Abstract
The past decade has seen tremendous progress in experimentally realizing the building blocks of quantum repeaters. Repeater architectures with multiplexed quantum memories have been proposed to increase entanglement distribution rates, but an open challenge is to maintain entanglement fidelity over longdistance links. Here, we address this with a quantum router architecture comprising many quantum memories connected in a photonic switchboard to broker entanglement flows across quantum networks. We compute the rate and fidelity of entanglement distribution under this architecture using an eventbased simulator, finding that the router improves the entanglement fidelity as multiplexing depth increases without a significant drop in the entanglement distribution rate. Specifically, the router permits channellossinvariant fidelity, i.e. the same fidelity achievable with lossless links. Furthermore, this scheme automatically prioritizes entanglement flows across the full network without requiring global network information. The proposed architecture uses presentday photonic technology, opening a path to nearterm deployable multinode quantum networks.
Introduction
Quantum networks distribute quantum information to enable functions that are impossible on classical networks. Key to these applications is the sharing of entanglement between many users over large distances, allowing quantum key distribution, distributed quantum computing, and quantumenhanced sensing. While entanglement distribution has been demonstrated over short distances^{1}, longdistance quantum networking is hampered by the exponential loss of photons in optical fibers^{2}. Quantum repeaters^{3} can overcome this problem by forming chains of entangled nodes.
Figure 1a shows a schematic of such a repeaterconnected quantum network. A graph of quantum repeaters connected by quantum links forms the backbone of the network. Client nodes are quantum computers that connect to the network through their nearest repeater node, while repeater nodes facilitate the sharing of entanglement between clients. Hidden under the linklayer abstraction^{4} lies a physical layer of repeater devices and lossy quantum channels, the components of which are illustrated in the Figure. A typical repeater device, shown in Fig. 1b, consists of memory capable of generating and storing entanglement with a photonic mode. The entanglement scheme used in a link between two repeater nodes determines the devices and channels used in the physical layer. Nitrogen vacancy (NV) centers in diamond have been entangled at a distance of 1.3 km^{1} using the emissionbased scheme proposed by Barrett and Kok^{5}, which uses a 50:50 beamsplitter to erase whichpath information and detectors to herald entanglement. A related scheme proposed by Cabrillo^{6} was used to provide ondemand entanglement between NV centers using a phasestabilized fiber link^{7}. Recently, the directtransmission (PLOB) bound^{2} for quantum communication was broken^{8} using a scheme proposed by Duan and Kimble^{9}, which relies on scattering photons from spins in highcooperativity cavities. These protocols are all inherently probabilistic, such that a single attempt at distributing entanglement succeeds with probability p_{distant}.
The latency due to twoway communication for entanglement distribution in firstgeneration repeater networks forms a bottleneck that can be resolved by multiplexing many quantum memories at each repeater node^{10}. To overcome this problem of latency, ref. ^{11} introduced a multiplexing scheme that is restricted to cavitybased entanglement protocols. Alternatively, ref. ^{12} proposed a scheme that is compatible with emissionbased protocols but does not maintain the fidelity of entanglement distribution. Previous papers considered switching to improve quantum key distribution rates, but have not quantified the effects of local connectivity on infidelities^{13,14,15}. Here, we introduce a quantum router scheme that uses multiplexing and alltoall conditional local switching to increase entanglement fidelities and maintain entanglement rates for leading memory types and entanglement protocols. Our router architecture bears similarities to the quantum key distribution protocol introduced in a contemporaneous proposal by Trényi and Lütkenhaus^{16}, but our paper considers spinphoton implementations of quantum repeaters in general quantum networks.
The router uses local, lowloss connections to link different users’ entangled qubits, thereby establishing entanglement across the channel. This quantum router architecture is motivated by recent advances in integrated photonics, with demonstrations of fast and lowloss onchip switching^{17} of many photonic modes and the integration of quantum emitters with integrated circuits^{18}.
In comparison with previous works, our proposed quantum router architecture is compatible with any entanglement generation protocol and quantum memory in the physical layer. Furthermore, our architecture extends naturally to chains of routers; in fact, we show that if brokered entanglement is used, the router can redirect entanglement flows to minimize latencies using only local information. We demonstrate these benefits by comparing the router architecture with a standard multiplexed architecture in which entanglement between users is generated serially^{19,20}, exploring the various impacts on repeater performance.
