottom pair), the horizontally striped red pair is connected around a topologically nontrivial loop on the torus. Although this last pair is annihilated as well, an uncorrected ${\sigma }_x$ error remains on the logical $Z$ string, having thereby introduced a logical error in the state stored in the memory.

Figure 2

Decay of the logical $Z$ operator in the noninteracting toric code. The simulation data are obtained for grid sizes $L$ increasing by powers of 2, from 16 (dotted blue) to 512 (solid red). All curves are ensemble averages over ${10}^4$ runs. The main plot displays $\langle Z_{\mathrm{ec}}\rangle$, which is the average value of $Z$ one would find if an error-correction scheme would be applied at the readout time $t$. The inset shows the expectation value of the bare (uncorrected) logical $Z$ operator. We have used $T/J=0.3$, and $\gamma (0)=\gamma (2J)$. See Sec. 3b for further details on the simulation.

Figure 3

Average of the corrected operator $Z_{\mathrm{ec}}$ for a model with independent ${\sigma }_x$ errors that occur with probability $f$ at each spin. The dashed-dotted, dashed, and solid curves refer to our numerical simulations with lattice sizes $L=40,100,200$, respectively. The error correction fails at a value $f_c\simeq 0.1$, which is slightly smaller than the value $0.11$ from Ref. [3]. In the inset, we plot the value of $f_c$ from simulations of the noninteracting toric code in contact with a bath at temperature $T$ and $\gamma (0)=\gamma (2J)$. The fraction $f_c$ is extracted at the time $\tau$ when $\langle Z_{\mathrm{ec}}\rangle$ decays to zero in the limit of large $L$ (see Fig. 2). This value is always smaller than $f=0.11$ and depends on $T$.

Figure 4

The values of $\tau$ extracted at the sharp transitions of the $\langle Z_{\mathrm{ec}}\rangle$ decay (circles). As in Fig. 2, we use $\gamma (0)=\gamma (2J)$. Comparison to Eq. (5) (dotted curve) gives good agreement for $f_c\simeq 0.1$.

Figure 5

Comparison of the equilibrium value of $N$ obtained numerically (crosses) with $N_{\mathrm{MF}}$ (curves) for different grid sizes. We have used the interaction exponents $\alpha =0$ (solid line), $\alpha =0.5$ (dashed line), and $\alpha =1.0$ (dotted line), and the temperature $T/J=0.5$.

Figure 6

Thermal stability of the interacting memory. The data in the top (bottom) panel were obtained for an Ohmic (super-Ohmic $n=2$) bath. Plotted as a function of $L$ are the numerically simulated times at which the expectation values of the bare (squares) and error-corrected (diamonds) logical $Z$ operator have decayed from $1$ to $0.9$. The dotted lines serve as a guide to the eye. The red dashed-dotted curves are calculated from Eq. (12) with $f_c=0.11$, where we have used the self-consistent values of $n_{\mathrm{MF}}$ and ${\epsilon }_{\mathrm{MF}}$ from Eqs. (6) and (8). Similarly, the green dashed lines are also due to Eq. (12), but here, $f_c$ is fit to the numerical data of the 90% threshold times, yielding $f_c=0.022$ for an Ohmic and $f_c=0.007$ for a super-Ohmic bath. The inset shows the decay of $\langle Z_{\mathrm{ec}}\rangle$ with time for $L=8,\dots ,128$ (from left to right), and the 90% threshold is illustrated by the dotted line. It is seen that to choose this particular value has no substantial influence on the scaling behavior with $L$. Parameters used in these simulations were $A/J=0.1$, and $T/J=0.3$. Times are in units of $({\kappa }_{1}J){}^{-1}$ and $({\kappa }_2{J}^2){}^{-1}$ for the first and second panels, respectively.

Figure 7

Short-time dynamics of the interacting memory in a super-Ohmic bath. In this case, the memory is in the nonsplit-pair regime. The curves refer to different values of $L$, which increase in powers of 2 from $L=64$ (lowest curves in both panels) to $L=2048$ (highest curves). Upper panel: The time dependence of the anyon number $N$ obtained from the simulations (solid lines) is compared to the solutions of Eq. (15) (dashed lines). The crosses are the exact values $N^{*}$ obtained from the partition function of pairs Eq. (16). Good agreement with $N^{*}$ is also obtained for the lower curves at longer times (not shown). Lower panel: The expectation value of the bare $Z$ obtained from the simulations (solid lines) is compared to $e^{-N^{*}/2L}$ (dashed lines), where $N^{*}(t)$ is obtained from the upper plot. Parameters used were $A/J=0.1$, and $T/J=0.3$. The time axes are in units of $({\kappa }_2{J}^2){}^{-1}$.

Figure 8

Thermal stability of the interacting memory with $\alpha \ne 0$ and an Ohmic bath. Data points refer to the numerically calculated times at which the error-corrected logical $Z$ operator has decayed from $1$ to $0.9$ in the cases $\alpha =0$ (diamonds), $\alpha =0.5$ (triangles), and $\alpha =1$ (squares). Note that we have replotted the data from $\alpha =0$ merely for comparison. The dashed lines are from Eq. (12) (as in Fig. 6), with a fit of $f_c$ yielding $f_c=0.027$ for $\alpha =0.5$ and $f_c=0.032$ for $\alpha =1$. Inset: Decay of $\langle Z_{\mathrm{ec}}\rangle$ as a function of time for different grid sizes, $L=8,16,\dots ,256$ (left to right), and $\alpha =1$. Parameters used in the simulations were $A/J=0.1$, and $T/J=0.3$. Times are in units of $({\kappa }_{1}J){}^{-1}$.

Figure 9

Decay of the bare and corrected expectation values of $Z$ due to a single pair of anyons in the memory. The dots show numerical data (averaged over ${10}^4$ samples), while the two curves are the continuum limit expressions Eqs. (B3) and (B5) for $\langle Z_{\mathrm{ec}}\rangle$ (solid) and $\langle Z\rangle$ (dashed). The numerical data have been obtained for $L=32,64,128$. All points collapse onto each other when plotted as a function of $\gamma (0)t/L^2$.