Implementing stabilizer codes in noisy environments

We propose an effective approach to implement stabilizer codes by using Rydberg blockade in open systems. Along the dynamics pathway toward the doubly Rydberg excited states, it is shown that an arbitrary initial state can be driven to multibody entanglement via stabilizer codes with the help of atomic spontaneous emission. The described system serves as a good candidate for coherent quantum-state manipulation.

Figure 1

The level configuration for each atom. See text for further details.

Figure 2

Fidelity of preparing a three-qubit GHZ state as a function of time for different values of ${\mathrm{\Omega }}_3$. The duration of each step is chosen as $\delta t=22.7/\mathrm{\Omega }$. The other fixed parameters are ${\mathrm{\Omega }}_f=0.1\mathrm{\Omega }, k=1, {\mathrm{\Delta }}_3=-80{\mathrm{\Omega }}_3, {\mathrm{\Delta }}_{3f}=80{\mathrm{\Omega }}_3, \tau =0.01\mathrm{\Omega }$, and ${\mathrm{\Delta }}_{33}={\mathrm{\Delta }}_{ff}=60{\mathrm{\Omega }}_3$.

Figure 3

Variation of the fidelity when $\delta t$ is changed from $22/\mathrm{\Omega }$ to $30/\mathrm{\Omega }$. The other parameters are the same as those in Fig. 2.

Figure 4

Fidelity versus time for various values of the decay rate $\tau$. The other parameters are the same as those in Fig. 2.

Figure 5

Fidelities of (a) three- and (b) four-qubit cluster states from Monte Carlo simulation of the master equation over 400 trajectories. The other parameters are ${\mathrm{\Omega }}_3=10\mathrm{\Omega }, {\mathrm{\Omega }}_f=0.1\mathrm{\Omega }, \tau =0.01\mathrm{\Omega }, k=1, {\mathrm{\Delta }}_2=-80{\mathrm{\Omega }}_3, \delta t=22.5/\mathrm{\Omega }, {\mathrm{\Delta }}_{33}={\mathrm{\Delta }}_{ff}=60{\mathrm{\Omega }}_3$, and ${\mathrm{\Delta }}_{3f}=80{\mathrm{\Omega }}_3$.

Figure 6

Eigenvalues $E_{1,2}$ for state ${|+\rangle }_1{|+\rangle }_2{|-\rangle }_3$ and excited state as functions of ${\mathrm{\Delta }}_{33}$ and $k$ for different cases: (a) ${\mathrm{\Delta }}_{33}={\mathrm{\Delta }}_{ff}<{\mathrm{\Delta }}_{3f}$; (b) ${\mathrm{\Delta }}_{33}={\mathrm{\Delta }}_{ff}>{\mathrm{\Delta }}_{3f}$; (c) ${\mathrm{\Delta }}_{33}<{\mathrm{\Delta }}_{3f}$ and ${\mathrm{\Delta }}_{ff}=90{\mathrm{\Omega }}_3$. The other parameters are ${\mathrm{\Omega }}_3=10\mathrm{\Omega }, {\mathrm{\Omega }}_f=0, {\mathrm{\Delta }}_2=-60{\mathrm{\Omega }}_3, k=1$, and ${\mathrm{\Delta }}_{3f}=60{\mathrm{\Omega }}_3$.

Figure 7

Probability amplitudes for system states versus time with (a) $k=2.58$ and ${\mathrm{\Delta }}_{33}={\mathrm{\Delta }}_{ff}=20{\mathrm{\Omega }}_3$; (b) $k=0.49$ and ${\mathrm{\Delta }}_{33}={\mathrm{\Delta }}_{ff}=80{\mathrm{\Omega }}_3$; (c) $k=5.10, {\mathrm{\Delta }}_{33}=10{\mathrm{\Omega }}_3$, and ${\mathrm{\Delta }}_{ff}=90{\mathrm{\Omega }}_3$. Here ${\mathrm{\Omega }}_f=0.1\mathrm{\Omega }$ and the common parameters are the same as in Fig. 6.