Results
Router architecture
We analyze the performance of our quantum router architecture for the NV center in diamond as the qubit platform and the Barrett–Kok^{5} scheme as the entanglement protocol. However, our architecture is agnostic to the specific physical memory and entanglementgeneration protocol.
In Fig. 2, we show the connectivity between photonic and stationary qubits in (a) a standard multiplexed repeater considered for diamond color centers^{19}, and (b) a multiplexed repeater with a router. In this representation, each qubit register within the router is linked by a distinct mode to a corresponding register at an adjacent neighboring node, here labeled left and right for generality. Different modes are shown separately, though they may be transmitted through the same physical channel, for example via temporal or spectral multiplexing. Each qubit register comprises a single NV center with two physical qubits: one electron spin (dark red), which can be used for opticallymediated entanglement, and one nuclear spin (light red), which has no optical transition but which can use spin–spin interactions to couple to and store quantum information from its local electron spin. This enables the use of “brokered entanglement”^{21}, where the electron spin serves as a shortterm “broker” of entanglement to the longerterm “storage” qubit of the nuclear spin.
In the routerless architecture (Fig. 2a), each of the m qubit registers first establishes entanglement with the left neighboring node via its electron spin and the corresponding optical mode. Letting p_{distant} be the success probability of one entanglement generation attempt, the process takes an average number of attempts 1/p_{distant}. Once successful, this entanglement is swapped to the nuclear spin for storage, at which point the electron spin attempts to establish entanglement with the right neighbor, again requiring an average of 1/p_{distant} attempts. Thus, the nuclear spin must idle while storing entanglement for an average time t_{idle} ~ t_{distant}/p_{distant}, where t_{distant} is the time needed per attempt of the entanglement protocol, typically limited by the roundtrip communication time between the repeater and a detector station. Once this succeeds, a Bell state measurement (BSM) on the joint electronnuclear spin state teleports entanglement to be shared between the left and right nodes. This protocol is performed simultaneously and independently on each qubit register.
In contrast, our router architecture (Fig. 2b) defines two “banks” of registers, one with \(\frac{m}{2}\) registers with optical links to the left, the other with \(\frac{m}{2}\) registers with optical links to the right. The router connects these two banks using a lowloss \(\frac{m}{2}\times \frac{m}{2}\) switchboard, over which entanglement protocols succeed with probability p_{local} ≫ p_{distant}. In this architecture, each qubit register first establishes entanglement with either the left or the right, as determined by their optical connectivity. As in the routerless case, each link requires an average of 1/p_{distant} attempts before succeeding, after which these “successful” registers will then swap entanglement to their nuclear spin. At this point, any successful registers enter a “pairing” stage where registers in opposite banks are paired up and subsequent clock cycles attempt entanglement between electron spins in these pairs. This requires an average number of attempts 1/p_{local}, taking a time t_{idle} ~ t_{local}/p_{local}; here, the local clock period t_{local} is likely dominated by any state initialization required by the entanglement protocol. Any successful but unpaired registers idle, waiting for a successful partner on the opposite side to become available. Once entanglement is formed within a pair, two electronnuclear spin BSM’s—one at each register—teleport entanglement to be shared across the full leftright link.
Figure 3 compares the timeline of activity for a router and a standard repeater. Even though the router requires an additional “pairing” stage, this stage requires a negligible amount of time compared to the time needed for a distant entanglement attempt t_{distant}, so the router and the standard repeater can have similar clock cycles. Moreover, the router uses the entanglement with its neighboring nodes more quickly than the standard repeater does, so the router reaches its steadystate rate in a smaller number of clock cycles than the standard repeater.
This router architecture has two primary advantages even at steady state. First, the idling time t_{idle} that a storage qubit must hold entanglement can be greatly reduced, decreasing errors due to decoherence. Second, the number of entanglement attempts a register must make while its storage qubit holds entanglement is reduced by a factor p_{local}/p_{distant}, thus sharply reducing the storagequbit decoherence due to brokerqubit entanglement attempts. In the remainder of this article, we quantitatively examine the impact the router has on the steadystate performance of a quantum network.
