Quantum repeaters based on two species trapped ions

We examine the viability of quantum repeaters based on two-species trapped ion modules for long distance quantum key distribution. Repeater nodes comprised of ion-trap modules of co-trapped ions of distinct species are considered. The species used for communication qubits has excellent optical properties while the other longer lived species serves as a memory qubit in the modules. Each module interacts with the network only via single photons emitted by the communication ions. Coherent Coulomb interaction between ions is utilized to transfer quantum information between the communication and memory ions and to achieve entanglement swapping between two memory ions. We describe simple modular quantum repeater architectures realizable with the ion-trap modules and numerically study the dependence of the quantum key distribution rate on various experimental parameters, including coupling efficiency, gate infidelity, operation time and length of the elementary links. Our analysis suggests crucial improvements necessary in a physical implementation for co-trapped two-species ions to be a competitive platform in long-distance quantum communication.


I. INTRODUCTION
Communication using information encoded into quantum states of single photons offers unconditional security since any interception can be detected. This is the primary motivation behind quantum communication and cryptography [1][2][3][4][5]. More generally, distributed entangled quantum states are a fundamental resource for quantum enhanced interferometry [6], metrology [7] and computation [8]. However, the attenuation in the optical fiber poses a significant challenge for the long-distance transmission of single photons [9,10]. Quantum repeaters [11,12] solve the attenuation problem by dividing the total communication distance into shorter channels connected by intermediate nodes, where the photon loss is detected [13][14][15][16][17][18] and can be corrected by using an active mechanism [19][20][21][22][23][24][25][26]. In addition to photon loss errors, operation errors accumulate over the quantum channel that degrades the quality of the transmitted entangled state. Depending on the methods used to overcome photon loss and operation errors, quantum repeaters make different demands on the quantum hardware for their construction [19,20,22].
Trapped ions offer one of the most mature technologies for scalable universal quantum computing [27][28][29] and quantum networking [30,31]. Construction of large-scale quantum networks require matter qubits with long lifetimes, fast quantum gates and measurement, efficient coupling to photons, and low error rates of quantum operations [29,[32][33][34][35][36][37][38][39][40]. Such a combination of characteristics is achievable if different elements with the desired properties are brought together in one technological platform. This is one of the reasons for the growing interest in two-species trapped ions (TSTI), for example, the pairs of 9 Be + -25 Mg + or 171 Yb + -138 Ba + ions [29,41]. Here a species of ions can be utilized as a communication qubit ( 25 Mg + or 138 Ba + ) that can be efficiently entangled to photons and a different longer lived species of ions ( 9 Be + or 171 Yb + ) that can be utilized as a quantum-memory qubit and for local processing. A high-fidelity transfer of quantum information between the two species of ions can also be achieved [29]. Furthermore, due to transition-frequency difference the optical operations with the communication ions do not disturb the quantum memory ions even though they are only a few microns away. Despite such experimental advances an analysis of quantum key generation rates, among the most promising uses of a quantum network, using two-species trapped ions for quantum repeaters is lacking.
We describe in this paper a quantum repeater architecture based on modules of two species of trapped ions. We study the viability of using these modules as building blocks for the construction of long distance quantum repeaters by analyzing quantum key distribution rates. The architecture can potentially work for any pair of communication and memory ions. For the numerical results later in the paper, however, we consider the advances made with the 171 Yb + -138 Ba + pair described in [41] with the following features: excellent isolation between 171 Yb + quantum memory ions and 138 Ba + communication ions was achieved and a state transfer between them was demonstrated; Ba + coherence times of 100 µs in unstabilized magnetic fields and a coherence time of 4ms with stabilized magnetic fields were measured; same-species twoqubit gates between 171 Yb + ions with 98% fidelity and crossspecies two-qubit gates between a 171 Yb + ion and 138 Ba + ion with 75% fidelty in 200 µs were demonstrated; light collection efficiency from the 138 Ba + ion of about 10% and fiber coupling efficiency of 17% in the absence of a cavity were reported. Our analysis suggests crucial improvements in the performance parameters that can maximize key generation rates. We focus, in this paper, on simple repeater architectures that comprise of a single TSTI module at every repeater station.
Based on the number of communication ions per module we classify TSTI modules as Type I and Type II. We then describe the modular operations necessary for quantum networking and highlight the operational differences between repeaters based on the module types in Sec. (II). The dependence of quantum key generation rates on experimental parameters such as gate error rates, operation time and coupling efficiency to fiber is presented in Sec. (III). We discuss the current experimental benchmarks and suggest the required improvements along with a concluding discussion in Sec. (IV).

