Towards long-distance quantum networks with superconducting processors and optical links

TL;DR

This paper proposes a quantum repeater architecture using microwave cat state qubits and optical links, enabling long-distance quantum communication and scalable quantum networks with enhanced entanglement distribution rates.

Contribution

It introduces a novel repeater design with microwave-to-optical transduction and spectral multiplexing, improving over existing methods for long-distance quantum networking.

Findings

Achievable entanglement distribution rates surpass direct transmission.

Proposed architecture maintains high fidelity despite noise and imperfections.

Potential for secure quantum communication and large-scale quantum networks.

Abstract

We design a quantum repeater architecture, necessary for long distance quantum networks, using the recently proposed microwave cat state qubits, formed and manipulated via interaction between a superconducting nonlinear element and a microwave cavity. These qubits are especially attractive for repeaters because in addition to serving as excellent computational units with deterministic gate operations, they also have coherence times long enough to deal with the unavoidable propagation delays. Since microwave photons are too low in energy to be able to carry quantum information over long distances, as an intermediate step, we expand on a recently proposed microwave to optical transduction protocol using excited states of a rare-earth ion ($\mathrm{Er^{3+}}$) doped crystal. To enhance the entanglement distribution rate, we propose to use spectral multiplexing by employing an array of cavities at each node. We compare our achievable rates with direct transmission and with two other promising repeater approaches, and show that ours could be higher in appropriate regimes, even in the presence of realistic imperfections and noise, while maintaining reasonably high fidelities of the final state. Thus, in the short term, our work could be directly useful for secure quantum communication, whereas in the long term, we can envision a large scale distributed quantum computing network built on our architecture.

Figures

Equations

$$H_0(t)=-K{a^{†}}^2{a}^2+({\mathcal{E}}_p(t){a^{†}}^2+{\mathcal{E}}_p^{\ast }(t)a^2)$$

$$H_x=H_0+{\mathcal{E}}_x(a+a^{†})$$

$$H_z=-K(a^{†}a{)}^2$$

$$H_c=H_{01}+H_{02}+{\mathcal{E}}_c(a_1^{†}{a}_2+a_1{a}_2^{†})$$

$$H=\sum _j\hslash \frac{{\omega }_j}{2}{\sigma }_z^{(j)}+\sum _j\hslash g^{\prime }(a^{†}{\sigma }_{-}^{(j)}+a{\sigma }_{+}^{(j)})+\hslash {\omega }_m{a}^{†}a$$

$${\mathrm{X}}_{\pi /2}^2{\mathrm{X}}_{-\pi /2}^1{\mathrm{Z}}_{\pi /2}^1{\mathrm{G}}_{\pi /2}{\mathrm{X}}_{-\pi /2}^1{\mathrm{Z}}_{-\pi /2}^1{\mathrm{X}}_{\pi /2}^1$$

$$H={\omega }_c{a}^{†}a+{\omega }_q{b}^{†}b-K_q{b^{†}}^2{b}^2+g(a^{†}b+a{b}^{†})$$

$${⟨T⟩}_L=(\frac{3}{2}{)}^n(\frac{L_0}{c}+T_o)\frac{1}{P_0{P}_1{P}_2....P_n}$$

$$F_{tot}=(F_{elem}{)}^l\times (F_{swap}{)}^{l-1}\times C_R$$

$${⟨T⟩}_L=(\frac{L_0}{c})\frac{f_0{f}_1{f}_2....f_n}{P_0{P}_1{P}_2....P_n}$$

$${⟨T⟩}_L=(\frac{3}{2}{)}^n(\frac{L_0}{c}+T_o)\frac{1}{P_0{P}_1{P}_2....P_n}$$

$$F_{tot}=(F_{elem}{)}^l\times (F_{swap}{)}^{l-1}\times C_R$$

Full text

I Introduction

Quantum networks will enable applications such as secure quantum communication based on quantum key distribution 84_Cryptography_Brassard; 02_Cryptography_Zbinden, remote secure access to quantum computers based on blind quantum computation 09_Universal_Kashefi; 12_Demonstration_Walther, private database queries 11_Practical_Zbinden, more precise global timekeeping 14_Quantum_Lukin and telescopes 12_Longer_Croke, as well as fundamental tests of quantum non-locality and quantum gravity 12_Fundamental_Terno. If the total distance through which entanglement needs to be distributed is greater than a few hundred kilometres, then direct transmission through fibers leads to prohibitive loss. This can be overcome using quantum repeaters 98_Quantum_Zoller; 11_Quantum_Gisin, which require small quantum processors and quantum memories at intermediate locations. The development of quantum computers has recently seen many impressive accomplishments 16_Scalable_Martinis; 17_Solving_Spiropulu; 18_Fault_Schoelkopf; 18_Quantum_Savage. Networking quantum computers is of interest both in the medium term, when individual processors are likely to be limited in size due to technical constraints, and in the long term, where one can envision a full-fledged quantum internet 08_Quantum_Kimble; 18_Internet_Hanson; 17_Towards_Simon, which would be similar to the classical internet of today, but much more secure, and powerful in certain aspects.

Out of different architectures for quantum computation being pursued 05_Resource_Rudolph; 02_Architecture_Wineland; 07_SQ_Blais; 99_QIP_Small; 98_Silicon_Kane, superconducting circuits are currently one of the leading systems. Easy addressability, high fidelity operations, and strong coupling strength with microwave photons 04_Strong_Schoelkopf; 10_CQED_Gross; 17_Microwave_Nori; 16_SQ_Semba are some of their most attractive features. Until recently, one challenge for using superconducting processors in a network context were the relatively short coherence times of superconducting qubits (SQs) 12_SQ_Steffen. Long absolute coherence times are important for quantum networks because of the unavoidable time delays associated with propagation. However, a recent development could help circumvent this limitation. Instead of the states of a SQ, one can use the states of the coupled microwave cavity as logical qubits, and manipulate these cavity states using strong SQ-cavity interaction 13_Hardware_Mirrahimi; 14_Dynamically_Devoret; 15_Confining_Devoret; 16_Extending_Schoelkopf; 16_Schrodinger_Schoelkopf; 16_Universal_Goto; 17_Engineering_Blais; 16_Bifurcation_Goto; 17_Robust_Tiwari; 17_Quantum_Blais; 18_CNOT_Schoelkopf. The coherence time of these cavities can be much higher than 100 ms 07_Ultrahigh_Visentin; 13_Reaching_Schoelkopf; 18_3D_Grassellino, whereas that of the best SQ is only close to 0.1 ms 12_SQ_Steffen. Moreover, the infinite dimensional Hilbert space of a cavity can allow new error correction codes which are more resource efficient than the conventional multi-qubit codes 01_Encoding_Preskill; 14_Dynamically_Devoret; 16_Extending_Schoelkopf; 17_Degeneracy_Mirrahimi; 18_Stabilized_Girvin. Since photon loss is the only prominent error channel here, unlike other architectures, it is also relatively easier to keep track of these errors.

Out of several cavity based architectures for quantum computation, the approach introduced in ref. 16_Universal_Goto; 17_Engineering_Blais seems to be one of the most promising candidates because of the relative experimental ease in preparing and manipulating their system. They use coherent states of a microwave cavity as qubits. A nonlinear superconducting element comprising of Josephson junction(s) is placed inside the cavity, and the whole system is driven using a two-photon drive, which generates superpositions of coherent states, also known as cat states, and stabilises them against amplitude decay, even in the presence of single photon loss. Gates can be performed using detunings and additional single photon drives. Two or more cavities can be coupled easily, e.g. using a transmon 14_Dynamically_Devoret; 16_Schrodinger_Schoelkopf; 18_CNOT_Schoelkopf; 19_Entanglement_Schoelkopf.