Simulations
We use the NetSquid discrete event simulator^{22} to compare the performance of repeaters with and without a router, computing the average rate and fidelity of entanglement distribution for a onerepeater network. In this network, a single repeater station connects Alice and Bob, as shown in Fig. 2 where Alice is the “left node” and Bob is the “right node.” We compare each architecture using the same number of total NV qubit registers m at the repeater, under the assumption that qubit registers will be a scarce resource in nearterm quantum networks. We choose the emissionbased Barrett–Kok protocol^{5} for our entanglement protocol. The physical parameters of our repeater memories correspond to a realization using photonic integrated circuits discussed below, using experimentally realized values reported in the literature (see Supplementary Note 1 for details).
Figure 4b plots the entanglement distribution rates for both repeater architectures for various link distances as a function of the number of qubit registers m at the repeater node. For the routerless case, the rate scales linearly with m since the qubit registers operate independently. For low m, the router exhibits comparatively lower rates; however, as m increases, the difference between the two protocols decreases, such that in the limit of large m the two architectures perform comparably. This difference can be attributed to mismatches in the number of successful entanglements in Alice’s and Bob’s banks after a given clock cycle. The delay in resolving these mismatches, the number of which scales with \(\sqrt{m}\), lowers the rate of the router below the linear scaling of the routerless architecture. However, the fractional impact of this effect on the rate is reduced for large m, and the rate of the router approaches the routerless rate (see Supplementary Note 2 for a lower bound on the router rate).
Figure 4c, d plot the infidelity of the distributed entanglement for repeaters with and without a router. We consider three sources of infidelities in the distributed entanglement. The first source of infidelity is the typical depolarizing and dephasing noise experienced independently by the electron and nuclear spins, as characterized by their T_{1} and T_{2} coherence times. We model this noise on a qubit after time t as:
For a repeater with no router, adding additional registers does not affect the amount of time an individual register must store entanglement; thus, increased multiplexing does not reduce this decoherence channel. However, for a repeater with a router, the number of registers has a dramatic effect on the fidelity. This observation can be understood by considering the mean idling time for memory in this scheme, which is determined by the number of clock cycles a given register in the Alice (Bob) bank must idle after establishing entanglement on its respective side of the link before a register in the Bob (Alice) bank also succeeds and is available for the pairing stage of the protocol. This maps to the success mismatch problem discussed earlier; as such, in the limit of large m, most successful registers will be paired with a register in the opposite bank with every clock cycle, such that the idling time approaches the duration of a single such cycle. In contrast, with no router, a register must idle for on average 1/p_{distant} ≫ 1 cycles. Thus, the addition of a router reduces the infidelity from this channel by a factor that approaches p_{distant}.
The second source of error stems from the coupling between the nitrogen nuclear spin and the NV electron spin. In particular, each excitation of the electron spin used to generate spinphoton entanglement in the BarrettKok protocol generates noise that decoheres the qubit stored in the nuclear spin. We can model this interactioninduced noise on the nuclear spin as^{19}:
where the dephasing a ≈ 1/4000 and the depolarization b ≈ 1/5000 for NV centers. Note that for other systems with stronger electronnuclear spin coupling such as siliconvacancy centers in diamond^{23}, this effect may be stronger. For both architectures, after a register has generated one entanglement link, it stores that entanglement in the nuclear spin while its electron spin attempts to generate entanglement on the other side—either directly to Bob in the case with no router, or to the opposite bank when using a router. In either case, each failed attempt decreases the fidelity of the final AliceBob EPR pair. Without a router, the mean number of failed attempts is equal to 1/p_{distant}; with a router, this number is instead 1/p_{local}, dramatically reducing the infidelity from this effect.
The third source of infidelity comes from imperfect twoqubit gates and electron spin readout. The proposed router needs to perform twice as many twoqubit gates and readouts for each distributed EPR pair as a standard repeater. The cumulative infidelity of these operations is independent of m, p_{distant}, and p_{local}. When the link between adjacent nodes is short, these operations dominate the infidelity of the AliceBob EPR pair, disadvantaging the router. However, when the link becomes lossier, the previous two sources of infidelity dominate, causing the router to increasingly outperform the standard repeater in entanglement fidelity (see Supplementary Note 3 for more details).
