Remote Entanglement of Superconducting Qubits via Solid-State Spin Quantum Memories

A superconducting qubit is one of the most promising building blocks for a quantum computer [1]. Recent developments in superconducting qubits enabled the integration of several tens of qubits onto a single chip and well-controlled quantum operation on them [2]. However, for practical calculations, integration on the order of one million qubits is estimated to be required [3], which is an extremely challenging task considering the current technologies. Although quantum memories may alleviate this requirement by several orders of magnitude [4], tens of thousands of qubits are still needed. To mitigate the severe demands concerning the integration of qubits, distributed quantum computing is suggested as an alternative [5, 6, 7]. Similar to a distributed classical computer, a distributed quantum computer is realized by linking small quantum computers using classical and quantum communication channels.

To realize quantum communication between superconducting qubits in remote dilution refrigerators, an optical photon must be used for communication outside the dilution refrigerators. Thus, the direct conversion of microwave photons to optical photons has been intensively investigated in the last decade [8, 9]. However, in the direct conversion scheme, high conversion efficiency is generally not compatible with the low optical pump power necessary to avoid heating up inside the refrigerator and to avoid generating noise. Instead, schemes using entangled photons to generate entanglements between superconducting qubits have recently gained attention [10, 11, 12]. These schemes use entangled-photon pairs, which have already been widely adapted for entanglement distribution at telecom wavelengths [13, 14, 15]. For quantum communication with optical photons, the application of a quantum memory is another choice to distribute entanglement between remote nodes [16, 17, 18, 19, 20]. Thus, using an analogy with communication at telecom wavelengths, we propose a quantum memory as an interface for microwave and optical photons, which works as a quantum repeater to generate entanglements between superconducting qubits in different dilution refrigerators.

In this article, we introduce a scheme to generate entanglement between superconducting qubits in different refrigerators via a quantum memory. To analyze the scheme’s performance, we consider a specific situation : generating entanglement between remote superconducting qubits using a quantum interface [21] consisting of a color center in diamond, a microwave resonator, and an optomechanical cavity. Assuming the current state-of-the-art technologies, we estimate the entanglement generation rate of the memory-based scheme. Since it is not trivial whether the memory-based scheme has any advantages over other schemes, we compare its performance with those of the direct conversion scheme and the entangled-photon scheme. We confirm that as long as the conversion efficiency of the direct conversion scheme does not exceed a certain threshold, it is beneficial to introduce the memory for the entanglement generation. Besides, though the entanglement generation rates of the memory-based and entangled-photon schemes are estimated to be on the same order of magnitude, heat generation inside the refrigerator could be reduced more in the memory-based scheme than in the entangled-photon scheme. Since decreasing heating in the refrigerator is crucial for scaling quantum devices, using a memory could not only deliver entanglement but also help the thermal design inside a refrigerator, contributing to the establishment of a distributed quantum computer.

This section overviews the entanglement generation protocol between superconducting systems in remote dilution refrigerators using a quantum interface based on the solid-state spin quantum memory. Figure 1 (a) shows a schematic of a quantum interface, consisting of a color center with nuclear spin memories in diamond and several resonators, that is connected to an optical fiber at the room temperature region and to a superconducting qubit via a waveguide. The device works as an interface for both microwave and optical photons. To obtain a single microwave photon coupling that is strong enough to transfer the quantum state efficiently, different kinds of resonators are integrated into the interface. Figure 1 (b) shows a simplified model of the coupled resonators. Since a color center with the orbital degree of freedom strongly interacts with the strain (phonon) due to the spin-orbit interaction [22, 23], a microwave photon is converted to a microwave phonon via the piezoelectric effect. Then, the phonon confined inside the one-dimensional optomechanical crystal cavity interacts with a color center through the strain (mechanical)-electron spin coupling. Here, we consider NV^-, NV⁰, and SiV^- as candidates that have high strain sensitivity in their ground or excited state. Similarly, coupling between a color center and an optical photon is enhanced by the confinement of the optical field inside the optomechanical cavity.