II. ARCHITECTURES FOR QUANTUM REPEATERS
In this section we describe simple QR architectures feasible with TSTI modules. A single TSTI module consists of one or more communication ions (Ba + ) and at least two memory ions (Yb + ), Figure 1. We classify these modules, and the respective architectures which use these modules, by the number of their communication ions. The number of communication ions per module is a natural characteristic for classifying the modules and architectures because it determines the repeater protocol. Moreover, the number of communication ions per module also determines the technological complexity of the required hardware. One or more modules can be stacked to form a single quantum repeater station. These modules interact with each other in a heralded manner only via photons emitted by the Ba + ions resulting in ion-ion entanglement. The entanglement of the communication ions is then coherently transferred to the memory ions in the respective modules using a quantum swap gate. A different entanglement swapping operation on distinct memory ions in the same module then yields entanglement length doubling.
We focus, in the present work, on repeater architectures with a single TSTI module per repeater node and defer the discussion of architectures with multiple modules per repeater node to later work. Multiple TSTI modules per node permit encoding of the entangled quantum states but are challenging in the near term. Our goal in this paper is to identify technical improvements necessary in individual TSTI modules for viable quantum networking for which single modules per repeater node suffice. The elements of network operation described above categorize the simple architectures we consider in this paper into the second generation of quantum repeaters without encoding [19,20,22]. In the following subsection (II A) we describe the modular operations necessary for key generation and then in subsection (II B) we describe the simple repeater architectures we consider in the rest of the paper.  Two memory ions in the same module undergo an entanglement swapping operation (blue arrows) to increase the range of entangled quantum states. ure 2. An optical setup collects the photons and channels it through a fiber coupler to an optical fiber. Appropriate frequency conversion to telecom wavelengths is performed for the fiber coupling. Photons from modules at neighboring repeater stations interfere at a beam splitter between the repeater stations. The coincident detection of two photons at the two outputs of the beam spiltter then projects the state of the Ba + ions into an entangled state with a success probability of 1 2 [42]. The success probability of generating entanglement between two Ba + ions located at neighboring stations in one trial is, p = 1 2 η 2 c e −L0 Latt , where L 0 is the spacing between neighboring repeater stations and η c is the coupling efficiency including emission, collection and coupling loses. Modules with multiple communication ions have a greater probability that at least one pair of ions across distinct modules are successfully entangled per trial. However, this is technically challenging since the ions within a module are separated only by a few microns. At such close separations, with a high probability a single photon emitted by a Ba + ion may be resonantly absorbed by a different Ba + ion in the same module instead of being collected by the optical setup. To mitigate such intramodular absorption the Ba + ions must be distributed with maximum spacing among the Yb + ions along with spatially resolved photon collection.

