One-Way Quantum Repeater Based on Near-Deterministic Photon-Emitter Interfaces

We propose a novel one-way quantum repeater architecture based on photonic tree-cluster states. Encoding a qubit in a photonic tree cluster protects the information from transmission loss and enables long-range quantum communication through a chain of repeater stations. As opposed to conventional approaches that are limited by the two-way communication time, the overall transmission rate of the current quantum repeater protocol is determined by the local processing time enabling very high communication rates. We further show that such a repeater can be constructed with as little as two stationary qubits and one quantum emitter per repeater station, which significantly increases the experimental feasibility. We discuss potential implementations with diamond defect centers and semiconductor quantum dots efficiently coupled to photonic nanostructures and outline how such systems may be integrated into repeater stations.

Popular Summary

Long-distance quantum communication, performed by sending photons between different places, is very challenging because of losses that diminish the quantum signal. To overcome these losses, a quantum repeater is needed, which consists of intermediate stations between the start and end point. In a one-way quantum repeater, quantum information is encoded in a suitable error-correcting code that is a special (entangled) state of many photons. This makes it possible to recover the encoded quantum information even if some of the photons are lost. We propose a one-way repeater protocol that can be implemented with significantly fewer resources than previous proposals, thus bringing one-way repeaters within reach of current experimental systems.

In particular, we show that our protocol can be constructed with as few as two spin qubits and one quantum emitter per repeater station. This is sufficient to reach communication rates on the order of 0.1 MHz over a 1000-km distance. Importantly, many of the required parameters are not far from current state-of-the-art performances, and our protocol could be implemented with diamond defect centers and semiconductor quantum dots efficiently coupled to photonic nanostructures.

Our proposal outlines a feasible route toward high-rate and long-distance quantum communication. It may serve as a motivation and future milestone for the further development of photonic quantum hardware based on solid-state spin-photon interfaces.

Figure 1

Sketch of a one-way quantum repeater with photonic tree clusters. A message qubit (orange dot) is encoded through a Bell measurement (dashed box) with the root spin qubit of a photonic tree cluster. An example with a [2,3,2] tree is shown where dots correspond to photonic qubits and solid lines indicate correlations. The encoded qubit (depicted in the orange boxes) is sent a distance $L_0$ to the next repeater station. At the next repeater station, the qubit information is reencoded through a Bell measurement between one of the first-level photonic qubits and the root spin qubit of a new tree. The remaining qubits of the incoming tree are measured with single qubit $z$ (red circle) or $x$ (orange circle) measurements. The reencoding can succeed despite multiple photons being lost in transmission (gray dots and dashed lines). At the end station, the message qubit can be transferred to, e.g., a receiving spin qubit by means of a controlled-phase (cz) gate.

Figure 2

Sketch of the principle behind counterfactual error correction in the tree-cluster encoding. A [2, 2] tree is considered for simplicity (a). A qubit $\alpha |0⟩+\beta |1⟩$ is encoded through a Bell measurement between this qubit and the root spin qubit (b). Loss of qubits (gray vertices) from one branch (c) still allows retrieving the encoded information by measuring at least one remaining second-level qubit in the $x$ basis and the second-level qubits of the intact branch in the $z$ basis (d) despite the fact that we do not get any information from (attempted) measurements on the lost photons (dashed circles).