Let us consider a possible scenario where the quantum processors at each node are based on the microwave cavity architecture discussed in ref. 16_Universal_Goto; 17_Engineering_Blais. Since the energies of the microwave photons exiting these cavities are lower than the thermal noise at room temperature, they cannot be used to carry quantum information over long distances. We need quantum transducers to convert these microwave photons to telecom wavelengths, which can then be transported via optical fibres or in free space. In addition, we also need a way to encode and send information that is robust to the realistic noise and losses incurred in long distance transmission. In this paper, we propose a novel scheme to generate robust entanglement between cat state qubits of distant microwave cavities mediated via telecom photons (see Fig. 1). There has been a great deal of work related to microwave-to-optical transduction by hybridising different physical systems with superconducting circuits and microwave resonators, e.g. NV centers in diamond 10_Strong_Esteve, cold gases 09_Strong_Schmiedmayer, rare-earth ions 12_Coupling_Johansson; 14_Interfacing_Fleischhauer; 14_Magneto_Longdell, and optomechanical systems 13_Nanomechanical_Cleland; 14_Bidirectional_Lehnert; 18_Harnessing_Regal; 18_Microwave_Groblacher. Some of us recently developed a protocol to achieve transduction in a rare-earth ion ($\mathrm{Er^{3+}}$) doped crystal 18_Electron_Thiel, which has an advantage of easier integration with the current telecom fibers, mainly because of its addressable telecom wavelength transition. We use that protocol in our scheme and discuss it in greater detail here.

Unlike in classical communication, where one can amplify signals at intermediate stations, in quantum communication, the no cloning theorem prohibits this kind of amplification. One solution is to use quantum repeaters, where the total distance is divided into several elementary links and entanglement is first generated between the end points of each elementary link, and then swapped between neighbouring links in a hierarchical fashion to ultimately distribute it over the whole distance 98_Quantum_Zoller; 11_Quantum_Gisin. Many repeater protocols rely on using atomic ensembles 01_Long_Zoller; 11_Quantum_Gisin at the repeater nodes. But the success probability of their swapping operation based on linear optics is limited to a maximum of $\mathrm{50\%}$ 99_Bell_Suominem, which hampers the performance over long distances and complex networks. It is possible to increase the success probability using auxiliary photons 11_Arbitrarily_Grice; 16_Efficiency_Simon. But this makes the system more complicated and error-prone, limiting its practical utility. Repeater approaches using deterministic swapping operations have been proposed in different systems to overcome this limitation, e.g. in trapped ions 09_Quantum_Simon, Rydberg atoms 10_Efficient_Zoller; 10_Quantumrepeaters_Simon, atom-cavity systems 15_Cavity-based_Rempe; 19_Telecom_Painter, and individual rare-earth ions in crystals 18_Quantum_Simon. Some components of such a repeater have been implemented experimentally in a few systems, e.g. NV centers in diamond 15_Loophole_Hanson, trapped ions 07_Entanglement_Monroe, and quantum dots 16_Generation_Imamoglu.

Deterministic two-qubit gates are easily achievable in the microwave cavity architecture proposed in ref. 16_Universal_Goto; 17_Engineering_Blais. So, instead of relying on one of the above systems to build a repeater, we leverage the quantum advancements of superconducting processors, and develop a new repeater scheme using microwave cavities and transducers. This could also be useful in allocating some resources of relatively nearby quantum computing nodes to serve as repeater links to connect more distant nodes. We calculate the entanglement distribution rates, and compare those with direct transmission, with the well-known ensemble-based Duan-Lukin-Cirac-Zoller (DLCZ) repeater protocol 01_Long_Zoller, and with a recently proposed single-emitter-based approach in rare-earth (RE) ion doped crystals 18_Quantum_Simon. We conclude that our approach could yield higher rates in suitable regimes. We also estimate the fidelities of our final entangled states in the presence of realistic noise and imperfections, and we find them to be sufficiently high to perform useful quantum communication tasks, even without entanglement purification or quantum error correction. The latter protocols are likely to be needed for more complex tasks such as distributed quantum computing, and we anticipate that it should be possible to incorporate them in the present framework.

The paper is organised as follows: In Sec. II, we briefly describe the qubit and the gates, following ref. 16_Universal_Goto; 17_Engineering_Blais. In Sec. III, we discuss the entanglement generation scheme between distant qubits, which includes a description of the transduction protocol. Sec. IV deals with our proposal for a quantum repeater with the same architecture. In Sec. V, we provide some additional implementation details pertinent to our proposal. In Sec. VI, we estimate the rates and fidelities of our final entangled states, and make pertinent comparisons with other schemes. In Sec. VII, we draw our conclusions and enumerate a few open questions and avenues for future research.

II The qubit and the universal set of gates

II.1 Theoretical model

An individual physical unit of the quantum processor is a nonlinear microwave cavity. The nonlinearity arises because of the interaction of the cavity with a nonlinear superconducting element, e.g. a transmon placed inside (see Fig. 1). The system is driven by a two-photon drive. The Hamiltonian of the system with two photon driving, in a frame rotating at the resonance frequency of the cavity is

[TABLE]

Here, $a$ is the annihilation operator of the cavity mode, $K$ is the magnitude of Kerr nonlinearity, and $\mathcal{E}_{p}(t)$ is the time dependent pump amplitude of the two photon parametric drive.

The coherent states $\ket{\alpha}$ and $\ket{-\alpha}$, where $\alpha=\sqrt{\mathcal{E}_{p}/K}$, are instantaneous eigenstates of this Hamiltonian. Under an adiabatic evolution of the pulse amplitude $\mathcal{E}_{p}(t)$, the photon number states (Fock states) $\ket{0}$ and $\ket{1}$ are mapped to the states $\ket{C_{\alpha}^{+}}$ and $\ket{C_{\alpha}^{-}}$ respectively. These so called cat states are equal superpositions of coherent states: $\ket{C_{\alpha}^{\pm}}=\mathcal{N}_{\alpha}^{\pm}(\ket{\alpha}\pm\ket{-\alpha})$, where the normalization constant $\mathcal{N}_{\alpha}^{\pm}=1/\sqrt{2(1\pm e^{-2|\alpha|^{2}})}$. Faster non-adiabatic evolution is possible with high fidelity using a transitionless driving approach 09_Transitionless_Berry; 17_Engineering_Blais (see Sec. V.1.1 for a detailed discussion).

There are several advantages of forming and manipulating cat states this way using Kerr nonlinearity, as compared to other techniques, e.g. using the engineered dissipation approach 14_Dynamically_Devoret; 15_Confining_Devoret. The system is easier to implement experimentally, has better stabilization against losses, has effectively longer coherence times, and since all the processes are unitary, it is possible to trace back a path in phase space, e.g. between cat states and Fock states, which is an important requirement for our entanglement generation scheme discussed in Sec. III. For conciseness, we shall refer to the mapping from photon number states to cat states as ‘driving’ and from cat to photon number states as ‘undriving’.

We pick $\ket{C_{\alpha}^{+}}$ and $\ket{C_{\alpha}^{-}}$ as our logical qubits, denoted by $\ket{\bar{0}}$ and $\ket{\bar{1}}$ respectively. We pick $\alpha$ large enough such that the coherent states $\ket{\alpha}$ and $\ket{-\alpha}$ have negligible overlap. Continuous rotation around X axis can be implemented using an additional single photon drive, described by the Hamiltonian

[TABLE]

Here $\mathcal{E}_{x}$ is the amplitude of the additional drive. Rotation around Z axis can be achieved by switching off the two photon drive, and evolving the system under the Hamiltonian

[TABLE]

To complete a universal set of gates, we need a two qubit entangling operation. The Hamiltonian for the system of two linearly coupled cavities is

[TABLE]

$H_{01}$ and $H_{02}$ are the Hamiltonians of the individual cavities (given by Eqn. 1), $a_{1}$ and $a_{2}$ are the annihilation operators for their respective microwave photonic modes, and $\mathcal{E}_{c}$ is the coupling strength. The cavities can be coupled using a transmon with two antenna pads, one in each of these cavities, overlapping with the microwave fields inside 16_Schrodinger_Schoelkopf; 18_CNOT_Schoelkopf; 19_Entanglement_Schoelkopf (see Fig. 1).

III Entanglement generation scheme between two distant nodes

Next, we discuss our scheme to generate entanglement between distant cat state qubits, which involves application of a few of the gates discussed in the previous section. The scheme is inspired from Barrett-Kok’s (BK) original theoretical proposal 05_Efficient_Kok for entanglement generation between distant atomic spins. The protocol is robust to path length fluctuations of the fiber connecting the distant spins. Minor variants of the BK protocol have been implemented in a few recent experiments 13_Heralded_Hanson; 15_Loophole_Hanson; 16_Robust_Devoret; probably the most notable one was the demonstration of loophole-free violation of Bell’s inequality 15_Loophole_Hanson. The novelty of our work is in bringing a promising contender for quantum computation, i.e. microwave cat qubits, and probably the only feasible carrier for long distance quantum communication, i.e. optical photons, together in a single platform, with a robust BK like entanglement generation protocol. We rely on microwave cavity cat states for computational and storage tasks, and on optical Fock states for communication.