Quantum state swap between communication and memory ions
A quantum swap gate is used to transfer the state of the Ba + communication ion to an available Yb + memory ion in a TSTI module. This operation is required to avoid entanglement swapping directly using the entangled communication ions. By doing so, we circumvent the difficulty of measuring the Ba + ions [41] and eliminate the effect of measurement errors. For pure states, ψ A ⟩ and ψ B ⟩, of two Ba + ions the quantum swap gate, S, exchanges the state of the two ions, A pair of swap gates, S c1m1 and S c2m2 , acting on two communication ions, c 1 and c 2 , in an entangled state β c1c2 ⟩ and two memory ions, m 1 and m 2 , in unentangled pure states ψ m1 ⟩ and ψ m2 ⟩ transfer the entanglement to the memory ions, This leaves the two Ba + ions in unentangled states ψ c1 ⟩ and ψ c2 ⟩. These communication ions may then be used for further rounds of heralded entanglement generation.
Gate operations on trapped ions are achieved via a sequence of laser pulses that controls dynamics of the ion wave functions. The desired swap gate in Eq. (1) can be decomposed into a quantum circuit comprised of a set of native gates allowed by the specific experiment. Most relevant among the several physical implementations of such gates are the Cirac-Zoller (CZ) [43,44] and the Molmer-Sorensen (MS) [45] gates. In the CZ-gate the motional mode of the ion crystal acts directly as a qubit transmitting quantum information. First, the internal state of one ion is mapped to the motion of an ion string followed by flipping the state of the target ion conditioned on the motion of the ion string. Finally the motion of the ion string is mapped back onto the original ion. On the other hand the main idea behind of the MS gate is to drive collective spin flips of the involved ions. Ions can change their state only collectively and by choosing an appropriate amount of interaction time the desired two qubit unitary can be implemented. In addition, for MS gates individual addressing of ions is not required and the gates do not fail completely even if the temperature of the ion string is not absolute zero. We refer the interested reader to reference [46] for an excellent introduction to these mechanisms. Here we mention that our numerical analysis of key generation rates in later sections is independent of the particular gate mechanism used.

Entanglement swapping between memory ions
A final step in our quantum repeater protocol is the entanglement swapping operation between Yb + memory ions within a TSTI module. By connecting one memory ion each from two adjacent entangled states, this operation extends the physical distance of the entangled state. Entanglement swapping on two memory ions can be achieved using a quantum circuit consisting of a CNOT operation followed by a X and Z measurement as shown in Figure 3. This procedure is equivalent to teleportation of the B ion state to the D ion. The en- tanglement swapping operation using coherent quantum gates on the memory ions is a deterministic process unlike the entanglement generation process which is probabilistic.

B. Overview of architectures
We now describe quantum repeater architectures based on the module configurations and modular operations described above. Repeaters with two different kinds of modular configurations are considered: repeaters using Type I modules with a single communication ion and Type II modules with two or more communication ions. More communication ions per module greatly enhance the key generation rates but the technological requirements to increase the number of Ba + ions per trap, in a manner useful for quantum networking, also get significantly more complex. We assume in this paper that in both cases the repeater stations comprise of a single TSTI module.
In a quantum repeater with Type I modules, entanglement is generated sequentially between the communication qubit of a module and another in its neighboring module -left followed by the right (or vice versa), Figure 4. Starting with heralded entanglement generation attempts to the left of a repeater station, two-way classical communication between neighboring stations confirms its success. A pair of quantum swap gates then act on a communication-memory ion pair in each module to yield a pair of entangled memory ions across the modules. This process is then repeated for the module in the neighboring repeater station to the right. Once a pair of entangled memory ions are available both to the left and right of a module, an entanglement swapping operation is performed that extends the physical range of the entangled state. While technologically Type I modules make modest demands, they are useful primarily for linear quantum repeaters. In quantum repeaters with Type II modules, the multiple communication ions in the same module allow simultaneous entanglement generation attempts with modules in neighboring repeater stations both to the left and right, Figure 5. Type II repeaters therefore obtain a significant boost in key generation rates. Moreover, such modules can be used in a true network configuration where the degree of a repeater node can be greater than two. However, Type II modules are technologically more demanding. Photons from a communication ion in a module can be resonantly absorbed by another communication ion in the same module. Therefore, spatially addressable excitation and photon collection is required. Operationally, Type II repeaters differ from Type I repeaters in that the entanglement generation attempts do not need to be sequential.

