Abstract
The vision to develop quantum networks entails multiuser applications, which require the generation of longdistance multiparty entangled states. The current rapid experimental progress in building prototypenetworks calls for new design concepts to guide future developments. Here we describe an experimentally feasible scheme implementing a twodimensional repeater network for robust distribution of threeparty entangled states of GHZ type in the presence of excitation losses and detector dark counts — the main sources of errors in realworld hardware. Our approach is based on atomic or solid state ensembles and employs builtin error filtering mechanisms peculiar to intrinsically twodimensional networks. This allows us to overcome the performance limitation of conventional onedimensional ensemblebased networks distributing multiparty entangled states and provides an efficient design for future experiments with a clear perspective in terms of scalability.
Introduction
The development of quantum networks holds the promise to realize quantum technologies such as secure communication schemes,^{1} distributed quantum computing,^{2} and metrological applications.^{3,4,5,6} This prospect resulted in intense efforts to build prototypenetworks with few nodes. The creation of longrange bipartite entangled states has been demonstrated^{7,8,9,10,11,12,13,14,15,16,17} and multiparty entangled states shared between several remote parties can be realized in the nearfuture. The ability to distribute such multiparty entangled states over longdistances^{18} will be an essential prerequisite for future quantum networks^{19,20,21,22,23,24} consisting of large twodimensional (2D) structures that allow for multiuser applications. This need and the current experimental possibilities raise the question which type of architecture can be used to distribute longrange multiparty entanglement in a practical and scalable fashion. We address this problem from an implementationoriented point of view and propose a networking scheme that allows for a robust and efficient realization with current and nearfuture experimental means.
Future quantum networks will require different types of multiparty entangled states suitable for different tasks. We concentrate on distributing threeparty GHZ states,^{25} e.g. \(\left(\left001\right\rangle +\left110\right\rangle \right)/\sqrt{2}\), between remote parties, that can be used for clock synchronization,^{3} quantum secret sharing,^{26,27} quantum secret voting,^{28} or for fundamental tests of nature.^{29} A key challenge for distributing quantum states over longdistances is the fact that losses and decoherence scale exponentially in the distance.^{30,31} To solve this problem, quantum repeaters^{32,33,34} have been introduced, which however are inherently onedimensional (1D) schemes aiming at generating a bipartite entangled state connecting two remote parties. Therefore, an intrinsically 2D network can be preferable for a multipartite entanglement distribution as it was shown^{35} to have a higher error threshold than a 1D counterpart under assumptions of a generic noise model and full Bell state analysis (or a universal gate set) availability.
In this work, we propose an implementationoriented 2D repeater scheme that, in contrast to,^{35} (i) can be realized using a constrained set of quantum operations available in atomic or solid state ensembles and (ii) has a mechanism to mitigate an excitation loss error and detector dark counts — major imperfections for this type of system. The resulting robust architecture is customtailored to implementations based on atomic or solid state ensembles, as these systems are promising for realizing longlived quantum memories.^{36,37} While atomic ensembles allow a limited set of operations for processing quantum information, record coherence lifetimes (\(2{T}_{{\rm{coh}}}\)) up to 6 h have been observed in rareearth doped crystals.^{38} This makes them particularly promising for entanglement distribution over continental distances using repeater schemes which inevitably requires memory times corresponding to the duration needed for classical communication between remote nodes.
Our scheme is inspired by the seminal DLCZ proposal for generating Bell pairs between two nodes,^{39} and requires optical cavities with good cooperativities, linear optical elements, and photodetectors. GHZ states are first generated over moderate distances and then merged to form GHZ states connecting increasingly remote parties [see inset in Fig. 1a and Fig. 2a]. Similar to the DLCZ proposal, our protocol does not require a universal gate set or a full Bellstate analysis. However, unlike the DLCZ scheme, the new protocol suppresses the propagation of socalled vacuum and multiexcitation errors. These errors are resulted from excitation loss or detection of a dark count during the generation of the elementary state or merging operation and lead to the preparation of a state containing more/fewer excitations than expected. In the DLCZ scheme, the vacuum and multiexcitation errors freely propagate decreasing fidelity dramatically with the increase of nesting level and therefore making the scheme not scalable (Sangouard et al.^{34} and Sec. A1 of Supplemental Material). In comparison, our 2D scheme results in a truly scalable architecture. As shown in Fig. 1 and explained below, this feature allows our repeater scheme to cover longer distances than networks based on combining conventional (onedimensional) quantum links.