III.1 Entanglement generation protocol

Our goal is to entangle one qubit at one node, let’s call it node ‘A’, with another qubit at another node, let’s call it node ‘B’. For this, we start with a set of two coupled cavities at each node, e.g. cavities ‘a’ and ‘b’ at node ‘A’ and cavities ‘c’ and ‘d’ at node ‘B’, as shown in Fig. 1. We shall perform a sequence of operations such that at the end, one of these cavities at each node (cavity ‘a’ at node ‘A’ and cavity ‘c’ at node ‘B’) are entangled. Our approach for remote entanglement generation is somewhat similar in spirit to those suggested for NV centers in diamond 07_Quantum_register_Lukin; 16_Robust_Markham, where the aim is to entangle distant long-lived nuclear spins of \ce^13 C (analogous to our cavities ‘a’ and ‘c’), by performing spin-photon operations on the optically addressable electronic spins of NV centers and using the hyperfine interaction between the nearby electronic and nuclear spins. A similar approach has also been proposed recently in a rare-earth ion based repeater architecture18_Quantum_Simon. Here, individual $\mathrm{Er^{3+}}$ spins in crystals are manipulated to generate spin-photon entanglement, and the state of $\mathrm{Er^{3+}}$ is then mapped to the long lived nuclear spins of the nearby $\mathrm{Eu^{3+}}$ ions, which serve as the storage qubits.

Following the BK’s proposal, we incorporate a two step protocol, depicted in Fig. 2. Step (II) helps in discarding the noise terms and in stabilizing against path length fluctuations. In step (I) of the protocol shown in Fig. 2, we first set out to generate entanglement between the two neighbouring cavities at each node. Let’s focus on cavities ‘a’ and ‘b’ at node ‘A’. The cavities are initially in vacuum state. They are driven using suitable microwave pulses and an $X_{\pi/2}$ rotation on cavity ‘a’ prepares the system in the state $\frac{1}{\sqrt{2}}(\ket{\bar{0}_{\mathrm{a}}}+\ket{\bar{1}_{\mathrm{a}}})\otimes\ket{\bar{0}_{\mathrm{b}}}$. As a reminder, $\ket{\bar{0}}$ and $\ket{\bar{1}}$ are the cat states $\ket{C_{\alpha}^{+}}$ and $\ket{C_{\alpha}^{-}}$ respectively. We then perform a CNOT operation (see Sec. V.1.2 for the sequence of gates needed), with the first qubit as the control and the second as the target, to reach the entangled state $\frac{1}{\sqrt{2}}(\ket{\bar{0}_{\mathrm{a}}}\otimes\ket{\bar{0}_{\mathrm{b}}}+\ket{\bar{1}_{\mathrm{a}}}\otimes\ket{\bar{1}_{\mathrm{b}}})$. Cavity ‘b’ is then undriven coherently (see Sec. V.1.1), such that $\ket{\bar{0}_{\mathrm{b}}}\rightarrow\ket{0_{\mathrm{b}}}$ and $\ket{\bar{1}_{\mathrm{b}}}\rightarrow\ket{1_{\mathrm{b}}}$, where $\ket{0_{\mathrm{b}}}$ and $\ket{1_{\mathrm{b}}}$ are the microwave Fock states of cavity ‘b’. A quantum transducer is then used to convert microwave photons to propagating optical photons at telecom wavelength. We discuss one possible transduction protocol in the next subsection. To avoid usage of more complicated notations, telecom Fock states obtained after transduction shall also be represented by $\ket{0_{\mathrm{b}}}$ and $\ket{1_{\mathrm{b}}}$; the distinction between microwave and telecom Fock states should be clear from the context.

The same procedure is followed in a distant set of two cavities ‘c’ and ‘d’ at node ‘B’. The cat states stored in the cavities ‘a’ and ‘c’ shall be referred to as the ‘stationary qubits’, and the telecom Fock states as the ‘flying qubits’. The combined state of the system, including the stationary and flying qubits at this point is $\frac{1}{2}(\ket{\bar{0}_{\mathrm{a}}\bar{0}_{\mathrm{c}}}\ket{0_{\mathrm{b}}0_{\mathrm{d}}}+\ket{\bar{0}_{\mathrm{a}}\bar{1}_{\mathrm{c}}}\ket{0_{\mathrm{b}}1_{\mathrm{d}}}+\ket{\bar{1}_{\mathrm{a}}\bar{0}_{\mathrm{c}}}\ket{1_{\mathrm{b}}0_{\mathrm{d}}}+\ket{\bar{1}_{\mathrm{a}}\bar{1}_{\mathrm{c}}}\ket{1_{\mathrm{b}}1_{\mathrm{d}}})$. The telecom photons are directed to a beamsplitter placed midway between those distant nodes. Detection of only a single photon in either of the output ports of the beamsplitter would imply measuring the photonic state $\frac{1}{\sqrt{2}}(\ket{1_{\mathrm{b}}0_{\mathrm{d}}}+e^{-i\phi}\ket{0_{\mathrm{b}}1_{\mathrm{d}}})$ or $\frac{1}{\sqrt{2}}(\ket{1_{\mathrm{b}}0_{\mathrm{d}}}-e^{-i\phi}\ket{0_{\mathrm{b}}1_{\mathrm{d}}})$. Here $\phi$ is the phase difference accumulated because of the different path lengths of those photons. The detection event projects the stationary qubits into either of the maximally entangled odd Bell states $\frac{1}{\sqrt{2}}(\ket{\bar{1}_{\mathrm{a}}\bar{0}_{\mathrm{c}}}+e^{-i\phi}\ket{\bar{0}_{\mathrm{a}}\bar{1}_{\mathrm{c}}})$ or $\frac{1}{\sqrt{2}}(\ket{\bar{1}_{\mathrm{a}}\bar{0}_{\mathrm{c}}}-e^{-i\phi}\ket{\bar{0}_{\mathrm{a}}\bar{1}_{\mathrm{c}}})$.

If the detectors can’t resolve the number of photons, or if both the nodes emitted a photon each and one of those photons was lost in transmission, then instead of the above pure entangled state of stationary qubits, the projected state would be a mixture, with contribution from the term $\ket{\bar{1}_{\mathrm{a}}\bar{1}_{\mathrm{c}}}$. To get rid of this term, and to make the protocol insensitive to phase noise which is greatly detrimental to the fidelity of the final state, we follow BK’s approach 05_Efficient_Kok, and perform step (II).

In step (II), we flip the stationary qubits ($\mathrm{X_{\pi}}$ rotation), drive the cavities ‘b’ and ‘d’ from vacuum states to $\ket{\bar{0}_{\mathrm{b}}}$ and $\ket{\bar{0}_{\mathrm{d}}}$ respectively, and then perform the same operations as in step (I), i.e. CNOT operation, followed by undriving of the cavities ‘b’ and ‘d’, followed by transduction to telecom photons which leave the cavity, and finally single photon detection in one of the output ports of the beamsplitter. Successful single photon detection heralds the entanglement, and the state of the stationary qubits is either $\frac{1}{\sqrt{2}}(\ket{\bar{1}_{\mathrm{a}}\bar{0}_{\mathrm{c}}}+\ket{\bar{0}_{\mathrm{a}}\bar{1}_{\mathrm{c}}})$ or $\frac{1}{\sqrt{2}}(\ket{\bar{1}_{\mathrm{a}}\bar{0}_{\mathrm{c}}}-\ket{\bar{0}_{\mathrm{a}}\bar{1}_{\mathrm{c}}})$ depending on whether the same or different detector(s) clicked during each step. Once the two nodes A and B share an entangled state, it can also be used to teleport a quantum state from one node to the other using only classical bits and local operations 93_Teleporting_Wootters.

The main reason that we undrive one of the two cavities at each node to generate Fock states, which we then transduce to telecom wavelengths, instead of the more obvious route of directly transducing these microwave cat states to telecom cat states and then transporting it over a fiber is that cat states dephase because of photon loss and it’s complicated to correct for those errors over long-distances where fiber loss could be greater than $90\%$ 17_Cat_Jiang; 16_Concurrent_Jiang.