Figure 1 (c) shows a protocol for the remote entanglement generation between superconducting qubits using the quantum interface. We consider nodes inside dilution refrigerators separated by several meters. First, entanglement between remote electron spins is generated using a beamsplitter that erases optical path information. Then, the entanglement is swapped to nearby nuclear spin memories. Next, entanglement between the superconducting qubit and the electron spin is generated in each node. Finally, Bell-state measurement (BSM) is performed between the electron and nuclear spins in each node, generating entanglement between remote superconducting qubits. In the following, we will analyze and discuss each part of the protocol in terms of the entanglement generation rate.

Entanglement generation between remote color centers in diamonds has already been demonstrated using two-photon and single-photon protocols [19, 24, 25, 26] (see Appendix A for details on single- and two- photon protocols). The two-photon protocol projects spins within different nodes (A and B) on one of the two Bell states,

where the sign depends on whether the same detector ($+$) or different detector ($-$) is clicked in the two-photon emission rounds [19]. $\ket{0}$ ($\ket{1}$) denotes the spin state in the computational basis, the subscript e indicates the electron spin, and the superscripts A and B correspond to the two nodes. Since the two-photon protocol is tolerant to the error caused by photon loss and does not require optical path length stabilization, in the following we consider only the two-photon protocol.

Then, we consider a practical limit of the entanglement generation rate between color centers separated by several meters. The entanglement generation rate between remote color centers, $R_{\textrm{e-e}}$, is expressed as

where $\eta_{\textrm{e}}^{\textrm{opt}}$ is the detection efficiency of the optical photon emitted from a color center considering all losses during the transmission and $r_{\textrm{e-e}}$ is the entanglement generation attempt rate of the electron spins. The factor 2 in the denominator of eq.(2) indicates that only two of the four Bell states can be heralded using a beamsplitter. Note that $\eta_{\textrm{e}}^{\textrm{opt}}\sim 1$ is already feasible considering the current state-of-the-art technologies (see Appendix B).

For color centers separated by 10 m, the propagation time of an optical photon in an optical fiber is 25 ns from a color center to a photodetector placed at the midpoint of the refrigerators (5 m). Then, the ideal entanglement attempt rate will be 40 MHz. However, in such a case, the actual entanglement attempt rate is limited by the gate operation time for the emission of a coherent photon. In a two-photon protocol that used the NV^- center, the typical operation time to generate a coherent photon was $\sim 10$ $\mu$s [19, 24], corresponding to $r_{\textrm{e-e}}=100$ kHz. Thus, assuming $\eta_{\textrm{e}}^{\textrm{opt}}=$ 0.9-1, $R_{\textrm{e-e}}$ would be 80-100 kHz. Actually, considering that the initialization of a color center and nuclear spin takes several tens of $\mu$s [24], the whole entanglement generation rate between the electron spins would be limited to a few tens of kilohertz.

In this section, we propose two protocols to generate entanglement between the superconducting qubit and the electron spin in diamond. We then consider the entanglement generation rate based on these protocols. In the following, we consider that $\ket{0}_{\textrm{e}}$ is the spin ground state in the computational basis that is excited by the microwave time-bin qubit.

Figure 2 (a) shows the first protocol, which uses an optical photon to herald the successful entanglement. The protocol can be applied to a color center that has high strain susceptibility in its ground or excited state. In Appendix D, we discuss the feasibility of the protocol considering NV^- and NV⁰ as an example.

