Multiplexed entanglement generation over quantum networks using multi-qubit nodes

TL;DR

This paper analyzes multiplexed entanglement protocols in quantum networks with multi-qubit nodes, highlighting their potential to overcome propagation bottlenecks and guiding future research directions.

Contribution

It provides a comparative analysis of multiplexed entanglement protocols using realistic models for quantum network hardware, identifying key parameters for performance enhancement.

Findings

Multiplexed protocols can significantly increase entanglement rates.

Nitrogen-vacancy centers are promising platforms for near-term implementation.

Improving specific network parameters yields substantial performance gains.

Abstract

Quantum networks distributed over distances greater than a few kilometers will be limited by the time required for information to propagate between nodes. We analyze protocols that are able to circumvent this bottleneck by employing multi-qubit nodes and multiplexing. For each protocol, we investigate the key network parameters that determine its performance. We model achievable entangling rates based on the anticipated near-term performance of nitrogen-vacancy centres and other promising network platforms. This analysis allows us to compare the potential of the proposed multiplexed protocols in different regimes. Moreover, by identifying the gains that may be achieved by improving particular network parameters, our analysis suggests the most promising avenues for research and development of prototype quantum networks.

Figures

Tables

Table 1. Table 1: Anticipated near-term parameters for a quantum network based on NV centers Bernien  et al. ; Cramer  et al. ; Bogdanovic  et al. ; Zaske  et al. ( 2012 ) . These parameters are also anticipated to be achievable using trapped ions  Hucul  et al. ( 2015 ) .

Variable	Description	Value
$N$	Total number of qubits at each node	2
$p_{\text{fc}}$	Frequency-conversion efficiency	0.3
$p_{\text{out}}$	NV-outcoupling efficiency	0.3
$t_{\text{eg}}$	Spin-photon entanglement generation time	1 $\mu$s
$t_{\text{sg}}$	NV-carbon swap gate time	200 $\mu$s
$p_{\text{em}}$	Midpoint-source photon-pair emission probability	0.01, 0.1

Equations

$$R_{\text{mBK}}\sim r_{\text{mBK}}\frac{{\eta }^2}{2}=\frac{1}{2}N{\eta }^2/t_c.$$

$$R_{\text{mEPL}}\sim \frac{N(N-1)}{2N-1}\frac{\eta }{8t_c}.$$

$$R_{\text{MPS}}\sim p_{\text{em}}{\eta }^2/(4t_{eg}),$$

Full text

Recent progress in the generation, manipulation, and storage of distant entangled quantum states has opened up an avenue to the construction of a quantum network over metropolitan-scale distances in the near future Hensen et al. (2015); Hucul et al. (2015). One of the key challenges in realizing such quantum networks will be to overcome the communications bottleneck induced by the long distances separating nodes. This occurs because probabilistic protocols require two-way communication and, for such distances, the entanglement generation rate becomes limited by the time required for quantum and classical signals to propagate.

It is unlikely that quantum networks will attain sufficient levels of complexity in the near future to support the transmission of complex multi-photon entangled states necessary to overcome this bottleneck through error correction Munro et al. (2015); Muralidharan et al. (2016). This motivates the development of alternative methods to circumventing this limited communication rate, of which the most promising near-term approach is through multiplexing entanglement generation Collins et al. ; Simon et al. ; Sangouard et al. ; Munro et al. (2010); Sinclair et al. (2014); Vinay and Kok .

Previous proposals have developed multiplexed entanglement-generation protocols for networks based on atomic-ensemble quantum memories and linear optics Simon et al. ; Sangouard et al. (2011); Sinclair et al. (2014) and for networks in which each node consists of many optically accessible qubits that can be temporally, spectrally or spatially multiplexed Collins et al. ; Sangouard et al. ; Munro et al. (2010); Vinay and Kok . However, these proposals are not effective for promising multi-qubit hybrid network node architectures Nickerson et al. (2013), in which one (or a few) optically accessible communication qubits in each node provide a communication bus to interface with multiple local memory qubits. Several platforms have demonstrated the key elements of such a system, including nitrogen-vacancy (NV) centres in diamond Bernien et al. ; Gao et al. (2015), trapped ions Hucul et al. (2015), and quantum dotsGao et al. (2015); Delteil et al. .

Here we focus on the scenario of efficiently generating heralded remote entanglement between two hybrid multi-qubit nodes separated by tens of kilometers in a quantum network (Fig. 1). We propose two strategies for multiplexing entanglement generation using multi-qubit architectures, identifying the scaling of the entangling rates with the distance between nodes. We compare these strategies to an alternative protocol based on the distribution of entangled photon-pairs Jones et al. (2016), modelling all three protocols analytically and with Monte Carlo simulations. This allows us to identify optimal protocols for different regimes of distance and node performance.