III. QUANTUM KEY GENERATION RATES AND ERROR MODEL FOR TSTI BASED REPEATERS
Here we present the error model used to calculate the quantum key generation rate achievable with TSTI module based quantum repeaters. Errors contribute to the quantum bit error rate of the key generation protocol and lead to a degradation of key generation rates. Our error model accounts for two kinds of errors: a) Photon loss errors incurred during the entanglement generation step between communication ions and b) operation errors of the quantum swap gate and the entanglement swapping operation. These errors are estimated in terms of the parameters of gate time, gate error rates and coupling efficiency between communication ions and fiber in the Type I and Type II repeaters. In addition we assume that the initial Bell states generated between the communication ions of two neighboring modules have a nonzero infidelity to account for experimental imperfections.

A. Error Model
The first kind of error we account for is the photon loss error which occurs because of fiber attenuation and a finite coupling efficiency, η c , between the communication ions and the fiber. This effect of this error is to reduce the success probability of entanglement generation between communication ions of two different modules given by, where L att = 20 km for conventional optical fibers. The factor of 1 2 occurs because of linear optical Bell state measurement [42]. Photon loss errors reduce the rate of obtaining remote entangled states and result in lower raw key generation rates. Second, we consider, the operation errors in the quantum gates and measurements used at the intermediate stations in the repeater protocols. This kind of error degrades the quality of the long distance entangled pair generated and contributes to the quantum bit error rate of the key generation protocol resulting in lower key generation rates. The quantum swap gate requires three Molmer-Sorensen type gates (or two if one of the qubits is in a prepared state). The error in the swap gate is therefore three (or two) times that of the CNOT gate to first order. We assume that these gate errors are parameterized by the larger of the gate errors among the swap and CNOT gates. We denote the effective error of the quantum channel, E, representing a pair of quantum swap gates that transfers the entangled state, ρ c1c2 , of two communication ions to two memory ions by g . This implies that the output of this channel, ρ ′ m1m2 = E(ρ m1 ρ c1c2 ρ m2 ) = Tr c1,c2 {S c1m1 S c2m2 (ρ m1 ρ c1c2 ρ m2 )S † c1m1 S † c2m2 }, obtained after tracing out the state of the communication ions is given by, where, {σ k } k=1,2,3 = {X, Y, Z}, are Pauli matrices and σ 0 = I is the identity operator on a single qubit. Further, the fidelity of the entangled pair reduces after performing entanglement swapping. Suppose, the input state generated between communication ions is given by, (4) where, φ ± ⟩ = ( 00⟩ ± 11⟩) √ 2 and ψ ± ⟩ = ( 01⟩ ± 10⟩) √ 2 are the complete set of the four perfectly entangled Bell states. Then, after performing a swap gate between memory and communication ions, the probability that there is either a X or a Y or a Z error in any one of the qubits in the Bell pair is given by (1−F0) 3 + g 4 . Note that entanglement swapping requires measurements in X and Z basis respectively. While, X and Y type errors are detected in Z measurements, Z and Y type errors are detected in X measurements. This gives the effective error measured at each repeater station to be Taking into account for odd number of errors in repeater stations, the quantum bit error rate is given by [19], We can define an effective quantum bit error rate Q as Q = (Q X + Q Z ) 2. We will use this expression for the calculation of key generation rates in the forthcoming section.

B. Rate calculation
We now calculate the key generation rates for Type I and II repeaters. A total repeater length of L tot between remote loca-tions, A and B, with intermediate repeater stations at a spacing of L 0 is considered. For the Type I repeaters, the probability of having a remote entangled pair between A and B after n eg steps of heralded entanglement generation between all neighboring repeater stations is given by, The total time taken for entanglement generation is given by where t 0 is the gate (swap or CNOT) and measurement time.
The raw key generation rates is given by, For the second architecture with multiple communication ions and memory ions, the raw key generation rates is given by is the probability to have i elementary links with m qubits with n 0 rounds of entanglement generation. Therefore, [1 − Prob(0,n eg )] is the probability to have at least one entangled pair after n eg attempts. The secure key generation rates for the two basis protocol is given by [2,4] where Q is the quantum bit error rate of the protocol. Q is probability that there is an error in the final Bell pair generated between end repeater stations.

