Protecting information via probabilistic cellular automata

Probabilistic cellular automata describe the dynamics of classical spin models, which, for sufficiently small temperature $T$, can serve as classical memory capable of storing information even in the presence of nonzero external magnetic field $h$. In this article, we study a recently introduced probabilistic cellular automaton, the sweep rule, and map out a region of two coexisting stable phases in the $(T,h)$ plane. We also find that the sweep rule belongs to the weak two-dimensional Ising universality class. Our work is a step towards understanding how simple geometrically local error-correction strategies can protect information encoded into complex noisy systems, such as topological quantum error-correcting codes.

Figure 1

Phase diagram of the sweep rule with noise parameters $p,q\in [0,1]$ in the $(T,h)$ plane, where $T=p+q$ and $h=(p-q)/(p+q)$ can be interpreted as temperature and bias. The shaded region corresponds to a region, where two stable phases coexist and the sweep rule can work as memory. The diagram is symmetric with respect to the $h=0$ axis.

Figure 2

(a) Neighborhood of the vertex $v$ comprises the highlighted spins $s_1,s_2,s_3,s_4$ situated on the (light and dark green) triangular faces. The deterministic rule updates $s_2,s_3$ (dark green) based on the restriction of the domain wall on the thick black edges. (b) Different configurations of the restricted domain wall (thick red edges) at time $t$ and the resulting updated spins at $t+0.5$.

Figure 3

Schematic model of classical memory that protects logical information via a probabilistic cellular automaton. The deterministic update removes noise (random spin-flips) added to the system in the probabilistic update.

Figure 4

Evolution of magnetization of the PCA with $N=512$ spins on a $16\times 16$ triangular lattice for $h=0$ at: (a), (d) low, (b), (e) moderate, and (c), (f) high temperatures. The plots in (d)–(f) depict magnetization after it is smoothened (see Appendix pp1 for details). Logical bit-flips are noted as occurring at time steps when the value of smoothened magnetization is zero.

Figure 5

Snapshots of spin configurations on a $128\times 128$ triangular lattice with $N=32768$ spins at $h=0$. (a) At low temperature $T=0.12$, small clusters of $-1$ spins (black) are isolated from each other. (b) At $T=0.19$, large clusters of $-1$ spins of arbitrary size appear, indicating critical behavior. As the system evolves, these clusters coalesce and cause logical bit-flips. (c) At high temperature $T=0.30$, the spin configuration is disordered, with roughly equal numbers of $+1$ and $-1$ spins.

Figure 6

Evolution of smoothened magnetization of the PCA with $N=512$ spins on a $16\times 16$ triangular lattice for $h=0.75$ at: (a) low, (b) moderate, and (c) high temperatures. Since $h>0$, the memory lifetime is limited by the lifetime of the logical state 0 (orange lower curve) that was initially encoded with $-1$ spins; the logical state 1 (blue upper curve) does not undergo logical bit-flips.

Figure 7

Plots of memory lifetime ${\tau }_{L,T}$ of the sweep rule as a function of the linear lattice size $L$ at various temperatures $T$. This data is fit into the ansatz in Eq. (6). (a) At $h=0$, we identify the critical temperature $T_c^0=0.197\pm 0.002$ as the point where $b=0$ (inset). The system behaves as a good memory below $T_c^0$. (b) At $h=0.75$, we similarly identify the critical temperature $T_c^{0.75}=0.0670\pm 0.0004$. Additionally, for $T>T_c^{0.75}$ we observe that ${\tau }_{L,T}$ increases with $L$ as long as $L<L_T^{*}$, where $L_T^{*}$ is the linear lattice size maximizing ${\tau }_{L,T}$; further increase of $L$ decreases ${\tau }_{L,T}$.

Figure 8

Critical behavior of the sweep rule at $h=0$. (a) The ensemble average of magnetization $\langle M\rangle$ drops continuously from a nonzero value to zero at the critical temperature, indicating a second-order phase transition. (b) The ensemble average of the magnetic susceptibility $\langle \chi \rangle$ captures the variance of the magnetization, which diverges at the critical temperature. (c) We can identify the critical temperature from the intersection of the plots of the ensemble average of two-point correlation functions $\langle \xi /L\rangle$ for different $L$. (d)–(f) The data collapse of $\langle M\rangle , \langle \chi \rangle$ and $\langle \xi /L\rangle$ into $L$-independent functions $\tilde{M}, \tilde{\chi }$ and $\tilde{\xi }$ respectively, is used to obtain the critical exponents $\beta , \gamma , \nu$ (see Table 1) with critical temperature $T_c^0=0.1973\pm 0.0001$.