III.2 Transduction protocol

Microwave to telecom transduction has attracted significant attention in recent times for various applications involving hybridising different systems 13_Hybrid_Nori. Coupling superconducting circuits with spin based systems e.g. NV centers 10_Strong_Esteve, cold gases 09_Strong_Schmiedmayer and rare-earth ions 12_Coupling_Johansson; 13_Anisotropic_Bushev; 14_Interfacing_Fleischhauer; 15_Interfacing_Morigi, or optomechanical systems 13_Nanomechanical_Cleland; 14_Bidirectional_Lehnert; 18_Harnessing_Regal; 18_Microwave_Groblacher are a few possible ways to achieve desired frequency conversions. Our entanglement generation protocol does not rely on a specific implementation of the transducer. However as a concrete example, we elaborate on the protocol some of us proposed in ref. 18_Electron_Thiel using a rare-earth ion ($\mathrm{Er^{3+}}$) doped crystal. High efficiency conversion due to ensemble enhanced coupling strengths, and easy integration with the current fiber optics owing to its available telecom wavelength transition, are a few of the attractive features of this system.

In this section, we shall explicitly describe only the one-way microwave to telecom frequency conversion, since we only need this in our entanglement generation scheme. The opposite conversion can be achieved by running the pertinent steps of our transduction protocol in reverse. The protocol is implemented in an $\mathrm{Er^{3+}}$ doped $\mathrm{Y_{2}SiO_{5}}$ crystal, which could be placed inside a microwave cavity (see cavities ‘b’ and ‘d’ in Fig. 1). $\mathrm{Er^{3+}}$ is a Kramers ion and as such has doubly degenerate energy eigenstates. An external constant magnetic field, $\vv{B_{0}}$, splits the ground state (\ce^4I_15/2) into the Zeeman levels $\ket{1}$ and $\ket{2}$, and the excited state (\ce^4I_13/2) into $\ket{3}$ and $\ket{4}$. The energy level diagram of the ensemble of $\mathrm{Er^{3+}}$ ions is shown in Fig. 3. The optical transitions between ground and excited states are at telecom wavelengths (around 1536 nm), whereas the Zeeman splitting is in GHz range for magnetic field strength ($|\vv{B_{0}}|$) of the order of tens to hundreds of mT depending on the field’s directions 18_Electron_Thiel; 13_Anisotropic_Bushev.

The transduction approach is inspired from the Controlled Reversible Inhomogeneous Broadening (CRIB) memory protocol 01_Complete_Kroll; 07_Analysis_Gisin and is similar to the previous transducer proposal by O’Brien et al. 14_Interfacing_Fleischhauer. The idea is to first map the microwave photon to a spin excitation and then map this spin excitation to an optical excitation followed by its eventual read-out. We assume that initially all the ions are in the ground state $\ket{1}$. As in the CRIB memory protocol, a narrow absorption line from the inhomogeneously broadened $\ket{1}-\ket{4}$ transition is prepared by transferring the rest of the population to an auxiliary level, e.g. level $\ket{2}$. At a later stage of the protocol, this narrow absorption line is artificially broadened using, for example, a one-dimensional magnetic field gradient, $\vv{\delta B}(z)$, where the ‘$z$’ direction can be suitably chosen.

In contrast to the original proposal 14_Interfacing_Fleischhauer, we propose to use the excited state Zeeman levels $\ket{3}$ and $\ket{4}$ (see Fig. 3 (b)) to transfer a microwave photon from the cavity into a collective spin excitation. The excited state spin transition of $\mathrm{Er^{3+}}$ is less susceptible to the various spin-dephasing sources, such as spin flip-flop and instantaneous spectral diffusion, and can thus have considerably longer coherence lifetimes than the ground state spin transition 18_Electron_Thiel. The spin transition $\ket{3}-\ket{4}$ is initially detuned from the microwave cavity by an amount $\delta$. A $\mathrm{\pi}$ pulse is applied to transfer all the population from $\ket{1}$ to $\ket{3}$. The spin transition is now brought in resonance with the cavity. The dynamics of the system can be described by the following Hamiltonian:

[TABLE]

where $\sigma_{-}^{(j)}$ is the spin flip operator, $\sigma_{z}^{(j)}$ is the spin population operator and $\omega_{j}$ is the spin transition frequency of the $j^{th}$ spin, and $a$ is the annihilation operator of the cavity mode and $\omega_{m}$ is the resonance frequency of the cavity. The collective spin $S_{-}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\sigma_{-}^{(j)}$ strongly couples to the cavity mode with an enhanced coupling strength $g^{\prime}\sqrt{N}$, where $N$ is the total number of participating spins. The free evolution of the coupled cavity-spin system transfers the cavity excitation into the collective spin excitation in time $T_{S}=\pi/(2g^{\prime}\sqrt{N})$. The efficiency of the process is primarily limited by the natural spin inhomogeneous broadening, denoted by $\Delta_{ns}$. For realistic values of $g^{\prime}\sqrt{N}=2\pi\times 34$ MHz, $\Delta_{ns}=2\pi\times 10$ MHz, the homogeneous linewidth of spin transition $\gamma_{2}=2\pi\times 100$ kHz, the spin decay rate $\gamma_{1}=2\pi\times 160$ Hz, and the cavity decay rate $\kappa=2\pi\times 10$ Hz 18_Electron_Thiel, the efficiency of transfer is $99.04\%$.

The emission of a telecom photon is controlled by the dephasing and rephasing of the collective spin and optical dipoles. The change of the dynamical phase as a function of time is depicted in Fig. 3 (c). After the microwave photon has been transferred to the spin excitation, the spin transition is detuned from the cavity, and the field gradient, $\vv{\delta B}(z)$, is applied. This artificially broadens the optical and spin transitions by $\Delta_{ao}$ and $\Delta_{as}$ respectively. After some spin-dephasing time $\tau_{1}$, a $\mathrm{\pi}$ pulse ($\pi_{1}$ pulse in Fig. 3 (c)) is applied to transfer all the population back from $\ket{3}$ to $\ket{1}$. The collective excitation now dephases at a rate $\Delta_{ao}$ on the optical transition. To initiate the rephasing process required for emission of a photon, the direction of the magnetic field gradient is reversed ($\vv{\delta B}(z)\rightarrow-\vv{\delta B}(z)$). The population from $\ket{1}$ is again taken to $\ket{3}$ ($\pi_{2}$ pulse) and is kept there for some time to allow spin rephasing. Afterwards, the population is brought back to $\ket{1}$ ($\pi_{3}$ pulse) and the optical dipoles rephase, leading to a collective emission of a single photon corresponding to the $\ket{1}-\ket{4}$ transition frequency, $\omega_{0}$. The efficiency of the overall transduction process also depends on the total time spent in the rephasing and dephasing operations, the finite optical depth of the crystal, and the coupling of the generated photon to the fiber. The overall efficiency can be greater than $85\%$ for realistic values 14_Interfacing_Fleischhauer. Higher efficiencies are possible with a more optimized $\pi$ pulse sequence. It is worth mentioning here that even though these transduction operations may look complicated, they can be compatible with the kind of 3-D microwave cavities we have in mind. See Sec. V.2.2 for a discussion on this.

The fidelity of the final entangled state between the nodes depends on the indistinguishability of the photons interfering at the beamsplitter. Interestingly, for photon-echo based quantum memories in big atomic ensembles, this value can approach unity, as decoherence predominantly affects only the efficiency of the photon generation process, but not the purity of the generated photons 07_Interference_Gisin. Possible imperfections in the setup, e.g. scattering of the pump laser, might cause some optical noise, but these could be almost entirely filtered out, as was shown in similar systems 15_Solid_Riedmatten; 13_Demonstration_Sellars. We realistically assume that the temperature in the dilution refrigerator is low enough ($\approx 20$ mK) to neglect contribution of thermal microwave photons 16_Schrodinger_Schoelkopf; 18_CNOT_Schoelkopf. However generation of spurious microwave photons possibly because of the noisy microwave circuit components and heating caused by lasers would adversely affect the overall process. This needs to be overcome with better experimental control or alternative theoretical approaches 18_Harnessing_Regal; 18_Microwave_Groblacher.