As a result, the router enables channellossinvariant entanglement fidelity; that is, regardless of the network link efficiency, the router can achieve fidelity that is limited only by the gates performed on the quantum memories and decoherence during a single roundtrip transit of the network. A threshold is crossed as the number of entanglement attempts per clock cycle exceeds 1/p_{distant}, where the average clock cycle produces at least one entanglement event on both sides of the router. This limits the decoherence a memory must endure to that of one clock cycle, and, for the case where the memory coherence time is much longer than the link roundtrip time, allows all routers to realize a fidelity that is purely limited by the fundamental gates necessary to perform the protocol (initialization, swapping, measurement, etc.). This asymptotic channellossinvariant fidelity is illustrated in Fig. 4d. While increasing the distance between adjacent nodes increases the roundtrip time and thus the decoherence in one clock cycle, Fig. 4c shows that the asymptotic infidelity increases less than it would have if it were to depend on the channel loss as well.
Discussion
Though the complete bipartite connectivity of routers may appear challenging to implement, a router can be added to a standard multiplexed repeater with relatively little physical resource overhead. Figure 5 illustrates one such realization using photonic integrated circuits (PICs). Quantum registers are integrated onto PICs that contain an array of Mach–Zehnder interferometers (MZIs)^{24}, forming a fast switching network to connect any register to any output channel. For a standard multiplexed kway repeater with no local connectivity, this requires an m × k switch to connect m repeaters to any of the possible k neighbors in the network, as shown in Fig. 5a. However, a router can be embedded in this architecture by extending this to an m × (k + 2) switch. The additional two ports lead to a detector station as shown in Fig. 5b, thus enabling photonmediated entanglement between any two registers in the repeater. Since typical singlephoton detectors have dead times of a few tens of nanoseconds, a single pair of detectors could facilitate many attempts at local entanglement in a single repeater clock cycle, which may be hundreds of microseconds long depending on the length of the link being connected. The additional complexity involved in adding these two additional channels is small, making this approach an attractive option for implementing alltoall local connectivity.
On a PIC, the m × k switch required for the routerless repeater can be realized on an MZI array of depth \({\log }_{2}(m)+{\log }_{2}(k)\)^{25}. However, this architecture does not permit simultaneous routing from multiple memories, which is desired for the local entanglement operations used in the quantum router. In Fig. 5c, we present an architecture that permits arbitrary m × k routing for connecting registers to the network, allows simultaneous routing from two memories to perform Bell state measurements for local entanglement generation, and maintains \({{{\mathcal{O}}}}(\log m)\) scaling of the array depth. The addition of a oneMZI interposer permits routing between the network and a Bell state measurement setup for local entanglement generation (see Supplementary Note 4 for more details). The simulations presented in Fig. 4 utilize this architecture assuming an aluminum nitride (AlN)based photonic chip; alternative implementations include using silicon nitride (SiN)^{26} or lithium niobate (LN)^{17}. The control requirements for such large MZI arrays are similar to those required by and demonstrated for optical neural nets^{27} and programmable photonic circuits^{28,29}, allowing us to take advantage of this existing technology for our router. Additional improvements include integrating detectors onchip^{30} to eliminate the offchip coupling inefficiency for local measurements and integrating the frequency conversion stage onchip, which is feasible in highly nonlinear materials such as LN^{31}.
This platform is an attractive path for realizing recent proposals for onchip polarizationtospin based networking^{32}. The PIC implementation presents an additional benefit if the links to Alice and Bob have different transmissivities. Since both the rate and fidelity of entanglement distribution are limited by the difference in the time needed to establish entanglement with both Alice and Bob, the repeater in Fig. 2b (with equal numbers of registers in the Alice and Bob banks) will be limited by the lossier side. In contrast, the PIC implementation allows dynamic allocation of registers to Alice and Bob to balance the rates of entanglement generation. This optimizes the entanglement rate by effectively increasing the multiplexing on lossier ports, and optimizes the entanglement fidelity by preventing a buildup of idling registers on a lowerloss port. Dynamic memory allocation is also of great importance for managing entanglement flows in general 2D quantum network geometries^{33,34,35}.