Figure 9

The plots of fidelity $F\left(t\right)$ of the logical bit encoded in the PCA at $h=0$. The width of the peak in the $F\left(t\right)$ plot captures the memory lifetime of the PCA at that temperature. (a) At $T=0.40$, the fidelity estimated from simulation agrees very well with the proposed model in Eq. (11). For a system with $N=200$ spins, we obtain parameters $N_{\text{eff}}=38.7\pm 1.2, \lambda =0.0884\pm 0.0001$ from the model, with goodness of fit parameters ${\chi }^2=335$ and degrees of freedom $\text{DOF}=398$. (b) For the same system at a lower temperature $T=0.21$, we obtain the parameters to be $N_{\text{eff}}=1.373\pm 0.006, \lambda =(1.105\pm 0.003)\times {10}^{-3}$ with ${\chi }^2=1181, \text{DOF}=998$. The ratio ${\chi }^2/\text{DOF}$ is found to increase as $T$ is lowered, indicating that the model best captures the behavior of the system at high temperatures [5].

Figure 10

(a) The plot of the mean flipping rate $\lambda$ as a function of $T$. $\lambda$ does not change noticeably with $L$. (b) The plot of the effective number of spins $N_{\text{eff}}$ as a function of $T$. In the range $T\in [0.25,0.5]$, this parameter scales linearly with temperature, as captured by Eq. (14). (c) We plot the slope $a$ and the $x$ intercept of this linear fit as a function of lattice size $L$. The $x$ intercept gives the temperature at which $N_{\text{eff}}=0$, subsequently indicating the breakdown of the model.

Figure 11

(a) Estimating parameters $a,b,c$ from the values obtained by fitting resampled data sets at $h=0.5$ into the ansatz Eq. (6). (b) By constraining $a,c$ to be constant over the range of temperatures we obtain a smoother plot of $b$ as a function of $T$ and, subsequently, can identify the phase transition as the point where $b=0$, i.e., $T_c^{0.50}=0.082\pm 0.001$.

Figure 12

Plots of memory lifetime ${\tau }_{L,T}$ of the sweep rule as a function of linear lattice size $L$. The dashed lines indicate the best fit into the ansatz in Eq. (6). The data in these plots have been fit by taking into account the statistical errors in the numerical estimation of memory lifetime. To avoid clutter, we do not display the corresponding asymmetric error bars.

Figure 13

Plot of parameter $b$ as a function of temperature $T$ for respective values of $h$ corresponding to the fit obtained in Fig. 12. We identify the critical temperature $T_c^h$ at each value of $h$ as the point where $b=0$. In each case, we constrain the parameters $a,c$ to be fixed in the neighborhood of $T_c^h$ and report the values of $a$ (polynomial scaling). The critical temperatures thus obtained are $T_c^0=0.198\pm 0.001, T_c^{0.125}=0.129\pm 0.001, T_c^{0.25}=0.106\pm 0.002, T_c^{0.50}=0.082\pm 0.001, T_c^{0.75}=0.0670\pm 0.0004$ and $T_c^{1.00}=0.0564\pm 0.0004$.

Figure 14

Phase diagram of the Toom's rule in the $(T,h)$ plane. The shaded region corresponds to a region where two stable phases coexist and the Toom's rule can work as memory. The diagram is symmetric with respect to the $h=0$ axis.

Figure 15

Plots of parameter $b$ as a function of temperature $T$ obtained from fitting memory lifetime of Toom's rule into the ansatz in Eq. (6). The critical temperatures obtained are $T_c^0=0.177\pm 0.005, T_c^{0.125}=0.125\pm 0.003, T_c^{0.25}=0.106\pm 0.002, T_c^{0.50}=0.080\pm 0.002, T_c^{0.75}=0.0662\pm 0.0008$, and $T_c^{1.00}=0.0556\pm 0.0006$.

Figure 16

The sweep rule is implemented on a triangular lattice with periodic boundary conditions. A lattice of linear lattice size $L=4$ is shown here. The $N=32$ spins situated at the face centres of the triangles form a honeycomb lattice (black edges) whose wave vector lies along the unit vector ${\widehat{k}}_{\text{min}}$ (red). The displacement vector ${\vec{R}}_{i,j}$ from $s_i$ to $s_j$ is depicted in blue.

Figure 17

Critical behavior of the asynchronous sweep rule at $h=0$. In panels (a)–(c), we plot ensemble averages of magnetization $M$, magnetic susceptibility $\chi$ and two-point correlation function $\xi /L$, whereas in panels (d)–(f), we plot their respective data collapses.

Figure 18

Critical behavior of synchronous Toom's rule at $h=0$. In panels (a)–(c), we plot ensemble averages of magnetization $M$, magnetic susceptibility $\chi$ and two-point correlation function $\xi /L$, whereas in panels (d)–(f), we plot their respective data collapses.

Figure 19

Critical behavior of asynchronous Toom's rule at $h=0$. In panels (a)–(c), we plot ensemble averages of magnetization $M$, magnetic susceptibility $\chi$ and two-point correlation function $\xi /L$, whereas in panels (d)–(f), we plot their respective data collapses.