Figure 1

(Color online) Comparison between the conventional and the repeater protocols in terms of the generation rate of Bell pairs or secret bit pairs (i.e., the sustained bandwidth of the repeater channel). The memory coherence time is assumed to be $t_{\text{coh}}=10\text{s}$, and the nearest-neighbor spacing is $l_0=10\text{km}$. (i) The blue solid curve is the Briegel-Dur-Cirac-Zoller protocol (with the maximum number of qubits per station increasing logarithmically with distance; see scheme C in Ref. [11]). The sharp decrease in rate is attributed to blinded connection and purification [9] when the memory error becomes dominant (i.e., $\text{time}\gtrsim 0.01{\tau }_{\text{coh}}=0.1\text{s}$). (ii) The red dashed curve is the parallel protocol (with the number of qubits per station increasing at least linearly with distance, see scheme B in Ref. [11]). (iii) The black dotted reference curve is the inverse of the classical communication time between the final stations. Since all conventional repeater protocols rely on two-way classical communication, their rates always stay below the reference curve unless parallel or multiplexed [8] repeater channels are used. (iv) The orange horizontal thick line is our repeater protocol with encoding (with the number of qubits per station increasing logarithmically or polylogarithmically with distance). Since our protocol runs in the one-way communication mode, the rate is independent of the communication distance and can reach above the black dashed curve. Our protocol is much more efficient over long distances than conventional protocols.Reuse & Permissions

Figure 2

(Color online) Idealized quantum repeater. There are $L=5$ repeater stations. Each intermediate station has two physical qubits. Step 1 (generation): Bell pairs between neighboring repeater stations are generated. Step 2 (connection): the qubits at the intermediate stations are measured in the Bell basis (see the inset). Step 3 (Pauli frame): the Pauli frame for qubits at the outermost stations is determined based on the outputs of intermediate Bell measurements. Finally, one remote Bell pair between the outermost stations is created.Reuse & Permissions

Figure 3

(Color online) Repeater protocol with encoding. Each repeater station has $2n$ memory qubits (blue dots) and $O\left(n\right)$ auxiliary qubits (gray dots). Here $n=3$. Step 1 (encoded generation): between two neighboring stations (upper-left panel) (i) memory qubits are fault tolerantly prepared in the encoded states $|\tilde{0}⟩$ or $|\widetilde{+}⟩=\frac{1}{\sqrt{2}}\left(|\tilde{0}⟩+|\tilde{1}⟩\right)$, (ii) purified physical Bell pairs are generated between auxiliary qubits (connected gray dots), and (iii) an encoded Bell pair ${|{\tilde{\Phi }}^{+}⟩}_{AB}=\frac{1}{\sqrt{2}}\left({|\tilde{0}⟩}_A{|\tilde{0}⟩}_B+{|\tilde{1}⟩}_A{|\tilde{1}⟩}_B\right)$ between neighboring stations is created using encoded CNOT gate (achieved by $n$ pairwise, teleportation-based CNOT gates [16, 17, 18]). Step 2 (encoded connection): encoded Bell measurements are simultaneously applied to all intermediate repeater stations via pairwise CNOT gates between qubits $a_i$ and $b_i$ followed by measurement of the physical qubits (the lower-left panel). Using classical error correction, the outcomes for the encoded Bell measurement can be obtained with a very small effective logical error probability $Q\sim q^{t+1}$ [Eq. (11)]. The outcome is announced as two classical bits (purple star) at each intermediate repeater station. Step 3 (Pauli frame): according to the outcomes of intermediate encoded Bell measurements, the Pauli frame [14] can be determined for qubits at the outermost stations. Finally, one encoded Bell pair between the final (outermost) stations is created.Reuse & Permissions

Figure 4

(Color online) From Eqs. (11, 13), the maximum number of connections $L^{\ast }$ is estimated as a function of the effective error probability $q$, assuming $F^{\ast }=0.95$, for various CSS codes [20] listed in Table . For $q<0.03$, $L^{\ast }$ scales as $1/q^{t+1}$.Reuse & Permissions

Figure 5

(Color online) Failure probability and unpurified Bell pairs. (a) The failure probability $P_{\text{fail}}$ decreases exponentially with the number of unpurified Bell pairs $N_0$ (when $N_0$ surpasses certain threshold) for $n=7$ (red solid line) and $n=23$ (blue dashed line). (b) For fixed $P_{\text{fail}}$, the ratio $N_0/n\sim 15$ for a wide range of $n$. The four curves from lower left to the upper right correspond to $P_{\text{fail}}={10}^{-3}$, ${10}^{-5}$, ${10}^{-7}$, and ${10}^{-9}$, respectively. For both plots, we assume unpurified Bell pairs with fidelity $F_0=0.95$ due to depolarizing error. The operational error probabilities are $\beta =\delta ={10}^{-3}$. After three levels of purifications, the fidelity of the Bell pair can be 0.9984.Reuse & Permissions