Results
Scheme
We propose a nested structure of quantum network^{35} consisting of segments in threeparty entangled states of increasing size as shown in Fig. 2a. The scheme requires the ability to generate GHZ states of elementary segments spanning a length \({L}_{0}\), and generalized entanglement swapping procedure to merge the GHZ states doubling the covered distance. We detail the generation and swapping operations separately in the next two paragraphs.
The generation of GHZ states at the elementary level is illustrated in Fig. 2b. We consider two types of ensembles: (i) ensembles with a \(\Lambda\)level scheme at nodes A and C, allowing for the efficient storage and readout of photons^{39,40,41} and (ii) ensembles with a double\(\Lambda\) configuration placed in a cavity at node B. Information is encoded in the absence or presence of a collective spin excitation in the ensembles, i.e. in the logical states \(\left0\right\rangle =\left{g}_{1}{g}_{2}...{g}_{N}\right\rangle\) and \(\left1\right\rangle =\frac{1}{\sqrt{N}}{\sum }_{i=1}^{N}\left{g}_{1}{g}_{2}...{s}_{i}...{g}_{N}\right\rangle\), where \(N\) is the number of emitters per ensemble and \(\leftg\right\rangle\), \(\lefts\right\rangle\) denote ground state levels of the emitters [see Fig. 2b]. Ensembles A and C are driven by weak coherent laser pulses resulting in entangled states of the ensembles and the corresponding forwardscattered photons,^{39}\(\left{0}_{A(C)}{0}_{a(c)}\right\rangle +{\epsilon }_{a(c)}\left{1}_{A(C)}{1}_{a(c)}\right\rangle +{\mathcal{O}}({\epsilon }_{a(c)}^{2})\), \({\epsilon }_{a(c)}\ll 1\) (the role of the parameter \(\epsilon\) is explained in Sec. B of Supplemental Material, our analysis includes also higher order terms). The capital and lowercase subscripts refer to the states of the ensemble and the corresponding emitted light field. Node B performs the gate operation \(\left{1}_{a}{0}_{B}{0}_{b}\right\rangle \to \left{0}_{a}{1}_{B}{1}_{b}\right\rangle\), \(\left{0}_{a}{0}_{B}{0}_{b}\right\rangle \to \left{0}_{a}{0}_{B}{0}_{b}\right\rangle\). The working mechanism and imperfections are detailed in Supplemental Material, Sec. G. The light fields emitted from nodes B and C are synchronously directed to a swapping station equipped with a 50/50 beamsplitter and two singlephoton detectors, as illustrated in Fig. 2b. Conditioning on detecting a single photon allows for a probabilistic projection onto the state \(\left{\Psi }_{{\rm{GHZ}}}\right\rangle =\left(\left{0}_{A}{0}_{B}{1}_{C}\right\rangle +\left{1}_{A}{1}_{B}{0}_{C}\right\rangle \right)/\sqrt{2}\). The path length fluctuation in fibers AB and CB can be solved by phase stabilization, similar to 1D repeater schemes.^{34} For example, it can be achieved by exciting memories in A and C in a Sagnac interferometer configuration,^{42,43} such that the excitation laser pulses for the two memories and the emitted photons travel the same path in a counterpropagating fashion.
For merging three GHZ states, the states of ensembles at adjacent nodes are projected onto the oneexcitation subspace. This operation is realized by reading out the atomic excitations^{39,40,41} and directing the emitted light fields to the swapping station described above [see also inset of Fig. 2a], where success is heralded by the detection of a single photon (see Sec. A of Supplemental Material for details), otherwise the resulting state is discarded.
Error filtering
Apart from the major challenge to mitigate photon transmission losses, experimental limitations at the individual nodes have to be taken into account. The most important local error sources are the readout inefficiency \(v\) and imperfect photon detectors (with a detection inefficiency \(f\) and a dark count probability \(d\) during a photon pulse detection). Moreover, quantum states stored in the ensembles are degraded over time. Due to the encoding used here, the relevant physical mechanism — dephasing of individual emitters — leads to an effective decay of the stored collective excitations (Supplemental Material, Sec. F). In the considered ensemblebased setting, the main imperfections are therefore errors of loss type and detector dark counts.