IV Quantum repeater architecture based on microwave cavities

Quantum repeaters are essential for distributing entanglement over long distances 98_Quantum_Zoller; 11_Quantum_Gisin for quantum computation and quantum communication applications. Many repeater proposals involve using atomic ensembles as stationary qubits mainly because good optical quantum memories can be built using them 10_Quantum_Young; 09_Optical_Tittel; 12_Efficient_Pan; 16_Efficient_Pan. However, one drawback is their probabilistic swapping operation, which hampers the entanglement distribution rate, and this prompted several single-emitter based approaches 09_Quantum_Simon; 10_Efficient_Zoller; 10_Quantumrepeaters_Simon; 15_Cavity-based_Rempe; 19_Telecom_Painter; 18_Quantum_Simon; 15_Loophole_Hanson; 07_Entanglement_Monroe; 16_Generation_Imamoglu. Since nonlinear microwave cavities have sufficiently long coherence times, are easy to address, and have deterministic two-qubit gates, in addition to a quantum computing architecture, one can also envision a quantum repeater architecture based on them. Moreover, this also allows relatively nearby quantum computing nodes to serve as repeater nodes to connect more distant quantum computers. Here we discuss a ground based quantum repeater architecture with microwave cavities and transducers.

IV.1 Single set of cavities in an elementary link

A repeater architecture with a few elementary links using microwave cavities and transducers, connected via optical fibers, is shown in Fig. 4 (a). Entanglement is distributed in multiple steps in a hierarchical way. In the first step, entanglement is generated between individual links of length $L_{0}$, possibly after several independent attempts, e.g. after the first step, stationary qubits ‘$1_{\mathrm{s}}$’ and ‘$2_{\mathrm{s}}$’, ‘$3_{\mathrm{s}}$’ and ‘$4_{\mathrm{s}}$’, and so on, are entangled. However, not all the links need to be successfully entangled at the same time. We keep trying until the neighbouring links are entangled, and in the subsequent steps, entanglement is distributed over the whole length by performing the entanglement swapping operations. For instance, in the second step, stationary qubits ‘$1_{\mathrm{s}}$’ and ‘$4_{\mathrm{s}}$’, ‘$5_{\mathrm{s}}$’ and ‘$8_{\mathrm{s}}$’, and so on are entangled by performing Bell state measurement (BSM) on qubits ‘$2_{\mathrm{s}}$’ and ‘$3_{\mathrm{s}}$’, ‘$6_{\mathrm{s}}$’ and ‘$7_{\mathrm{s}}$’, and so on. It takes ‘n’ such steps (known as nesting levels) to distribute entanglement over the total length, $L$, comprising of $\mathrm{2^{n}}$ elementary links, $L=2^{n}L_{0}$.

IV.2 Multiplexed repeater architecture

An important metric for quantifying the usefulness of a repeater architecture is the rate of distribution of entangled states between distant nodes. One way to increase this rate is to use a spectrally multiplexed architecture 07_Multiplexed_Kennedy; 11_Quantum_Gisin; 14_Spectral_Tittel; 18_Quantum_Simon. A part of that architecture is shown in Fig. 4 (b). An array of cavities is employed at each end of the elementary link. There are ‘$m$’ number of the pair of coupled cavities (the storage cavity and the cavity generating the flying qubit) in this array. Each of the cavity generating the flying qubits at each node is coupled to a corresponding cavity at the next node independently via a quantum channel. To distinguish each of these $m$ channels spectrally, a frequency translator which spans tens of GHz can be optically coupled to the output ports of the cavities generating the flying qubits 17_Heralded_Tittel. These photons with different carrier frequencies are then coupled to the same spatial mode of a fiber using a tunable ridge resonator filter with $\approx 1$ MHz resonance linewidths 18_Bridging_Vahala, thus allowing a very large multiplexing in principle. At the measurement station, there is a beamsplitter, two sets of ridge resonators for demultiplexing, and $2m$ single photon detectors.

Independent attempts are made to distribute entanglement along each set of identical channels of frequency $\omega_{j}$. There are $m$ such sets and each set comprises of $2^{n}$ identical channels. Our approach requires entangling operations (for entanglement swap) only between neighbouring storage cavities, as shown in Fig. 4 (b)). A more complicated multiplexing approach that requires all-to-all connectivity between the storage qubits at each node would yield higher rates 07_Multiplexed_Kennedy, but we do not adopt that approach here because all-to-all connectivity is difficult to be realised experimentally with good fidelities 17_Quantum_Blais; 17_Robust_Tiwari. Our protocol is successful if entanglement is successfully distributed over the total distance $L$ in at least one set of identical channels.

In sec. VI, we discuss our achievable rates for distribution of entanglement, compare those rates with direct transmission and with two other repeater approaches, and provide the fidelities for our final entangled states. Before proceeding to that section, we list the realistic values adopted for some of the experimental parameters and calculate the fidelities of the relevant operations in the next section. This would be necessary for a proper discussion on the achievable rates and fidelities, and hence the slight detour.

V Some implementation details

V.1 The qubit and the gates

One way to realise the Hamiltonian (1) using 3-D cavities is to place a non-linear element, e.g. a transmon inside the cavity (see Fig. 1), and drive it using suitable microwave tones 17_Engineering_Blais; 14_Dynamically_Devoret; 15_Confining_Devoret. In 2-D, one could use a $\mathrm{\lambda/4}$ transmission line resonator (TLR) terminated by a flux-pumped SQUID 17_Engineering_Blais; 08_Flux_Tsai.

Next, we quantify the performance of the qubits in the presence of realistic losses. Most of the simulations incorporating such losses have been carried out using an open source software, QuTiP (Quantum Toolbox in Python) 12_Qutip_Nori. The most prominent loss mechanism for the cavities is individual photons leaking out of the cavity at a rate $\kappa$.

Through most of the paper, we have tried to illustrate our points with the help of explicit examples. We have chosen 3 sets of experimentally realisable values for Kerr nonlinearity $K$ 12_Josephson_Blais, and cavity decay rate $\kappa$ 07_Ultrahigh_Visentin; 13_Reaching_Schoelkopf; 18_3D_Grassellino, which are listed in Table 1-2. Although the individual values of $K$ and $\kappa$ are all different, these are chosen such that the ratios of $K/\kappa$ are $10^{3}$, $10^{4}$ and $10^{5}$ in increasing order. These will often represent the different rows of our tables. The fidelities of most of the operations depend on these ratios, rather than their individual magnitudes. For the sake of brevity, we’ll often refer to these values by referring to the ratios.

V.1.1 Driving and undriving of the cavity

The pulse amplitude $\mathcal{E}_{p}(t)$ can be increased adiabatically to evolve the state of cavity from the Fock states $\ket{0}$, and $\ket{1}$ to the cat states $\ket{C_{\alpha}^{+}}$ and $\ket{C_{\alpha}^{-}}$ respectively, preserving parity, where $\alpha=\sqrt{\mathcal{E}_{p}/K}$. As an example suggested in ref. 17_Engineering_Blais, we take the pulse $\mathcal{E}_{p}(t)=\mathcal{E}_{p}^{0}(1-e^{(-t/\tau)^{4}})$, $\mathcal{E}_{p}(t=0)=0$, $\mathcal{E}_{p}(t)=\mathcal{E}_{p}^{0}=2K$ for $t\gg\tau$ to create a cat state with $\alpha=\sqrt{2}$. For $K/\kappa=10^{3}$, $\ket{0}$ is mapped to $\ket{C_{\alpha}^{+}}$ with $\mathrm{99.62\%}$ fidelity in the duration $1.3\tau=0.04$ ms. To undrive the cavity, we simply apply the time reversed control pulse, to get back to $\ket{0}$ with the same fidelity.

Since this was just a smooth pulse obtained after some guess work, it need not be optimized to reach the final state with the highest possible fidelity in the fastest possible way. Faster mapping with higher fidelity is possible with an additional orthogonal two photon drive $i\mathcal{E}_{p}^{\perp}(t)({a^{\dagger}}^{2}-a^{2})$ 17_Engineering_Blais using the non-adiabatic transitionless driving approach propopsed in ref. 09_Transitionless_Berry. The shape of both of these pulses ($\mathcal{E}_{p}(t)$ and $\mathcal{E}_{p}^{\perp}(t)$) is optimized with Gradient Ascent Pulse Engineering (GRAPE) algorithms 05_Optimal_Glaser; 11_Comparing_Herbruggen. Fig. 5 (a) shows the two orthogonal drive pulses applied for mapping $\ket{0}$ to $\ket{C_{\sqrt{2}}^{+}}$ in $t=0.5/K$ with $99.97\%$ fidelity for $K/\kappa=10^{3}$. Fig. 5(b) shows the pulses required for undriving the cavity from $\ket{C_{\sqrt{2}}^{+}}$ to $\ket{0}$ in $t=0.5/K$. Table 1 gives the drive times and fidelities for different ratios of $K/\kappa$.