C. Numerical results
We cast the numerical evaluation of secure key generation rates of Type I and II repeaters as a multi-parameter optimization problem. The secure key generation rate given in Eq. (11) depends on the following parameters: total repeater length, L tot ; repeater spacing, L 0 ; fiber coupling efficiency, η c ; quantum swap gate and CNOT gate error, g ; gate and measurement time for quantum swap gate and CNOT gate, t 0 ; initial infidelity of Bell pair generation. A total repeater length of L tot = 1000 km and an initial infidelity of Bell pair generation of 10 −4 is assumed. In all the plots we report the maximum obtainable secure key generation rates by optimizing over the number of entanglement generation steps, n eg .
To start, we study the optimal repeater spacing -a crucial repeater design parameter that determines the key generation rate of Type I and Type II repeaters. On one hand, with small repeater spacings, i.e., many repeater stations, the operation errors dominate: multiple applications of the quantum swap gate and CNOT gate lowers the fidelity of the generated Bell pairs and increases the quantum bit error rate, Q. On the other hand, with large repeater spacing, i.e., small number of repeater stations, the loss errors dominate: a large number of entanglement generation steps are required leading to low secure key generation rates. This suggests that there exists an optimal repeater spacing for which the secure key generation rate is the highest at every value of η c . We show the dependence of key generation rates with respect to repeater spacing in Figure 6(a). Next, we study the variation of key generation rate with respect to coupling efficiency, η c , at the optimal repeater spacing as shown in Figure 6(b). Reasonable key generation rates (∼100 bits/s) are obtained for coupling efficiencies of about 70% for Type I repeaters with 1 communication and 2 memory ions (Type I -1C/2M). However, the rate achievable with Type II repeaters with 10 communication and 2 memory ions (Type II -10C/2M) ions are about an order of magnitude higher. We note the sharp increase in the secure key generation rate by two orders of magnitude as η c increases to 100% from 10%. We note that R sec scales as η c 2 . This suggests that higher coupling efficiencies are necessary to achieve high secure key generation rates for all TSTI module based repeater architectures.
Further, the numerics accounts for the effect of operation errors by letting operation errors accumulate linearly with the number of repeater stations. As the operation errors adds up, the quantum bit error rate of the key generation protocol in-creases and the secure key generation rate reduces rapidly. The variation of key generation rates with respect to gate error rate is shown in Figure 6(c). As it can be seen in the figure, the architectures tolerates operation errors up to 2.5 × 10 −3 Notice that reasonable secure key generation rates can still be achieved at error values well above the error correction threshold of 10 −4 [47].
Finally, the dependence of key generation rates on the gate operation time, t 0 , is shown in Figure 6(d). The quantum swap gate needed for transferring the quantum state from the communication to the memory ions and the CNOT gate required for entanglement swapping (including the X and Z measurements) take a finite amount of time which appears in the rate evaluation. Note that the secure key generation rate plateaus for faster gate times. Therefore, it is relatively insensitive to the gate operation time as long as gates are faster than a few hundred microseconds.
To highlight the necessity of higher coupling efficiency, η c , we present the optimized values of repeater spacing and the number of entanglement generation steps for η c = 100% and η c = 10% in Table I. The data reveals that while the optimal repeater spacing does not change much with the increase of η c , the secure key generation rate shows dramatic improvement. For both Type I and Type II repeaters the secure key generation rate improves by a factor of two orders of magnitude.   ). Ltot = 1000 km, g = 10 −4 , t0 = 1 µs, initial Bell state infidelity of 10 −4 .
Our numerical results present an optimistic scenario since the experimental state-of-art coupling efficiency is about 1.7% between the Ba + communication ion and fiber in the absence of a cavity. However, this can be significantly enhanced using adequate cavity QED effect (Purcell enhancement). Further, photon collection optics with a higher effective numerical aperture needs to be used. The state-of-art gate error rate is about 2% for two-qubit gates between ions of different species for a Be-Mg pair [29]. At these error rates secure quantum key generation rates are highly suppressed for a total distance of L tot = 1000 km. However, these error rates for cross-species gates are sufficient to obtain reasonable key generation rates across a shorter total repeater length, L tot = 500km, as shown in Figure 7(a) assuming perfect coupling between ion and fiber. Interestingly, for such large gate error rates while the key generation rates are low, the optimization picks higher repeater spacings ∼50-100 km, as shown in Figure 7(b), and large number of entanglement generation attempts, n eg . Secure key generation rate with respect to gate error rates for ηc = 30%, t0 = 1 µs and b) Optimal repeater spacing, L0, with respect to operation error g for ηc = 30%, t0 = 1 µs with Ltot = 500 km for Type I (1C/2M, blue curve) and Type II (10C/2M, red curve) repeaters.