The router architecture has additional advantages in multirepeater quantum networks. The high local connectivity of the router enables chains of repeaters to automatically connect the longest available entanglement links, favoring the generation of a single EPR pair shared between distant nodes over that of multiple EPR pairs shared between relatively proximate nodes. Specifically, the firstin, firstout behavior of a router guarantees that entanglement will always grow from the longest established chain. In contrast, even if we allow adjacent standard repeaters to have alltoall connectivity between registers in adjacent nodes (e.g., by spectral shifting or timebin reordering), there is no guarantee that the longest links will be generated first.
This behavior is illustrated in Fig. 6, which compares the dynamics of a standard network and a router network under similar scenarios. Before each clock cycle, a repeater or router will determine which links to attempt on the subsequent clock cycle given the current link state. An example is shown in Fig. 6a. If we further allow the standard repeater to have knowledge of the global network state, it can adjust its transmission sequence to prioritize a longer link, e.g., by attempting to entangle registers B5 and C6 (instead of B5C5 and B6C6). In contrast, the router does not require any information about the network state; each register simply attempts entanglement with a fixed partner in a neighboring node.
Figure 6b shows a possible network state after the entanglement attempts in a clock cycle. Each router uses local entanglement and Bell state measurements to connect its left and right banks of registers. Following a firstinfirstout strategy, it connects the first available entangled register in the left bank with the first available entangled register in the right bank and repeats until only unentangled registers remain in one of the banks. Using only local information, the router network thus automatically connects as many long links as possible, as demonstrated by the link state in Fig. 6c. On the other hand, the standard repeater may not be able to connect long links even if there are enough successful entanglements to do so. As a result, the state of the standard network after this time step yields some long links, but also many singular short links that cannot be merged.
The router’s strategy reduces the idling time between local entanglement generation and the delivery of this entanglement to the end users, thereby increasing the fidelity of the delivered entanglement. We highlight that this router behavior is realized without any need for knowledge of the global network state. In a true deployed network, communication latencies prevent individual nodes from knowing the state of the total network in realtime, making the router’s ability to optimize fidelity with localonly information all the more notable.
In conclusion, we have addressed a critical problem in quantum repeater architectures: minimizing memory latencies to maintain entanglement fidelity across repeater networks. Specifically, the quantum router architecture enables entanglement fidelity that is invariant to the channel losses in the network. Our architecture applies to leading entanglement protocols and automatically prioritizes entanglement flows using only local information. An important next step will be to exploit the local connectivity for entanglement distillation^{36} and local error correction^{37} to further improve the fidelity of generated entanglement. In this way, the entanglement rate can be traded for entanglement fidelity, without modification to the router architecture. While the above analysis considered NV centers in diamond, the architectural benefits should apply to other quantum memory modalities, including solidstate artificial atoms, trapped ions, and neutral atoms. In suitable parameter regimes, routers can also incorporate multiplexing into realizations of distanceindependent entanglement rates using local GHZ projective measurements^{38}. Our results emphasize the importance of local connectivity in designing multiplexed quantum repeaters for highrate and highfidelity entanglement distribution across quantum networks.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Code availability
The NetSquid simulation package is available for download by request at https://netsquid.org/. The additional code used to generate data in this work is presently available at https://github.com/leeyuan13/NetSquidSim.
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Acknowledgements
Y.L. acknowledges funding support by the MIT School of Engineering’s SuperUROP program. E.B. was supported by a NASA Space Technology Research Fellowship and the NSF Center for Ultracold Atoms (CUA). A.D. and S.W. were supported by an ERC Starting grant, EU Flagship on Quantum Technologies, Quantum Internet Alliance, an NWO VIDI grant, and the Zwaartekracht QSC. D.E. acknowledges support from the NSF EFRIACQUIRE program Scalable Quantum Communications with ErrorCorrected Semiconductor Qubits and the NSF ERC Center for Quantum Networks (CQN). We also thank Ian Christen, Kevin Chen, and Robert Knegjens for valuable discussions.
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Y.L. and E.B. designed and performed the simulations. D.E. conceived the idea and supervised the project. A.D. and S.W. developed the code framework to run simulations using NetSquid. All authors contributed to writing and revising the manuscript.
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Lee, Y., Bersin, E., Dahlberg, A. et al. A quantum router architecture for highfidelity entanglement flows in quantum networks. npj Quantum Inf 8, 75 (2022). https://doi.org/10.1038/s41534022005828
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DOI: https://doi.org/10.1038/s41534022005828
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