Relaxation dynamics of the toric code in contact with a thermal reservoir: Finite-size scaling in a low-temperature regime

We present an analysis of the relaxation dynamics of finite-size topological qubits in contact with a thermal bath. Using a continuous-time Monte Carlo method, we explicitly compute the low-temperature nonequilibrium dynamics of the toric code on finite lattices. In contrast to the size-independent bound predicted for the toric code in the thermodynamic limit, we identify a low-temperature regime on finite lattices below a size-dependent crossover temperature with nontrivial finite-size and temperature scaling of the relaxation time. We demonstrate how this nontrivial finite-size scaling is governed by the scaling of topologically nontrivial two-dimensional classical random walks. The transition out of this low-temperature regime defines a dynamical finite-size crossover temperature that scales inversely with the log of the system size, in agreement with a crossover temperature defined from equilibrium properties. We find that both the finite-size and finite-temperature scaling are stronger in the low-temperature regime than above the crossover temperature. Since this finite-temperature scaling competes with the scaling of the robustness to unitary perturbations, this analysis may elucidate the scaling of memory lifetimes of possible physical realizations of topological qubits.

Figure 1

The vertex $\left(A_v\right)$ and plaquette $\left(B_p\right)$ operators of the toric code as defined in (2). Edges marked in red are operated on by ${\sigma }^x$ while those marked in blue are operated on by ${\sigma }^z$.

Figure 2

A torus with a self-correcting error string (left) and an uncorrectable error string (right).

Figure 3

A depiction of the operation of the Lindblad operators in the master equation describing the interaction of the toric code with an external bath, as defined in (10).

Figure 4

The eight starting geometries (translucent green, blue) for Monte Carlo simulations of random walks. The solid green site denotes the origin at which the “fixed” quasi-particle sits. Blue configurations were sampled twice as often as green configurations, owing to the different likelihoods of different starting geometries. The simulation was terminated when the traveling quasiparticle reached one of the translucent red vertices—i.e., an annihilation geometry.

Figure 5

Probability of annihilation $p^{\mathrm{t}}\left(n\right)$ after $n$ steps on the torus for the model described in Sec. 4, as computed via Monte Carlo. We see that the value of $p^{\mathrm{t}}\left(n\right)$ agrees with the planar value $p^{\mathrm{p}}\left(n\right)$ [see (34) and 46] up until a characteristic value of $n$ where it is possible for the walker to make topologically nontrivial walks on a finite-size lattice.

Figure 6

The finite-size scaling of the characteristic departure $\left(n_d\right)$ and crossing $\left(n_c\right)$ times from an analysis of the Monte Carlo data shown in Fig. 5. The lines represent the best-fit power laws to the three largest system sizes.

Figure 7

The probability of topologically nontrivial annihilations on a torus, $P_{2\mathrm{D}}^{\Omega }\left(L\right)$ as a function of system size, as computed by Monte Carlo simulations. Also shown are the bounds $P_c$ and $P_d$, which are computed from $p^{\mathrm{t}}\left(n\right)$ and $p^{\mathrm{p}}\left(n\right)$, as described in the text. The lines represent fits to ${(lnL)}^{-1}$ of the three largest system sizes.

Figure 8

Time evolution of the expectation value of the ${\Pi }_{++}$ operator (defined in Sec. 5) computed via Monte Carlo, where $1/4$ has been subtracted to reveal the exponential decay. These simulations were initialized to a pure ground state with $L=128$ with $T=\{0.02,0.08,0.14,0.2\}$, respectively, for (a)–(d). The black lines represent exponential fits to the Monte Carlo data. Note the stretched exponential behavior for early times in (c). ${\gamma }_0$ has been set to 1.

Figure 9

Ground-state relaxation rates as a function of temperature for system sizes $L=\{16,32,64,128\}$ corresponding to (a)–(d), respectively. The solid lines are the low-temperature phenomenological model, $T_{\mathrm{C}}$ (26), and the high-temperature fit, $T_H$ (27). The vertical dashed line represents the dynamical crossover temperature $T_{\mathrm{dyn}}^{*}$. Note that ${\Gamma }_{++}\left(T\right)$ is a monotonic, increasing function of $T$; the rescaling by $exp\left(-\Delta /T\right)$ generates the nonmonotonicity. $\Delta$ has been set to 1.

Figure 10

Low-temperature regime: Finite-size scaling of ${\Gamma }_{++}$ in the low-temperature regime where we have collapsed the temperature dependence of the data according to (26). The solid lines are the low-temperature model $\left(T\right)$ predictions as described in (28); these lines nearly completely overlap due to the weak residual temperature dependence in (26). The dotted line indicates the best fit to purely linear scaling in $L$ which is expected in the high-temperature regime $\left(T_{\mathrm{H}}\right)$.

Figure 11

High-temperature regime: finite-size scaling of ${\Gamma }_{++}$ in the high-temperature regime. The solid line represents a fit of all high-temperature data $\left(T_{\mathrm{H}}\right)$ to a linear scaling in $L$. The dotted line indicates a best fit to the polylog scaling $L^2/lnL$, which is expected in the low-temperature regime $\left(T_{\mathrm{C}}\right)$.

Figure 12

The dynamical crossover temperature $T_{\mathrm{dyn}}^{*}$ as a function of system size $L$. The line represents the fit to $ln{\left(L\right)}^{-1}$ scaling for the largest system sizes. Also shown is the equilibrium crossover temperature $T_{\mathrm{eq}}^{*}$ as defined in Ref. [5].

Figure 13

Various configurations of domain walls in one dimension. (a) An “annihilation geometry,” where the domain walls (in red) are separated by a single spin. (b) A “free” pair of domain walls. This configuration will annihilate trivially with probability 1/2. (c) Domain walls are further separated. (d) Domain walls are separated by half the system size. The rightmost domain wall is the same as the leftmost domain wall due to periodic boundary conditions.