IV. DISCUSSION AND CONCLUSION
We studied quantum repeater architectures based on TSTI modules for their utility towards secure quantum key generation. Three elementary modular operations necessary for this application were identified: heralded entanglement generation between communication ions, quantum state swap between communication and memory ions and entanglement swapping between memory ions. We classified the modules into Types I and II based on the number of communication ions and associated technological complexity. Secure quantum key generation rates were calculated for repeater protocols based on the modular operations assuming a general error model. Finally, the variation of the key generation rate with respect to repeater and error model parameters was studied.
Our analysis revealed that repeater designs using TSTI modules possess a combination of characteristics that is promising for applications such as quantum key generation. The numerical analysis pointed out the improvements in the experimental parameters necessary to obtain reasonable secure key rates over long distances. Increasing the coupling efficiency was found to have the most beneficial effect on the rates while faster gate times resulted in more modest gains. With adequate cavity QED effect, the fiber coupling efficiency can be further improved over 30% and the collection probability of over 15% is possible [35]. In any case, the gate errors need to be considerably improved to 99.9% levels especially for the quantum swap gate that acts on ions of two different species. Finally, coherence time of the memory ion has to be further increased to allow for multiple rounds of entanglement generation [33].
Besides trapped ions, there are several other competitive platforms in contention for the construction of quantum repeaters such as ensemble of atoms, Nitrogen-Vacancy (NV) centers and superconducting qubits. While an ensemble of atoms can be coupled to photons with a high efficiency of 85% the only known approach to apply quantum gates is through linear optics which is limited by a finite success probability of 1 2. This causes the key generation rates to drop exponentially with the number of entanglement swapping operations. NV centers can be coupled to photons with an efficiency of 15%, which is higher than the currently experimentally achievable coupling efficiency with communication ions. However, the spin-photon entanglement success probability is very low, about 10 −6 , for NV centers compared to ions which is about 0.07. While superconducting qubits are another attractive candidate, the wavelength conversion efficiency between a single microwave photon and a telecom photon is expected to be very low.
For future work, we will focus on quantum repeater architectures with more than one module per repeater station. These designs will harness the improvements in the experimental parameters of TSTI modules anticipated in the near term. Multiple interconnected modules, with higher coupling and low error gates, at the same repeater station will permit encoding of quantum states allowing robust key generation rates over long distances. CM acknowledges support by the ARO with funds from the IARPA LogiQ program, the AFOSR project on Quantum Networks and MURI on Quantum Measurement and Verification, the ARL Center for Distributed Quantum Information, and the National Science Foundation Physics Frontier Center at JQI.