Figure 3

Generation of photonic tree-cluster states. Two memory spin systems ($S_1$, $S_2$) and a quantum emitter ($E$) are all prepared in state $|+{⟩}_s$, and cz gates entangle the spins as shown in (a). In the second step (b), all third-level photons (purple dots) of one sub-branch of the tree cluster are emitted (operation $C$) followed by the emission of the corresponding second-level photon (blue dot), which detaches the emitter from the preliminary tree cluster (operation $M$). Operations $M$ and $C$ can be implemented using a sequence of $\pi$ pulses on various transitions in the emitter, as indicated by the color code in (b) referring to the color of the transitions in the emitter level diagram shown in the inset. Here driven transitions are indicated by solid lines, whereas decay ($\gamma$) is represented by the wiggle line. To emit more second-level qubits, $E$ is again prepared in $|+{⟩}_s$ followed by a cz gate with $S_2$ and step (b) is repeated. In step (c), the state of the memory spin is first swapped to the emitter and then emitted as the first-level photon of the branch. Here the emission of the first-level photon is achieved by an operation $\mathit{M}$, which is similar to $M$ except that the red $\pi$ pulse and subsequent decay is replaced by a weak driving of the $|2{⟩}_s-|e{⟩}_s$ transition to ensure that the photons have a narrow bandwidth. The steps are then repeated until the entire tree has been emitted.

Figure 4

The basic elements of the reencoding operation at the repeater stations. (a) The reencoding is performed in a loss tolerant manner by first performing a heralded storage of the message qubit in an auxiliary memory spin (dashed box). This is obtained through a spin-photon controlled-phase (cz) gate with a first-level photon of the encoded tree. The transfer is heralded by the detection of the photon in the $x$ basis. If unsuccessful (photon loss), the auxiliary spin is initialized and the operation is tried again with another first-level photon. Once the storage is successful, a deterministic Bell measurement is performed between the auxiliary spin and the root spin qubit of the new tree cluster, thereby completing the reencoding operation. (b) The experimental implementation of the reencoding operation. The spin-photon cz gate is performed by reflecting the photonic qubit off a one-sided cavity (shown as a photonic crystal cavity) strongly coupled to a quantum emitter, as described in the text. The parameters $g$, $\gamma$, and $\kappa$ are the single-photon Rabi frequency of the optical transition, the spontaneous emission rate of the emitter, and the decay rate of the cavity field, respectively. The same physical setup is used to generate the tree-cluster state requiring an extra spin qubit and one auxiliary level ($|2{⟩}_s$) in the quantum emitter (see Fig. 3).

Figure 5

Illustration of an on-chip photonic circuit for performing $z$- and $x$-basis measurements with integrated single-photon detectors. Mach-Zehnder interferometers with fast switching rate, for example, based on electro-optic switching [28, 29, 30, 31, 32, 33], are assumed to switch fully between the top and bottom path (labeled switch) by controlling the relative phase of the two interferometer arms ($\varphi$). For a $z$-basis measurement, the photons are guided directly to the detectors for time-resolved recording. For an $x$-basis measurement, the switch guides the first wave package of the time-bin encoding to the delay line and the second to the short arm. For first-level photons the delay needs to be longer and is replaced with a fiber delay line [26].

Figure 6

(a),(b) Minimized cost parameter $C$ as a function of distance $L$ assuming spin entangling gate times of ${\tau }_{\mathrm{CZ}}=10{\tau }_{\mathrm{ph}}$ (a) and ${\tau }_{\mathrm{CZ}}=100{\tau }_{\mathrm{ph}}$ (b) for tree-cluster state generation. The markers correspond to times symbol, ${\epsilon }_r=0.1‰$; filled circle, ${\epsilon }_r=0.3‰$; circle, ${\epsilon }_r=0.5‰$; square, ${\epsilon }_r=0.1\%$. Here, ${\epsilon }_r$ is the error probability of the reencoding operation at the repeater stations. (c),(d) Corresponding secret bit rate assuming a photon emission time of ${\tau }_{\mathrm{ph}}=1\mathrm{ns}$ and gate time of ${\tau }_{\mathrm{CZ}}=10\mathrm{ns}$ (c) and ${\tau }_{\mathrm{CZ}}=100\mathrm{ns}$ (d). We have restricted the minimization to trees with $n\le 300$ photons and repeater station spacings $\ge 1\mathrm{km}$. We have assumed a detection efficiency of ${\eta }_d=95\%$. For comparison, the secret bit rate of a two-way repeater (dashed line) with similar resources (see main text) is also plotted together with the rate of direct transmission (dot-dashed line) assuming a 1 GHz single-photon source.