Robust implementations of quantum repeaters

We show how to efficiently exploit decoherence free subspaces (DFSs), which are immune to collective noise, for realizing quantum repeaters with long-lived quantum memories. Our setup consists of an assembly of simple modules and we show how to implement them in systems of cold, neutral atoms in arrays of dipole traps. We develop methods for realizing robust gate operations on qubits encoded in a DFS using collisional interactions between the atoms. We also give a detailed analysis of the performance and stability of all required gate operations and emphasize that all modules can be realized with current or near future experimental technology.

Figure 1

Illustration of the entanglement purification scheme. Pairs of entangled flying qubits (open circles) are created via entanglement swapping. These pairs are then used to purify an entangled pair of DFS qubits (solid circles; see text). The framed steps (iv)–(vi) are repeated until a desired entanglement fidelity is achieved.

Figure 2

The quantum repeater scheme. The dashed boxes indicate the subroutine shown in Fig. 1. The created entangled DFS qubit pairs are used to purify further DFS qubit pairs [in line (iii)]. The resulting pairs are then used to purify the pairs of line (iv), and so on.

Figure 3

Quantum circuit for the state transfer of the auxiliary qubit to the DFS qubit.

Figure 4

Quantum circuit for entanglement purification. The procedure is performed on both sides, $A$ and $B$.

Figure 5

Quantum circuit for a CNOT operation between two logical qubits.

Figure 6

Measurement of the logical qubit.

Figure 7

Schematic picture of a repeater node. The four register atoms per qubit (black circles) are stored in a one-dimensional array of dipole traps, e.g., an optical lattice. Between the third and fourth atom there is a free lattice site in which the auxiliary atom (gray) can be moved, e.g., via an optical conveyor belt.

Figure 8

Numerical studies of the phase difference $\varphi$ between states ${\mid 0⟩}_L$ and ${\mid 1⟩}_L$ after applying the rotation $R_L^z\left(\pi \right)$ versus the ratio $U∕J$ calculated with the full Hamiltonian Eq. (15) and the parameters Eqs. (18, 19) given by adiabatic elimination.

Figure 9

Fidelity Eq. (27) for the rotation $R_L^z\left(\pi \right)$. In (a) the hopping term between the second and third atom is detuned, in (b) the interaction between register atoms of kind $a$ and $b$. The other parameters are (a) $U_{a,b}=U_{ab}=100J_1$, (b) $U_{a,b}=U=100J$.

Figure 10

Numerical studies for a fast rotation with no or only weak interaction between the atoms. The fidelity Eq. (27) of the rotation $R_L^z\left(\pi \right)$ and the phase difference between the two logical qubits after this rotation versus the ratio $U∕J$ is calculated with the full Hamiltonian Eq. (15). The accuracy of the gate gets better with a smaller interaction $U$ between the atoms. Note that here no adiabatic elimination has been used.

Figure 11

Numerical studies of the fidelity Eq. (28) of the rotation $R_L^x\left(\pi \right)$ versus the ratio $U∕J$ calculated with the full Hamiltonian Eq. (15) and the parameters Eqs. (23, 24) given by adiabatic elimination.

Figure 12

Fidelity Eq. (28) for the rotation $R_L^x\left(\pi \right)$. In (a) the hopping term between the second and third atom is detuned, in (b) the interaction between register atoms of kind $a$ and $b$. The other parameters are (a) $U_{a,b}=U_{ab}=100J_1$, (b) $U_{a,b}=U=100J$.

Figure 13

(a) Numerical results for the fidelities Eqs. (30, 31), where the auxiliary atom has a nonvanishing rest interaction with the register atoms. (b) Numerical results for the fidelity Eq. (35) due to the phase difference, where the auxiliary atom has a nonvanishing rest interaction with the register atoms. In both cases we took for the register atoms $U∕J=100$.

Figure 14

(a) Fidelity after evolving the state for a time $t=\pi \hslash ∕\sqrt{2}J$ for the case of noninteracting register atoms. (b) Fidelity Eq. (35) of state ${\mid 1⟩}_L$ after the time evolution, where a perfect fidelity of the atom swap operation has been assumed and the error is only due to the phase difference, see text.