In order to be able to effectively assess the potential of these network protocols, it is vital to incorporate the known and anticipated limitations of potential platforms from the start. In this paper we therefore use network parameters representing the expected near-term performance of NV centre nodes. These centres are promising nodes for such a network, combining a robust and long-lived 13C nuclear-spin quantum register Cramer et al. ; Reiserer et al. (2016) with a photonic interface (Fig. 1). Our conclusions are nonetheless broadly applicable to other platforms with comparable system performances, particularly including trapped ions Hucul et al. (2015).

I Quantum network protocols

We begin by briefly introducing the three candidate protocols that we consider for a metropolitan-scale quantum network. For each network, we identify the scaling of the entanglement generation rate with the system transmission efficiency and the distance between nodes.

I.1 Multiplexed Barrett-Kok protocol

The first scheme is a multiplexed version of the Barrett-Kok (BK) protocol. In this scheme, entanglement is generated at both nodes locally between the spin state of the communication qubit and the modal occupation of a single photon (typically the photonic state is time-bin encoded for NVs). This procedure constitutes a single attempt to generate remote entanglement. The two photons are then transmitted to a remote beam splitter, where a probabilistic joint Bell state measurement (BSM) on the photons projects the two distant communication qubits into an entangled state upon measurement of the appropriate outcomes Barrett and Kok (2005).

In this protocol each photon needs to be transmitted over a distance $d/2$ from the nodes to the central BSM station. This is followed by the transmission of classical information over the same $d/2$ distance heralding to the nodes the success or failure of the entangling attempt. Hence in the standard BK protocol, the entanglement attempt rate $r_{BK}$ is limited by the combined quantum and classical communication time ($t_{c}=d/c$) required to establish whether the protocol succeeded: $r_{BK}\sim t_{c}^{-1}$. Even for modest distances, this time delay is sizable; e.g. for $d=50$ km the delay is $t_{c}=250\,\mu$s, limiting the attempt rate to 4 kHz.

This rate limitation can be mitigated by using a multiplexed version of the BK protocol (Fig. 2), in which the spin state of the communication qubit is swapped to a memory qubit directly after spin-photon entanglement generation, freeing up the communication qubit for additional entanglement generation attempts. For the NV system, naturally occuring nearby 13C nuclear spins provide robust memory qubits Reiserer et al. (2016); Blok et al. (2015). The state is stored in this memory qubit until information about the success of the attempt arrives. In the meantime, spin-photon entanglement generation and subsequent state swapping to other memories can continue until all of the memories are occupied. The multiplexed protocol allows $N$ qubits per node to be utilised, where $N$ includes both the communication qubit and the memory qubits.

The maximum number of qubits per node that can be usefully employed in this protocol is given by $N_{\mathrm{max}}=\lceil t_{c}/t_{sg}\rceil$ 111$\lceil x\rceil$ denotes ceil($x$) where $t_{sg}$ is the duration of the swap gate (typically much longer than the duration of entanglement generation attempts $t_{eg}$). The attempt rate of the multiplexed Barrett-Kok (mBK) protocol is therefore a factor $N$ larger than for the standard BK scheme: $r_{\text{mBK}}\sim N/t_{c}$ for $N\leq N_{\mathrm{max}}$. This rate is upper bounded by $r_{\text{mBK}}\leq 1/t_{sg}$.

The success of each attempt of the BK scheme is conditioned on the detection of both the photons emitted by the communication qubits in the BSM. As a result, the system transmission efficiency $\eta$ appears quadratically in the entanglement success rate $R_{\text{mBK}}$. Hence for $N\leq N_{\mathrm{max}}$:

[TABLE]

The factor of half corresponds to the probability of a successful BSM at the beam splitter.

I.2 Multiplexed Extreme-Photon-Loss Protocol

In the case of high levels of photon loss $(\eta\ll 1)$, a protocol based on entanglement distillation can be more effective than the BK protocol. In this protocol, instead of directly trying to generate a maximally entangled state $\left|\Psi\right>=(1/\sqrt{2})(\left|01\right>+\left|10\right>)$, two weakly entangled states of the form $\rho\approx\frac{1}{2}\left|\Psi\right>\!\left<\Psi\right|+\frac{1}{2}\left|00\right>\!\left<00\right|$ are efficiently generated conditional on the detection of only a single photon at the beam splitter station Campbell and Benjamin (2008); Nickerson et al. (2013). Here $\left|0\right>$ ($\left|1\right>$) denotes the state of the communication qubit from which a photon is (is not) emitted. These weakly entangled states contain a contribution $\left|00\right>\!\left<00\right|$ from the case in which both communication qubits emitted a photon, but only one was detected. After the two states are successfully generated, an entanglement distillation procedure is performed using local operations and classical communication. This distillation produces a pure entangled state with a $1/8$ probability. Since two raw states are consumed to generate a final entangled state, this extreme-photon-loss (EPL) protocol requires at least two qubits per node, as the first state has to be stored in a memory qubit until the second entangled state is generated.

