Robustness of quantum memories based on Majorana zero modes

We analyze the rate at which quantum information encoded in zero-energy Majorana modes is lost in the presence of perturbations. We show that information can survive for times that scale exponentially with the size of the chain both in the presence of quenching and time-dependent quadratic dephasing perturbations, even when the latter have spectral components above the system's energy gap. The origin of the robust storage, namely the fact that a sudden quench affects in the same way both parity sectors of the original spectrum, is discussed, together with the memory performance at finite temperatures and in the presence of particle exchange with a bath.

Figure 1

A qubit ${\widehat{\rho }}_q$ is encoded into a topological system of size $N$ and stored for a time $t$. Perturbations act on the system during the storage and induce a nonunitary time evolution that destroys the encoded information. We model this process with a decoherence channel ${\mathcal{D}}_t(·)$. A recovery operation attempts at retrieving the information and reconstructs a qubit ${\widehat{\rho }}_q^{\prime }$ which should be as similar as possible to ${\widehat{\rho }}_q$.

Figure 2

Robustness of the quantum memory against Hamiltonian perturbations: ${\mu }_0=0$, ${\Delta }_0=J$ and (a) and (c) ${\mu }_{-}=J$, ${\mu }_{+}=1.5J$; (b) ${\mu }_{-}=2.5J$, ${\mu }_{+}=3J$. Upper panels (a) and (b): Fidelity as a function of time for different chain lengths $N=8$, 16, 24, 32, 40, and 48. Lower panel (c): Memory time $t_0$ as function of $N$ for different values of the fidelity threshold $F_0$: $F_0=0.9995$ blue line (x marker), $F_0=0.999$ green line ($+$ marker), $F_0=0.9985$ red line (triangular marker). We considered $N_{\mathrm{d}}=101$ realizations uniformly distributed in the range $[{\mu }_{-},{\mu }_{+}]$. Data show convergence behavior for $N_{\mathrm{d}}\to \infty$ (see Appendix ) and can therefore be considered as an approximation of the continuum situation.

Figure 3

Robustness of the quantum memory against time-dependent Hamiltonian perturbations. (a) ${\mu }_0=0$, ${\Delta }_0=J$ and ${\mu }_{-}=J$, ${\mu }_{+}=1.5J$. The Hamiltonian time evolution is swapped between $\widehat{H}\left({\mu }_{-}\right)$ and $\widehat{H}\left({\mu }_{+}\right)$ every $\delta t=J^{-1}/4$. $N_{\mathrm{d}}=101$. (b) ${\mu }_0=0$, ${\Delta }_0=2J$. The parameters of ${\widehat{H}}_0+\widehat{V}$ oscillate as $\mu =\overline{\mu }+\delta \mu ·\text{sgn}\left[\sin \left(2\pi \omega t\right)\right]$ and $\Delta =\overline{\Delta }+\delta \Delta ·\text{sgn}\left[\sin \left(2\pi \omega t\right)\right]$, $\omega =2J$, $\overline{\mu }=1.1J$, $\overline{\Delta }=1.9J$. We consider $N_{\mathrm{d}}=144$ realizations, with $\delta \mu$ and $\delta \Delta$ uniformly distributed in the “square” regions $\delta \mu \in [-0.1J,0.1J]$ and $\delta \Delta \in [-0.1J,0.1J]$. For every realization we add a (different) site-dependent and time-dependent randomness in the chemical potential ${\mu }_i\left(t\right)={\mu }_i^{\prime }\{\frac{1}{2}+\frac{1}{2}\text{sgn}\left[\sin \left(2\pi \omega t\right)\right]\}+{\mu }_i^{\prime \prime }\{\frac{1}{2}+\frac{1}{2}\text{sgn}[-\sin \left(2\pi \omega t\right)]\}$. ${\mu }_i^{\prime }$ and ${\mu }_i^{\prime \prime }$ are taken randomly with a flat distribution in $[-0.1J,0.1J]$. Panels show the optimal fidelity as a function of time for different sizes of the chain $N$. (c) Memory time $t_0$ as function of $N$ for different values of the fidelity threshold $F_0$ for the case of (b): $F_0=0.97$ blue line (x marker), $F_0=0.98$ green line ($+$ marker), $F_0=0.99$ red line (triangular marker).

Figure 4

Optimal Gaussian fidelity as a function of time for different system sizes $N$: ${\mu }_0=0$ and ${\mu }_{-}=J$, ${\mu }_{+}=1.5J$. We considered at most $N_{\mathrm{d}}=101$ realizations uniformly distributed in the range $[{\mu }_{-},{\mu }_{+}]$; plotted values for $N_{\mathrm{d}}\to \infty$ are obtained via scaling as $1/N_{\mathrm{d}}$ (see Appendix ).

Figure 5

Upper bound to the optimal fidelity in presence of thermal modes. (Left) Temperature is fixed: ${\beta }^{-1}=2.5J$ and the figure shows the upper bound as a function of time for different lengths of the chain $N=8$, 16, 24, 32, 40, and 48. (Right) Size is fixed: $N=32$ and the figure shows the upper bound as a function of time for different temperatures ${\beta }^{-1}=0.5J$, $1.0J$, $2.5J$, $5J$, and $10J$. We take $N_{\mathrm{d}}=101$ realizations.

Figure 6

Effects of particle losses according to Eq. (13); ${\mu }_0=0$ and $\Gamma =J$. (Left) Upper and lower bound to the optimal fidelity as a function of time, showing a clear exponential scaling. (Right) Upper bound to the optimal fidelity as a function of time for different system sizes: $N=8$, 16, 24, and 32. Results coincide exactly and are indistinguishable.

Figure 7

Dependence of $F_t^{*}$ on $N_{\mathrm{d}}$. These data refer to the topological perturbation of Fig. 2: $N=32$, ${\mu }_0=0$, ${\mu }_{-}=J$, ${\mu }_{+}=1.5J$. The scaling is shown for five times. From up to down: $24J^{-1}$, $54J^{-1}$, $174J^{-1}$, $354J^{-1}$, $474J^{-1}$. Data show a convergence behavior for $N_{\mathrm{d}}\to \infty$.

Figure 8

Dependence of $F_{\mathrm{G}t}^{*}$ on $1/N_{\mathrm{d}}$. These data refer to the topological perturbation of Fig. 2: $N=32$, ${\mu }_0=0$, ${\mu }_{-}=J$, ${\mu }_{+}=1.5J$. The scaling is shown for five times. From up to down: $12J^{-1}$, $24J^{-1}$, $54J^{-1}$, $174J^{-1}$, $354J^{-1}$, $474J^{-1}$. Data show a $N_{\mathrm{d}}^{-1}$ dependence: the $N_{\mathrm{d}}\to \infty$ value is extrapolated with a linear fit (thin red lines).

Figure 9

Singular values of the matrix in Eq. (C4). (Left) The quench Hamiltonians are in the topological phase: ${\mu }_0=0$, ${\mu }_j=J$, ${\mu }_k=1.5J$. (Right) The quench Hamiltonians are not in the topological phase: ${\mu }_0=0$, ${\mu }_j=2.5J$, ${\mu }_k=3.0J$.