Values of $K\mathrm{\geq 2\pi\times 10}$ MHz, much higher than those we have considered, are possible, and would make the process much faster 12_Josephson_Blais. However, the ideal platforms for very large $K$ are usually the 2-D TLRs. But, at present, since the quality factors of 3-D cavities are reported to be much higher than those of 2-D TLRs, they seem to be a better choice for long distance entanglement distribution as long coherence time is more important than the speed of operations (see Sec. VI). Our entanglement generation scheme, however, is independent of the type of cavity.

V.1.2 Single and two qubit gates

Rotation around X axis can be implemented with an additional single photon drive $\mathcal{E}_{x}$ (see Eqn. (2)). The drive amplitude $\mathcal{E}_{x}$ should however be much smaller than the two photon drive drive strength $\mathcal{E}_{p}^{0}=\left\lvert\alpha\right\rvert^{2}K$ to stay within the qubit space. Under that approximation, $X_{\pi/2}$ is accomplished in $t=\pi/(8\left\lvert\alpha\right\rvert\mathcal{E}_{x})$. In the absence of noise due to single photon loss, taking a very small value of $\mathcal{E}_{x}$ would yield a very high fidelity, but noise forces one to be fast to minimise decoherence, even if it means going a little bit out of the qubit space. Thus one has to optimise between reduction in fidelity because of noise and because of leaking out of the qubit space, keeping the operations sufficiently fast. See Table 1 for the fidelities and gate times, taking $\mathcal{E}_{x}=\mathcal{E}_{p}^{0}/10$, $\mathcal{E}_{x}=\mathcal{E}_{p}^{0}/20$, and $\mathcal{E}_{x}=\mathcal{E}_{p}^{0}/45$ for the three $K/\kappa$ ratios respectively.

$Z_{\pi/2}$ is achieved in time $\pi/(2K)$ by the free evolution under the Kerr Hamiltonian in Eqn. 3. Tables 1 shows the fidelities and times for our parameters.

For a two qubit entangling gate (see Eqn. (4)), we start with the state $\mathrm{\ket{\bar{0}}\otimes\ket{\bar{0}}}$ and evolve it under the Hamiltonian described by Eqn. 4. The rotation in the two-qubit space is denoted by $G_{\theta}$, where $\theta$ is the same angle as that disscussed in ref. 16_Universal_Goto. When $\theta=\pi/2$, we arrive at the maximally entangled state $\mathrm{\frac{(1+i)\ket{\bar{0}}\otimes\ket{\bar{0}}+(1-i)\ket{\bar{1}}\otimes\ket{\bar{1}}}{2}}$ in $t=\pi/(8\left\lvert\alpha\right\rvert^{2}\mathcal{E}_{c})$, where $\mathcal{E}_{c}$ is the coupling strength between the two cavities. As was the case for $X_{\phi}$ rotation, to remain in the Hilbert space of the two qubits, a small $\mathcal{E}_{c}$ is preferable, but time and noise considerations pushes us to be fast. See Table 1 for the times and fidelities for $\mathcal{E}_{c}=\mathcal{E}_{p}^{0}/15$, $\mathcal{E}_{c}=\mathcal{E}_{p}^{0}/25$, and $\mathcal{E}_{c}=\mathcal{E}_{p}^{0}/55$ for the three ratios of $K/\kappa$ respectively.

This set of values for the single photon driving amplitude $\mathcal{E}_{x}$ and the coupling strength $\mathcal{E}_{c}$ has been chosen after running many simulations, and taking different values in each of these simulations to arrive at a set which gave reasonably good fidelities. But optimum amplitudes, possibly in the form of time dependent functions, can be obtained using GRAPE, and they might yield slightly higher fidelities.

In addition to the two-qubit rotations $G_{\theta}$, we need additional single qubit operations to perform a CNOT gate in our chosen basis of $\ket{C_{\alpha}^{+}}$ and $\ket{C_{\alpha}^{-}}$. One of the sequence of operations which yields a CNOT gate with the first qubit as the control and the second qubit as the target is

[TABLE]

Here $\mathrm{X_{\phi}}$ and $\mathrm{Z_{\phi}}$ are the respective single qubit rotations around those axes. The superscript denotes the qubit on which those gates act. Table 1 shows the resulting fidelities and time taken for different parameters.

V.2 Compatibility of 3-D microwave cavities with additional elements and operations

V.2.1 Effect of a Kerr nonlinear element on the decay rate of a microwave cavity

The Hamiltonian of a weakly anharmonic SQ coupled to a cavity can be expressed in the form

[TABLE]

Here, $a$ is the annihilation operator for the cavity mode with frequency $\omega_{c}$, and $b$ is the annihilation operator for the SQ mode with $\omega_{q}$ as the transition frequency between its ground state and the first excited state. $K_{q}$ is the anharmonicity of the SQ mode, and $g$ is the coupling strength between the cavity and qubit modes.

The qubit-cavity interaction hybridizes these modes, and in the dispersive regime, where the coupling $g$ is much less than the detunings between the cavity and qubit transition frequencies, the cavity acquires a bit of the SQ component and vice-versa. Consequently, a part of the nonlinearity of the SQ mode ($K_{q}$) is also inherited by the cavity and in the dressed picture, there appears a Kerr nonlinearity term of the form $-K{a^{\dagger}}^{2}a^{2}$ (see the Hamiltonian in Eqn. 1), where the mode $a$ now has a small SQ component. $K$ can be estimated by observing the energy spectrum of the eigenstates of the full Hamiltonian given in Eqn. 7. We do so numerically by diagonalising the above Hamiltonian and observing the energy difference between consecutive cavity levels $\omega_{\ket{i+1,0}}-\omega_{\ket{i,0}}$. Here, $\ket{i,j}$ denotes an eigenstate of the above Hamiltonian with $i$ and $j$ number of excitations in the dressed cavity and SQ mode respectively. We verify that the energy difference between consecutive cavity levels follow the trend expected from a Kerr nonlinearity term in the Hamiltonian. Our way of obtaining the Kerr nonlinearity numerically agrees well with those seen in experiments 13_Observation_Schoelkopf.

The coherence times mentioned in ref. 07_Ultrahigh_Visentin; 13_Reaching_Schoelkopf; 18_3D_Grassellino are for a bare cavity. But due to the hybridization of the cavity-SQ modes, the lifetimes of both the cavity and the SQ change. Since a SQ has a shorter coherence time than our desired kind of cavities, the SQ-cavity interaction will shorten the lifetime of the cavity. We therefore need to take this effect into account.

To the lowest order in $g/\Delta$, the ‘inverse-Purcell’ enhanced decay rate of the cavity $\kappa\approx(1-(g/\Delta)^{2})\kappa_{c}+(g/\Delta)^{2}\gamma$ 07_Generating_Schoelkopf; 16_Quantum_Schoelkopf; 09_Dispersive_Blais. Here, $g$ is the qubit-cavity coupling strength, $\Delta=\omega_{q}-\omega_{c}$ is the detuning between the cavity and the qubit, and $\kappa_{c}$ and $\gamma$ are the decay rates of the bare cavity and the SQ respectively. Taking the above $\kappa$, the effective coherence time of a cat state in the cavity in the presence of the two photon drive, $\kappa_{eff}$, is numerically found by fitting the evolution of coherence as a decaying exponential. It is approximately equal to $2\kappa\left\lvert\alpha\right\rvert^{2}$, where $\left\lvert\alpha\right\rvert$ is the size of the coherent state.

For experimentally conceivable values of $\kappa_{c}$ 07_Ultrahigh_Visentin; 13_Reaching_Schoelkopf; 18_3D_Grassellino, $\gamma$ 19_SQs_Oliver; 12_SQ_Steffen, $g$ 07_cQED_Schoelkopf; 13_Observation_Schoelkopf; 17_Microwave_Nori, $\Delta$ 13_Observation_Schoelkopf; 16_Quantum_Schoelkopf, and $K_{q}$ 12_Black_Girvin; 12_Josephson_Blais; 13_Autonomously_Devoret; 16_Quantum_Schoelkopf, the calculated $K$ and $\kappa$ are listed in Table 2.