The proposed 2D repeater protocol is designed to prevent the propagation of this kind of errors by employing an intrinsic redundancy of the native 2D network to filter out the errors in the merging process at each nesting level. The filtering mechanism, illustrated in Fig. 3, works as follows. At each nesting level one has to perform three merging operations to complete the entanglement generation between the three outermost nodes (X, Y, Z in Fig. 3). The third merging operation is redundant and it shows, if unsuccessful, that there is a lack or an excess of excitations in the generated state. This indicates an occurrence of either the loss type error or the multiexcitation error at previous stages of the protocol. In our protocol, the errors have, therefore, to conspire within one nesting level to pass undetected. In contrast, in the 1D benchmark scheme the errors can conspire within all levels of the bipartite state preparation to pass filtration in the last generation step. As a result, the 2D architecture exhibits a slower rate of error probabilities growth with the increase of the network nesting level (see Supplemental Material, Sec. A).
However, the filtering is a passive operation, therefore, it is important to start the entanglement distribution with initial states of high fidelity. As detailed in Sec. A and B of Supplemental Material, the protocol for generating entanglement at the elementary level and the relative orientation of GHZ states in the scheme is chosen such that the involved measurements allow for (i) identifying imperfect merging operations for ideal input states and (ii) filtering out imperfect states for ideal merging operations. The initial states of high fidelity at the elementary level and the subsequent error filtering at each nesting level result in the scheme robust against the vacuum and multiexcitation errors.
Key features
In the following, we highlight some of the main properties of our scheme and summarize the effect of limited memory coherence times in the presence of experimental imperfections. Figure 1a shows the fidelity of the generated state \(\rho\) with respect to the target state \(\left{\Psi }_{{\rm{GHZ}}}\right\rangle\), \(F=\sqrt{\left\langle {\Psi }_{{\rm{GHZ}}}\right\rho \left{\Psi }_{{\rm{GHZ}}}\right\rangle }\), for the proposed architecture. In Fig. 1b, for comparison, we introduce a fidelity benchmark representing the performance of a 1D scheme based on longdistance bipartite entanglement distribution. The 1D benchmark uses conventional ensemblebased repeater scheme which creates three longdistance Bell pairs to be subsequently merged to generate the desired target state. The benchmark estimates an upper bound of the fidelity as ideal local GHZ states [shown with double lined triangles in the inset in Fig. 1b] are used for the final merging. In contrast to the 2D approach, which relies on multiparty entanglement at all network levels, the 1D scheme exploits the distribution of longdistance Bell pairs and involves the error filtering mechanism only in the end (Methods section).
The most striking feature of Fig. 1 is the fact that the 2D protocol can distribute GHZ states over increasingly large distances by increasing the memory coherence time (or alternatively, by using socalled multiplexing approaches^{44,45,46,47,48} in which several quantum memories or memory modes are used in parallel to compensate for limited coherence times). An increase of \({T}_{{\rm{coh}}}\) significantly extends the distance that can be covered, since our protocol profits from using higher nesting levels for larger distances. This property makes the scheme practical and scalable. This is not the case for the 1D benchmark based on a regular DLCZ scheme, which is hampered by fundamental difficulties associated with the creation of longrange bipartite links (see Supplemental Material, Sec. A3 and Sec. C for details). Therefore, the 1D strategy reaches its performance limit at much shorter distances than the 2D scheme as can be seen in Fig. 1b in comparison with Fig. 1a.
To further characterize the potential of our approach, we analyze the case of infinite memory coherence time. Figure 4 shows the fidelity achievable by a network for which the number of nesting levels is optimal for a given distance in the presence of typical experimental imperfections for \({T}_{{\rm{coh}}}\to \infty\). A network with more nesting levels divides the target distance into shorter segments what reduces the probability to lose photons during transmission at the cost of additional errors generated by the repeater scheme itself. For each target distance, there is an ideal number of nesting levels beyond which the addition of a repeater station becomes undesirable. The comparison of the 2D approach (black line) with the 1D benchmark (green line) for generating threeparty GHZ states shows that the latter suffers from severe limitations in the achievable fidelities for large distances even for perfect quantum memories. This is the result of the faster errors accumulation with the growing network size in 1D scheme which lacks the filtering mechanism of the native 2D protocol, as discussed in the previous section.
Here we emphasize that although the 1D benchmark uses fewer resources (memory cells) than our 2D scheme (see Supplemental Material Sec. A2) its performance is saturated at much shorter distances and can not be improved by a simple increase of resources invested into multiplexing. The presented 2D architecture allows us to convert the extra resources into longer entanglement distribution distances with higher target fidelities. Moreover, the additional memory cells improve coverage of intrinsically 2D networks by allowing the multiparty entanglement generation between arbitrary nodes of the network (see Supplemental Material Sec. A1), while 1D based schemes entangle only the outermost ensembles.