The advantage of this scheme over the BK protocol is that it does not require the detection of coincident photons, instead allowing for multiple attempts to generate the second state. This results in a success probability that is proportional to $\eta$ rather than $\eta^{2}$ and thus an entangling rate $R_{\text{EPL}}~{}\sim\eta/(16t_{c})$, where a factor $1/8$ corresponds to the probability that the distillation operation succeeds, and a factor $1/2$ reflects the need to generate two entangled states.

Analogously to the BK protocol, a multiplexed version of the scheme can be envisioned in which multiple entanglement generation attempts are performed within one communication cycle. Since, in the second stage of the protocol one memory is continuously occupied by the first entangled state, the maximum number of qubits that can be effectively utilised is one more than in the BK protocol: $N_{\mathrm{max}}=\lceil t_{c}/t_{sg}\rceil+1$. The resulting entanglement success rate $R_{\text{mEPL}}$ for the multiplexed extreme-photon-loss protocol for $N\leq N_{\mathrm{max}}$ is proportional to the inverse of the sum of the time spent in the first stage $(t_{c}/(\eta N))$ and second stage $(t_{c}/(\eta(N-1)))$ of the protocol:

[TABLE]

The entangled state fidelity in this protocol is sensitive to decoherence of the memories during entanglement attempts. In order to ensure a minimum fidelity, stored entangled states can be discarded after a set number of subsequent entanglement attempts, at the expense of decreasing the entanglement rate. Entanglement generated from a single photon detection event is expected to succeed within at most a few hundred attempts ($\sim$100 attempts at 50 km, $\sim$1000 attempts at 100 km) for the range of parameters considered here. For nitrogen-vacancy centre nodes, recent results indicate that 13C nuclear-spin memories may effectively preserve quantum states over this number of attempts Reiserer et al. (2016), and so this effect is not expected to significantly impact our conclusions.

I.3 Midpoint-Source Protocol

The final configuration that we consider is the midpoint-source (MPS) protocol following Ref.  Jones et al. (2016). In addition to the two nodes, this protocol requires an entangled-photon source (which emits pairs of photons with probability $p_{\text{em}}$) positioned midway between the nodes (Fig. 3). In this protocol, pairs of entangled photons generated by the photon source are split and one is sent to each of the two nodes. At each of the nodes, a BSM is performed between this photon and a photon generated by the local communication qubit. Entanglement swapping succeeds only if both BSMs succeed (requiring the detection of four photons in total).

Since the successes of the BSMs can be reported to their local nodes immediately, in the case of local failure the nodes can quickly proceed to a new entanglement generation attempt. In this way the entanglement attempt rate can be significantly increased. The attempt rate is upper bounded by $r_{\text{MPS}}\leq t_{eg}^{-1}$, where $t_{eg}$ is the duration of the spin-photon entanglement generation.

This upper bound is saturated if the number of successful local BSMs per communication time $t_{c}$, $n=p_{\text{BSM}}\,t_{c}/t_{eg}\approx(1/2)p_{em}\eta\,t_{c}/t_{eg}$, satisfies $n\ll 1$. In this limit the protocol can be effectively run with a single qubit per node, and the rate is therefore insensitive to the swap gate time $t_{\text{sg}}$. When operating the MPS protocol in this low $n$ regime, the entanglement success rate is given by

[TABLE]

where the factor of $1/4$ arises because both BSMs must succeed in the same round, and $\eta$ includes the system losses for both the photon from the entangled photon source and the locally generated photon.

This scaling is different to that identified in Ref. 16 since, for the system parameters that we consider, $t_{\text{eg}}$ is not small enough to ensure that the expected number of successes $n$ per communication time $t_{c}$ approaches unity. As shown in Fig 5, for a shorter $t_{eg}$, the network could leave this low-success-probability regime. If the attempt rate is high enough to ensure that at least one attempt succeeds locally per $t_{c}$, the overall entanglement success rate will only primarily depend on whether there was a simultaneous success at the other node; the scaling is thus effectively proportional to $\eta$, which is the scaling described in Ref. 16. However, achieving this limit clearly requires a shorter $t_{\text{eg}}$ as the loss (1-$\eta$) increases.

For $n\sim 1$, the inclusion of additional memory qubits becomes beneficial to prevent idle time. In this case, after a local success, the communication qubit state is swapped to a memory qubit. This swapping operation therefore prevents the node from performing further entanglement generation attempts during a time $t_{\text{sg}}$, limiting the overall attempt rate.