We note that due to the difference of as much as four orders of magnitude between $\kappa_{c}$ and $\gamma$, $\kappa$ is practically determined by $\gamma$. On the one hand, this relaxes the requirement on the quality factor of the bare cavity that is needed for our implementation. As an example, for $K/\kappa=10^{4}$, instead of taking $\kappa_{c}=2\pi\times 0.32$ Hz, if we assume an order of magnitude larger $\kappa_{c}=2\pi\times 3.2$ Hz, the resulting $\kappa$ stays almost the same (aprox. $5\%$ variation). But on the other hand, large $\gamma$ makes achieving large $K/\kappa$ ratios difficult, as both $K$ and (the effect of $\gamma$ on) $\kappa$ increase simultaneously, though not exactly in the same way. Since the dependence of $K$ and $\kappa$ on the experimental parameters is not identical, it is still possible to obtain high $K/\kappa$ ratios with appropriate values for these parameters (see Table 2). It should be pointed out that our selection of these values is based on running multiple simulations, ensuring that we stay in the dispersive regime, and from the intuition derived indirectly from approximate perturbative expressions in ref. 07_Charge_Schoelkopf; 12_Black_Girvin; 13_Observation_Schoelkopf. It is likely that a proper optimization of the relevant parameters might yield slightly better values for $K$ and $\kappa$ than the ones we have used. However, even with the values we have taken, we still manage to obtain high entanglement generation rates and fidelities, as discussed in the next section (Sec. VI).

It is worth noting that improving $\gamma$ of the SQs will significantly improve the achievable $K/\kappa$ ratios. If the SQs were as long-lived as the cavities, then $K/\kappa$ ratios of the order of $10^{7}$ should be achievable with the parameters in Table 2.

V.2.2 Compatibility of 3-D microwave cavities with the transduction operations

Here we discuss the compatibility of high quality factor (high Q) 3-D superconducting cavities with the transduction operations, specifically those involving magnetic fields necessary for our protocol.

There is an increasing effort towards designing 3-D microwave cavities that would permit magnetic field manipulation of SQs 16_3D_Fedorov; 18_Fast_Kirchmair, and spin systems 18_Applying_Wallraff; 18_Loop-gap_Kubo inside them. The two most common elements used to build high Q superconducting cavities are Aluminium (Al) and Niobium (Nb) 07_Ultrahigh_Visentin; 13_Reaching_Schoelkopf; 18_3D_Grassellino. The demand for magnetic field at least in tens of mT for our transduction protocol 18_Electron_Thiel; 13_Anisotropic_Bushev is difficult for cavities made out of pure Al, which loses its superconductivity at a criticial magnetic field strength, $H_{c}$ $\approx 10$ mT 54_Superconducting_Eisenstein, unless one includes a pathway for the magnetic field lines to escape without interacting much with the Al walls, e.g. by making a hybrid cavity including Copper (Cu) 16_3D_Fedorov. However, many of these designs would still result in slightly lower Q cavities for high magnetic fields. As a better alternative, one can opt for cavities made out of Nb, whose $H_{c}$ $\approx 200$ mT 54_Superconducting_Eisenstein; 01_Critical_Saito can sustain the magnetic field strengths needed in our protocol. Another concern here is that the SQs (placed inside these cavities) themselves are predominantly made of Al, and it is important to shield them from strong magnetic fields. The fields generated in some experiments, e.g. in ref. 18_Applying_Wallraff have a gradient like character, and we can therefore place SQs close to regions of weak magnetic field while the transducer close to the more intense regions.

It should be pointed out that since the transducer is placed in the cavity generating the flying qubits, and not the one which is supposed to act as a memory, the requirement of long coherence time is less stringent on this cavity. If, however, these operations turn out to be too difficult to be performed in the cavity while maintaining reasonably good coherence, one can adopt an alternative route. After the undriving operation to a microwave Fock state, one could transfer the microwave photon to a lower finesse cavity, possibly even a planar resonator, and perform transduction there. To make the operation faster, it should be possible to tune the decay rate of the original cavity to approximately 4 orders of magnitude higher than its intrinsic value 13_Catch_Martinis; 18_On-demand_Schoelkopf.

VI Entanglement distribution rates and fidelities

In the previous section, we presented some experimental values which will be relevant here to calculate the possible entanglement distribution rates for our repeater protocol, and to estimate the fidelities of the final states. We shall compare the rates with those possible with direct transmission, with the popular ensemble based DLCZ repeater approach 01_Long_Zoller, and with a recently proposed repeater approach using single rare-earth (RE) ions in crystals 18_Quantum_Simon.

The average time to distribute entanglement over length $L$ using a quantum repeater scheme with $2^{n}$ elementary links, each of length $L_{0}$ is 09_Quantum_Simon; 11_Quantum_Gisin; 07_Quantum_repeaters_Gisin; 18_Quantum_Simon (see the Supplementary Information for a brief explanation of the formula)

[TABLE]

Here, $T_{o}$ is the time taken for the local operations at one node for one elementary link, $P_{0}$ is the success probability of generation of entanglement in an elementary link, and $P_{i}$ is the success probability of entanglement swapping in the $i^{th}$ nesting level. The local operations include the gates and the driving, undriving, and transduction operations (see Fig. 2) at each node. For our protocol, $P_{0}=\frac{1}{2}\eta_{t}p{\eta_{o}}^{2}$. The transmission efficiency in the fiber is $\eta_{t}=e^{-\frac{L_{0}}{L_{att}}}$, where $L_{att}$ is the attenuation length. The probability of emitting a telecom photon from a cavity into the fiber mode is denoted by $p$. For our system, it is primarily the transduction efficiency. And $\eta_{o}$ is the single (optical) photon detection efficiency. $P_{i}$ is mainly governed by the (stationary) qubit readout efficiency. Cat states can be read out in a Quantum Non-Demolition (QND) fashion with close to unit efficiency by measuring the parity 14_Tracking_Schoelkopf; 16_Extending_Schoelkopf (see section S1.G in the Supplementary Information for a brief discussion on parity measurements). Another way to read-out cat states, is to un-drive them to Fock states, and then read out the number of photons non-destructively, using a similar technique as before 10_QND_Schoelkopf. Since two such measurements are needed for entanglement swap, $P_{i}={\eta_{m}}^{2}$, where $\eta_{m}$ is the QND efficiency. Values close to unity are expected for our long storage-time cavities.

The entanglement distribution rate is the inverse of the average time needed to distribute entanglement. For the multiplexed architecture the rate is multiplied by $m$, since there are $m$ independent sets of channels operating in parallel.

In Fig. 6, we plot entanglement distribution rates for our proposed protocol as a function of the total distance $L$ and compare these with the rates possible using direct transmission and a source producing entangled photons at 1 GHz 18_Highly_Schmidt, with the rates achieved using the DLCZ protocol 01_Long_Zoller; 05_Measurement_Kimble, and with those achieved using single RE ions (RE) in crystals 18_Quantum_Simon. We choose $K/\kappa=10^{5}$ and $n=3$ for our repeater protocol. In the Supplementary Information, we discuss the rationale behind choosing this particular nesting level, and also show the rates and fidelities for other nesting levels and other ratios of $K/\kappa$. We take realistic values for other pertinent experimental parameters, e.g. $p=0.8$, $P_{i}=0.9$, $\eta_{d}=0.9$, and $L_{att}=22$ km. The relevant parameters for DLCZ include the single photon generation probability and the memory efficiency, which too have been taken realistically as 0.01 and 0.9 respectively 01_Long_Zoller; 05_Measurement_Kimble; 11_Quantum_Gisin. Since the operation times for DLCZ protocol can in principle be fast enough to be ignored 05_Measurement_Kimble, we do not include those here. For the single RE ion based approach, apart from the parameters already considered for the rate calculation in ref. 18_Quantum_Simon, we have taken into account a realistic operation time of 0.1 ms. The figure also depicts the increased rates obtained by multiplexing ($m=200$) for all three repeater protocols. Our protocol yields higher rates than the other two repeater protocols.