A detailed analysis of the 2D protocol performance, including an assessment of the achievable rates, is provided in Supplemental Material (Sec. A3 and Sec. C). In the following, we discuss means to further improve the performance by temporal filtering.
Temporal filtering protocol
Limited coherence times of quantum memories are generally an important restricting factor for quantum repeaters. To mitigate their effect, we introduce a temporal filtering mechanism by defining a time window \(\tau\), after which quantum states are discarded. During the probabilistic generation and merging procedure shown in Fig. 2, ensembles storing excitations longer than \(\tau\) are reset to their ground state, such that the entanglement generation process involving these nodes starts anew. As a result, the influence of decoherence is decreased at the expense of a reduced rate. The time \(\tau\) can be changed dynamically by the control software without changing the hardware in a quantum network. Due to this added flexibility, different types of applications — requiring either high fidelities or high rates — can be accommodated. The corresponding tradeoff has been calculated semianalytically (as described below) and is shown in Fig. 5.
Diagrammatic technique
We used a numerical Monte Carlo algorithm and developed a new semianalytical technique to analyze the performance of our scheme in the presence of realistic imperfections. The Monte Carlo method is very flexible and simulates the full repeater protocol step by step.^{49,50} The strength of this exact simulation is also its weakness — the runtime is proportional to the entanglement generation time in the quantum network, growing quickly with the network’s scale.
The analytical method overcomes this difficulty while still incorporating all relevant error sources, including finite memory coherence times. It takes time delays due to classical communication into account and can be used to analyze a large class of repeater schemes, including 1D protocols. The main idea is to determine the density matrix distribution \(\varrho (t)\) for the ensemble of states generated by the network at time \(t\). More precisely, we obtain the Laplace image of this distribution, \(\tilde{\varrho }(s)\), which fully describes the statistics of the network and allows one to infer the average generated state \(\rho \equiv {\int }_{0}^{\infty }\varrho (t){\mathrm{d}}t=\tilde{\varrho }(s){ }_{s=0}\), corresponding generation time \(T\equiv {\int }_{0}^{\infty }t\ {\rm{Tr}}\varrho (t){\mathrm{d}}t=\frac{d}{ds}{\rm{Tr}}\tilde{\varrho }(s){ }_{s=0}\), and other relevant statistical quantities.
We assume that the probability \({p}_{1}\) to generate an entangled state of an elementary segment is small for each time step \(\Delta t\), such that the continuous probability density \(p(t)\equiv \nu {e}^{\nu t}\) with the rate \(\nu \equiv {p}_{1}/\Delta t\) can be introduced. As an example, we consider a linear network consisting of two links, where the density matrix distribution for generating the \({\alpha }^{{\rm{th}}}\) link in a state \({\rho }_{\alpha }\) before the \({\beta }^{{\rm{th}}}\) link has been generated at time \(t\) in a state \({\rho }_{\beta }\) is given by \({\varrho }_{\alpha \beta }(t)={p}_{\beta }(t){\int }_{0}^{t}{\mathrm{d}}{t}^{\prime}\ {p}_{\alpha }\)(\({t}^{\prime}\))\({e}^{(tt^\prime){\mathcal{L}}_{\alpha}}{\rho}_{1}\otimes{\rho}_{2}\) with \(\alpha \beta \in \{12,21\}\). Here we sum over all intermediate times \(t^{\prime}\) of the \({\alpha }^{{\rm{th}}}\) link creation. The degradation of the link during the waiting time \(t{t}^{\prime}\) due to finite memory life times is taken into account using the superoperator \({{\mathcal{L}}}_{\alpha }\). The corresponding Laplace image of the distribution is given by \({\tilde{\varrho }}_{\alpha\beta}(s)={\nu}_{\alpha}{\nu}_{\beta}/[({s}^{\prime}+{\nu}_{\alpha })({s}^{\prime}{{\mathcal{L}}}_{\alpha })]\cdot {\rho }_{1}\otimes{\rho}_{2}{}_{{s}^{\prime}=s+{\nu }_{\beta}}\).