First, the superconducting qubit, which is prepared in the state, $(1/\sqrt{2})(\ket{g}_{\textrm{sc}}+\ket{e}_{\textrm{sc}})$, emits a coherent microwave time-bin qubit [28, 27], resulting in the state,

Here, $\ket{g}_{\textrm{sc}}$ and $\ket{e}_{\textrm{sc}}$ denote the superconducting qubit basis states, and $\ket{L}_{\textrm{mw}}$ ( $\ket{E}_{\textrm{mw}}$) is the microwave late (early) time-bin qubit. The electron spin is prepared in the state $(1/\sqrt{2})(\ket{0}_{\textrm{e}}+\ket{1}_{\textrm{e}})$. Then, $\ket{0}_{\textrm{e}}$ is excited by $\ket{E}_{\textrm{mw}}$, which is converted to a microwave phonon via the piezoelectric effect. The spin is subjected to an optical pump at the same time as the absorption of $\ket{E}_{\textrm{mw}}$ by the spin (see Appendix D for details). The detuning of the optical pump is equal to the frequency of $\ket{E}_{\textrm{mw}}$. As a result of the successful absorption of $\ket{E}_{\textrm{mw}}$, the spin is excited to an optically excited state, $\ket{\phi}_{\textrm{e}}^{\textrm{opt}}$. Then, the spin emits a photonic early time-bin qubit, $\ket{E}_{\textrm{opt}}$. After the absorption of $\ket{E}_{\textrm{mw}}$, $\ket{0}_{\textrm{e}}$ and $\ket{1}_{\textrm{e}}$ are flipped. The procedure of photon emission is repeated for $\ket{L}_{\textrm{mw}}$. The spin emits a photonic late time-bin qubit, $\ket{L}_{\textrm{opt}}$, conditioned on the successful absorption of $\ket{L}_{\textrm{mw}}$. After that, $\ket{0}_{\textrm{e}}$ and $\ket{1}_{\textrm{e}}$ are flipped again. Consequently, the spin entangles with $\ket{E}_{\textrm{opt}}$ and $\ket{L}_{\textrm{opt}}$ as

Using a delay line and a beamsplitter, photonic time-bin qubits are measured on the basis $(1/\sqrt{2})(\ket{0}_{\textrm{opt}}^{\textrm{A}}\ket{1}_{\textrm{opt}}^{\textrm{B}}\pm\ket{1}_{\textrm{opt}}^{\textrm{A}}\ket{0}_{\textrm{opt}}^{\textrm{B}})$. Detection of a single photon at the photodetectors heralds the successful interaction between the microwave time-bin qubit and the spin. The spin state is then corrected depending on the measurement outcome. Consequently, the state of the microwave time-bin qubit is teleported into the spin state with the heralding of the optical photon as

It is noteworthy that different protocols have been suggested to transfer the state of a superconducting qubit to the electron spin (NV^-) in diamond [29, 30]. The scheme proposed in Ref.[29] uses a superconducting flux qubit coupled to a single NV^-. To maximize the interaction between the superconducting qubit and NV^-, NV^- is placed in close proximity to the flux qubit ($\sim 15$ nm). However, such a configuration may hinder integration of an optical cavity, since the electrode that is too close to the NV^- center contribute to the loss of the optical photon, degrading the quality factor of the optical cavity. Thus, the protocol is not compatible with our scheme, which uses a photonic (optomechanical) cavity to enhance spin-photon interaction. In another approach, a microwave photon emitted from a superconducting qubit is converted to an optical photon using an electro-optic effect and driving laser [30, 31, 32, 33, 34]. Using the converted optical photon and a driving laser, the ground state of NV^- is controlled through the interaction between a $\Lambda$-type three-level system ($\ket{m_{s}=\pm 1}$ and $\ket{A_{2}}$ of NV^-). Since their scheme already integrates whispering-gallery-mode photonic cavity into the device structure, it could be another choice to transfer the state of a superconducting qubit to the electron spin.

Figure 2 (b) shows the second protocol for the entanglement generation, which uses a microwave time-bin qubit emitted from the spin. In this scheme, a color center having high strain susceptibility in the ground state is required (e.g., NV⁰ and SiV^-). In the Appendix D, we consider the detail of the protocol, using NV⁰ as an example.