Quantum repeaters would be practically more useful when the rate of distribution of entanglement surpasses that of direct transmission at the so called ‘cross-over’ point. From the plot, the cross-over points for our protocol for $m=1$ and $m=200$ are 387 km and 244 km respectively. The fidelity of the final entangled state can be estimated by subsequently multiplying the fidelities of different operations and the residual coherence of the storage cavities.

[TABLE]

Here, $F_{tot}$ is the fidelity of the final entangled state, $F_{elem}=\prod_{i}F_{O_{i}}$ is the fidelity of generation of entanglement in an elementary link, where $O_{i}$ are the various local operations, e.g. the gates, the driving and undriving of the cavities, and the transduction from microwave to optical frequencies. $F_{swap}$ is the fidelity of entanglement swap, and $l=2^{n}$ is the number of elementary links, where $n$ is the nesting level; the number of entanglement swaps needed to distribute entanglement over the whole length, $L$, is $l-1$. $C_{R}$ is the residual coherence of the storage cavities, only taking into account the decoherence because of the non-zero waiting times.

The residual coherence ($C_{R}$) for cat states is approximated as $e^{-\kappa_{eff}T}$, where $k_{eff}$ is the cat’s decoherence rate and $T$ is the average time needed to distribute entanglement. One relatively simple way to have a larger $C_{R}$ is by undriving the storage cavities from the cat basis to the Fock basis before the waiting times. The decoherence rate in the Fock basis is approximately $\kappa$, which is less than the decoherence rate ($k_{eff}\approx 2\kappa|\alpha|^{2}$) of our cat states. The cavities would then be driven back to the cat basis when the additional operations need to be performed. Moreover, since even $\kappa$ is much greater than the decay rate of the bare cavity, $\kappa_{c}$, because of the relatively lossy SQ coupled to it, one could increase $C_{R}$ further by transferring the Fock state to an effectively longer lived microwave cavity for storage 13_Catch_Martinis; 18_On-demand_Schoelkopf; 18_Deterministic_Wallraff; 18_Perfect_Sun. The residual coherence can also be enhanced with appropriate error-correction codes 01_Encoding_Preskill; 13_Hardware_Mirrahimi; 14_Dynamically_Devoret; 15_Confining_Devoret; 16_Extending_Schoelkopf; 17_Degeneracy_Mirrahimi; 18_Coherent_Devoret; 18_Fault_Schoelkopf; 18_Stabilized_Girvin. For suitably encoded cat states, error-correction has been experimentally implemented to increase the coherence time by a few times 16_Extending_Schoelkopf, and error detection has been performed in a fault-tolerant way 18_Fault_Schoelkopf. Note that our present protocol would require quantum error correction for qubits encoded in states of different parity. See the Supplementary Information for a discussion on different approaches to increase the residual coherence.

In addition to the fidelities of the different operations calculated explicitly using QuTip (see the Supplementary Information for a list of all the operations considered), if we take the fidelity for transduction as $99.95\%$, comparable to the fidelities of several of the gate operations in Table 1, the highest fidelities of the final entangled states calculated for $m=1$ and $m=200$ at the cross-over points are approximately $91\%$ and $92\%$ for $m=1$ and $m=200$ respectively (see the Supplementary Information for details).

Considering only factors such as a non-zero single photon generation probability, and imperfect memory and detection efficiencies, and using the already mentioned realistic values for them (0.01, 0.9 and 0.9 respectively), the fidelity of the final state for DLCZ repeater protocol with $n=3$ would be roughly $75\%$ 11_Quantum_Gisin. However this should be treated as kind of an upper bound, since we have not considered the finite fidelities of other operations, e.g. the read and write processes, the finite coherence times of the memories, and the effect of the phase fluctuations of the fibers, which would all contribute towards bringing down the final fidelity. For the RE ion based approach, an upper bound on fidelity, estimated considering only entanglement swapping and spin mapping ($\mathrm{Er^{3+}-Eu^{3+}}$) operations, is roughly $80\%$.

Using established protocols of entanglement purification, the fidelity of our final states can be significantly increased as they cross the threshold above which purification is possible 07_Entanglement_Briegel. Here, however, we do not incorporate error-correction or entanglement purification in our architecture, but we believe that those should be possible (see the Supplementary Information for a brief discussion on error-correction).

VII Conclusion and Outlook

Recently, orthogonal cat states in microwave cavities 16_Universal_Goto; 17_Engineering_Blais have been proposed as promising qubits for quantum computation. Their long coherence times also make them attractive in a quantum network context. Here, we proposed a robust scheme to generate entanglement between distant cat-state qubits, mediated by telecom photons in conventional optical fibers. As an important step of this entanglement generation scheme, we expanded on a specific microwave to optical transduction protocol some of us recently proposed using a rare-earth ion ($\mathrm{Er^{3+}}$) doped crystal 18_Electron_Thiel. We also designed a quantum repeater architecture using these cat state qubits and rare-earth ion based transducers. Our calculations show that higher entanglement distribution rates than that possible with direct transmission or with two other repeater approaches, including the well-estalished DLCZ approach, can be achieved, while maintaining a high fidelity for the final state in the presence of realistic noise and losses, even without entanglement purification or quantum error correction. By increasing the size of the local quantum processing nodes, our proposed approach can naturally be extended to a full-fledged distributed quantum computing architecture, thus paving the way for the quantum internet 08_Quantum_Kimble; 18_Internet_Hanson; 17_Towards_Simon.

We hope that our results will stimulate new experimental and theoretical work involving these cat state qubits and transducers. It would be useful to think about ways to generate higher Kerr nonlinearities and to diminish the effect of the SQ on the decay rate of the cavity, possibly by engineering novel Hamiltonians and by designing novel forms of filters respectively 19_Fast_Buisson; 12_Josephson_Blais; 19_Quantum_Oliver. Regarding cat state qubits, future work may include developing and incorporating efficient quantum error-correction codes 15_Confining_Devoret; 17_Degeneracy_Mirrahimi; 18_Coherent_Devoret, and possibly envisioning schemes to protect against phase decoherence. It is worth mentioning that although we have chosen microwave cat-state qubits, rather than the more obvious (microwave) Fock state qubits as our computational units, since quantum computation with error correction seems to be currently more mature with cat states, our entanglement generation protocol, and hence the overall repeater framework can straightforwardly be applied to Fock state qubits. Recently, new error correction codes have been proposed for Fock state qubits 16_New_Girvin; 18_Performance_Jiang. There have consequently been a few experiments to demonstrate error-correction 18_Demonstration_Sun, as well as single and two qubit gates 18_CNOT_Schoelkopf; 19_Entanglement_Schoelkopf; 18_Demonstration_Sun on suitably encoded Fock state qubits. The recent work thus tightens their competition with cat state qubits, and it will be interesting to see how the field progresses.

Within the areas of modular and distributed quantum computing, it is of interest to think about how best to connect different computational units to efficiently distribute an arbitrary computational task 13_Efficient_Stather; 15_Efficient_Brierley; 14_Freely_Simon; 14_Large-scale_Monroe. Regarding the quantum communication aspect, one would need to figure out efficient repeater architectures for a truly global quantum network, possibly using both ground based and (quantum) satellite links 05_Experimental_Pan; 15_Entanglement_Simon; 17_Towards_Simon; 17_Satellite_Pan.

Acknowledgements.

We thank F. Kimiaee Asadi, M. Falamarzi Askarani, S. Barzanjeh, A. Blais, S. Goswami, J. Moncreiff, D. Oblak, S. Puri, A. Saxena, W. Tittel, and S. Wein for useful discussions. We are further thankful to A. Blais for providing detailed comments on an earlier version of the manuscript, to S. Puri for providing some of the codes used in ref. 17_Engineering_Blais, and to A. Saxena for his help in the initial phase of the project. S.K. was supported by an Eyes High Doctoral Recruitment Scholarship (EHDRS) and an Alberta Innovates Technology Futures (AITF) Graduate Student Scholarship. N.L. was supported by an Eyes High Post-Doctoral Fellowship (EHPDF) and by the Alliance for Quantum Technologies’ (AQT’s) Intelligent Quantum Networks and Technologies (INQNET) research program. C.S. acknowledges funding from the Natural Sciences and Engineering Research Council (NSERC) through its Discovery Grant and CREATE grant “Quanta”, and from Alberta Innovates through “Quantum Alberta”.