The probability to generate and successfully merge states of two segments during the time window \([t,t+dt)\) is given by \({\rm{Tr}}\ {\mathcal{M}}{\varrho }_{\alpha \beta }(t)dt\), with \({\mathcal{M}}\) the merging superoperator. The summation over all possible combinations of unsuccessful mergings leading to the generation of an entangled state results in a sum of multiple convolutions in the time domain. In the Laplace domain, the sum converges to
with \({\sum }_{\alpha \beta }\) the sum over all possible orders in which links are generated and \({\mathcal{I}}\) the unit superoperator. The Laplace image (1) is used to find the average density matrix \(\rho ^{\prime}\) and generation time \(T^{\prime}\) for the given nesting level of the network, as described above. To address the next network level, we apply the approximation that the segments are generated timeindependently in the state \(\rho ^{\prime}\) with the rate \(\nu ^{\prime} =1/T^{\prime}\). In a recursive procedure, the state \(\rho\) and generation time \(T\) of an arbitrary network level can be found. As detailed in Sec. E of Supplemental Material, we introduce diagrams to conveniently deal with probabilistic processes in networks of arbitrary complexity.
Discussion
We proposed a scalable 2D architecture for generating longdistance multiparty entanglement and provided a full performance analysis with emphasis on the effect of finite memory lifetimes. Covering increasingly longer distances as shown in Fig. 1 can be achieved either using memories with long coherence times or using multiplexing approaches,^{44,45,46,47,48} in which several memorymodes are used in parallel. The presented scheme provides a flexible structure for creating GHZ states between arbitrary nodes of the 2D network, not only between the outermost ensembles. The logical topology of the network [shown in Fig. 2a] is thereby not identical to the topology of the required fiber links, which can easily be adjusted to accommodate urban constraints. We are here primarily interested in metropolitan distances and applications requiring only moderate bitrates. Important examples include secret voting and the protection of classified information that requires several parties for decryption.^{26,27}
The presented scheme can be modified to work without cavities at the expense of using a larger number of ensembles (see Supplemental Material, Sec. H). This modification uses polarizationtype qubits^{34,39} and twoclick conditioning,^{51} enhancing its resistance with respect to fiberlength fluctuations. It will be interesting to develop similarly robust schemes for other platforms such as trapped ions,^{8,52} in which a universal gate set can be implemented. In this case, more complicated 2D repeater protocols involving quantum error correction^{33} instead of error filtering could be envisaged.
Methods
Parameters of the simulated network
The results of simulations presented in Figs 1, 4, and 5 are obtained with the following parameters of the network: detection inefficiency \(f=0.05\), readout inefficiencies \(v=0.05\), fiber attenuation length \(L_{\rm{att}}=22\) km, gate efficiency (see text) \(\eta =0.6\), signal length \(l=1{0}^{4}\)s, and dark count probability in the signal measurement operation \(d=10\,{\rm{Hz}}\times 1{0}^{4}{\rm{s}}=0.001\).
Onedimensional benchmark scheme
Here we present an alternative, onedimensional (1D), repeater approach for distribution of multipartite GHZ state, which provides us with the fidelity benchmark used in the present work. This 1D benchmark scheme is based on the generation of longrange bipartite quantum states that are created using regular DLCZ quantum repeaters. A comparison of various nesting levels structures for the 1D benchmark scheme and the propose 2D scheme is presented in Fig. 6a.
While two links are enough to distribute entanglement between three parties, we equip the 1D scheme with the third link, which is used to implement error filtering procedure in the final step, similar to the original DLCZ protocol, as explained below. To ensure the robustness of the benchmark we assume that the distant parties X, Y, and Z of the 1D scheme have access to ideal GHZ states that can be generated on demand. This provides an upper bound on the performance of 1D schemes of DLCZ type for distribution of tripartite GHZ states. Therefore, the corresponding fidelity of the 1D scheme serves as a benchmark for our 2D scheme.
Figure 6b shows the final merging step of the 1D scheme acting as an error filtering operation. In this step, five swapping operations are followed by the last one, which is deterministic in the ideal case. A failure of the last swapping operation indicates an erroneous state. More details about the 1D benchmark scheme are presented in Sec. A of Supplemental Material.
Note added
Recently we have become aware of Santra et al.^{53} discussing a similar temporal filtering technique
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The code that supports the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank M. Afzelius and P. Jobez for fruitful discussions on solidstate ensemblebased implementations and J. Wallnöfer for his input on 2D architectures. Research was sponsored by the Swiss National Foundation (SNSF) through grant number PP00P2179109, by the Austrian Science Fund (FWF): P28000N27, P30937N27, and by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF1520060. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. This research was undertaken thanks in part to funding from CFREF.
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Kuzmin, V.V., Vasilyev, D.V., Sangouard, N. et al. Scalable repeater architectures for multiparty states. npj Quantum Inf 5, 115 (2019). https://doi.org/10.1038/s4153401902303
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DOI: https://doi.org/10.1038/s4153401902303
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