Rate-distance tradeoff and resource costs for all-optical quantum repeaters

We present a resource-performance tradeoff of an all-optical quantum repeater that uses photon sources, linear optics, photon detectors, and classical feedforward at each repeater node, but no quantum memories. We show that the quantum-secure key rate has the form $R\left(\eta \right)=D{\eta }^s$ bits per mode, where $\eta$ is the end-to-end channel's transmissivity, and the constants $D$ and $s$ are functions of various device inefficiencies and the resource constraint, such as the number of available photon sources at each repeater node. Even with lossy devices, we show that it is possible to attain $s<1$, and in turn outperform the maximum key rate attainable without quantum repeaters, $R_{\mathrm{direct}}\left(\eta \right)=-{log}_2(1-\eta )\approx (1/ln2)\eta$ bits per mode for $\eta \ll 1$, beyond a certain total range $L$, where $\eta \sim e^{-\alpha L}$ in optical fiber. We also propose a suite of modifications to a recently proposed all-optical repeater protocol that ours builds upon, which lower the number of photon sources required to create photonic clusters at the repeaters so as to outperform $R_{\mathrm{direct}}\left(\eta \right)$, from $\sim {10}^{11}$ to $\sim {10}^6$ photon sources per repeater node. We show that the optimum separation between repeater nodes is independent of the total range $L$ and is around 1.5 km for assumptions we make on various device losses.

Figure 1

Attaching a $\{3,2,2\}$ tree to a node of a photonic cluster.

Figure 2

Panels (a) and (b) show schematics of one elementary link, and a chain of them connecting Alice and Bob, respectively, for a repeater architecture that employs quantum memories, Bell pair sources, probabilistic BSMs, and multiplexing over $m$ orthogonal parallel channels. Panel (c) depicts the construction of a photonic cluster state that can subsume the roles of the quantum memory and the Bell pair sources, thereby resulting in a quantum repeater architecture based solely on “flying” qubits. The outer (white) photonic qubits are transmitted on the fiber channels, and the inner (black) qubits are held locally in a (lossy) waveguide at the repeater node. See text for a detailed description.

Figure 3

(a) The naive multiplexing scheme. A dashed rectangle represents a cluster that has some probability of having been created after a probabilistic fusion step (red circle) or at the output of creating GHZ states using linear optics starting from six single photons (labeled “GHZ Factory”). A solid rectangle represents a cluster state that is successfully created with high probability by choosing a successful outcome (blue square) out of several identical copies attempted (dashed boxes). (b) The improved multiplexing scheme. A box surrounding clusters of the same type represents a bank of clusters and any operation applied to the bank is applied to all the clusters in it.

Figure 4

The tree cluster $C^k$ (and the final cluster $C^{km}$ after the $X$ and $Y$ measurements), shown in Fig. 2, are created by a sequence of probabilistic linear-optical fusion-II operations, starting from three-photon maximally entangled (GHZ) states.

Figure 5

The probability that all $n=250$ major nodes are simultaneously successful in creating clusters of size $k=7$ fusion steps (i.e., $2^k+2=130$ photon clusters), using the naive and the improved multiplexing schemes.

Figure 6

The key rate (in bits per mode) $R_n^{(m,\vec{b})}\left(L\right)$ achieved by an $n$-node repeater chain shown as a function of range $L$, for $n=1,10,24,56,133$, and 314 (magenta dotted plots), with $m=4$ parallel channels and $\vec{b}=\{7,3\}$ trees. The analytical lower bound to the rate-distance envelope $R_{\mathrm{LB}}^{(m,\vec{b})}\left(L\right)$ (black solid plot) is seen to surpass the best-possible repeaterless-QKD rate $R_{\mathrm{direct}}\left(L\right)$ (blue dashed plot) at $L=87$ km.

Figure 7

The bits per mode rates $R^{\left(k\right)}\left(L\right)$ plotted for different values of $k$, the numbers of fusion steps, for the (a) naive scheme and (b) with the improvements of this paper. The repeater-less rate bound $R_{\mathrm{direct}}\left(L\right)$ is the pink dashed line. $N_{\mathrm{cluster}}=2^k+2$ is the total number of photons in the cluster generated at each repeater in every clock cycle.

Figure 8

Single-qubit measurements can be applied before fusion operations. (a) $X$ and $Y$ basis measurements can be moved before conditional $Z$ operators. (b) $Z$ operators before $Z$ basis measurements can be removed. (c) Hadamard gates followed by measurement in the $X, Y$, or $Z$ basis is equivalent to direct measurement in a different pauli basis. (d) Single-qubit measurements on the final cluster can be moved before fusion operations.