II Modelling

We model each of the protocols described in the previous section with an approximate analytical approach as well as with Monte Carlo simulations. We use system parameters that are expected to be achievable for NVs and trapped ions in the near term (Tab. 1). The outcoupling efficiency of the NV centre is assumed to benefit from coupling to an optical cavity (with outcoupling efficiency $p_{\text{out}}=0.3)$, and emitted photons are assumed to be frequency-converted to telecom-wavelength photons with efficiency $p_{\text{fc}}=0.3$. Fiber losses are therefore limited to standard telecom values of $\alpha=0.2$ dB/km. Hence the overall system transmission efficiency is given by $\eta=p_{\text{out}}\,p_{\text{fc}}\,10^{-\alpha d/20}$ where the last term corresponds to the fiber losses over a distance of $d/2$.

It is as yet unclear how much progress will be made in the near term in overcoming the technical challenges necessary to demonstrate an entangled-photon-source with a high brightness and with spectral properties that are well-matched to the node emission. We therefore consider two possible values for $p_{\text{em}}$ (0.1 and 0.01), taking 0.01 to be more technically feasible Clausen et al. (2014); Loredo et al. (2016).

II.1 Scaling with distance

The modelled dependency of the entangling rate on the node separation is shown in Fig. 4. As expected from Section I, the scaling with distance is most favorable for the mEPL protocol ($R_{\text{mEPL}}\sim 10^{-\alpha d/20}\,d^{-1}$), whereas the BK protocol scales worst ($R_{\text{mBK}}\sim 10^{-\alpha d/10}\,d^{-1}$). Even for an MPS protocol with an extremely efficient source ($p_{em}=0.1$), the mEPL protocol outperforms it for distances greater than $\sim 100$ km since $R_{\text{MPS}}$ scales less favourably with distance as $R_{\text{MPS}}\sim 10^{-\alpha d/10}$.

In Fig. 5 we justify our claim that the MPS protocol will not benefit from more than a single qubit per node. We plot the expected number of successful BSMs $n$ during the communication time as a function of distance, and observe that for our network parameters this stays well below one even for the case of a very efficient source ($p_{em}=0.1$).

II.2 Scaling with number of memories

Notably, for these near-term parameters, scaling up to a large number of qubits per node does not speed up the entanglement rate. As previously noted, the MPS protocol always operates in the low success probability regime in which only the communication qubit is actively used. For the mBK and mEPL protocols, the duration of the swap gate significantly limits the number of qubits per node that can be used over relevant node separations. We investigate the rate dependency of the mEPL protocol on the number of memory qubits in Fig. 6 for a fixed node separation of $d=50$ km and a varying swap gate duration $t_{\text{sg}}$. For $t_{\text{sg}}\ll t_{c}$ the rate scales linearly with the number of qubits. However, as explained in Section I, once $Nt_{\text{sg}}\approx t_{c}$ is reached, adding more memory qubits does not boost the entangling rate.

III Conclusions

Our analysis highlights the potential of multiplexed distillation-based schemes to provide high rates of remote entanglement generation and the most favourable scaling with respect to losses. For such schemes, we have identified the swap gate time $t_{\text{sg}}$ between the communication and the memory qubits as the key parameter in constraining the achievable entanglement generation rate, as this limits the number of quantum memories that can be used. This highlights the importance of developing methods to increase this storage rate while ensuring that memories remain robust to decoherence. One promising approach for nitrogen-vacancy centre nodes may be to use pairs of strongly coupled carbons to encode quantum memories in decoherence protected subspaces that combine rapid gates (due to their strong coupling) with long memory lifetimes Reiserer et al. (2016).

We find that the midpoint-source protocol has a different dependence on the system parameters, with its performance only weakly constrained by the memory storage time. However, its increased sensitivity to losses hinders its performance over long distances. In addition, there is considerable uncertainty in the projected performance of entangled-pair sources in the near-term, particularly with regard to the source brightness. Until brightnesses on the order of 0.1 per attempt can be achieved, our analysis suggests that these schemes will not perform as effectively as the multiplexed distillation-based protocols.

Acknowledgements.

We acknowledge stimulating discussions with David Elkouss, Kenneth Goodenough, Norbert Kalb, Gláucia Murta, Tim Taminiau and Le P. Thinh. This work was supported by the Dutch Organization for Fundamental Research on Matter (FOM), Dutch Technology Foundation (STW), the Netherlands Organization for Scientic Research (NWO) through a VICI grant (RH), a VIDI grant (SW) and the European Research Council (ERC) through a Starting Grant (RH and SW).