The initial state of the spin is prepared in $(1/\sqrt{2})(\ket{0}_{\textrm{e}}+\ket{1}_{\textrm{e}})$. Then, $\ket{0}_{\textrm{e}}$ is excited to $\ket{\phi}_{\textrm{e}}^{\textrm{mw}}$ by a microwave driving pulse supplied from an external signal generator. According to the model shown in Fig. 1 (b), the excited spin emits a microwave phonon to the optomechanical cavity (See Appendix C for the detailed model). Repeating the emission with bit-flip between $\ket{0}_{\textrm{e}}$ and $\ket{1}_{\textrm{e}}$ generates the state (see Appendix D for details),

The microwave time-bin qubit is transferred to the superconducting qubit via a waveguide coupled to the microwave cavity. The superconducting qubit, which is prepared in $\ket{g}_{\textrm{sc}}$, absorbs the microwave time-bin qubit, resulting in the entangled state,

By measuring population in the second excited state, $\ket{f}_{\textrm{sc}}$ of the superconducting qubit, we can confirm whether the superconducting qubit successfully absorbs the microwave time-bin qubit or not, heralding the generation of entanglement [27].

A difference between these two protocols is whether the superconducting qubit or the spin is the emitter (receiver) of the microwave time-bin qubit. Efficient ($>90\%$) microwave time-bin qubit generation and absorption by the superconducting qubit have been demonstrated [27, 35], suggesting that the superconducting qubit can be used in both ways with similar efficiency. Thus, the efficiency of microwave emission or absorption by the color center spins determines which protocols to choose. However, this efficiency depends on various parameters: the choice of color center, coupling between the spin and the phonon, coupling between the microwave phonon and the microwave photon, and the quality factor of the optomechanical crystal and the microwave resonator. Thus, in the following, we will treat the microwave photon generation or absorption efficiency of the spin as a parameter. Regarding the efficiency based on feasible device parameters, we calculate and discuss it in the Appendix C.

Another difference is that the first protocol uses the two-photon absorption process at the spin while the second protocol uses single-photon absorption at the superconducting qubit. In the first protocol, the frequencies of the superconducting qubit and the spin need not be in perfect resonance, since tuning the frequency of an optical driving laser can compensate the frequency difference between them. Besides, it is easier to detect a single optical photon than a single microwave photon. However, the use of an optical driving laser may cause heating inside the refrigerator without a high-quality photonic crystal cavity to enable the optical drive by a single optical photon. On the other hand, in the second protocol, an optical driving laser is not necessary, thus mitigating the possible heating due to the drive laser and simplifying the measurement system compared to the first protocol. We should note that measurement of the spin on $\ket{+}_{\textrm{e}}=(1/\sqrt{2})(\ket{0}_{\textrm{e}}+\ket{1}_{\textrm{e}})$ basis directly converts a microwave time-bin qubit to an optical time-bin qubit in the first protocol. Here, however, we focus only on the protocol using the spin as a quantum memory.

On the basis of both schemes, we estimate an actual entanglement generation rate between the superconducting qubit and the spin. For a superconducting qubit coupled to a cavity, the generation or absorption of a microwave time-bin qubit takes several hundreds of nanoseconds to a few microseconds [27, 35]. Here, we assume that it takes 1 $\mu$s to generate or absorb a microwave time-bin qubit for the superconducting qubit. Similarly, for the electron spin, $\sim$1 $\mu$s is thought to be needed for the operation based on the simulation of the master equation (see Appendix C). Then, ignoring the propagation time in the waveguide between the superconducting qubit and the electron spin, it is possible to attempt entanglement generation at a rate of $r_{\textrm{sc-e}}\sim$500 kHz, where $r_{\textrm{sc-e}}$ is the entanglement trial rate. Considering the success probability of microwave photon emission or absorption, the successful entanglement generation rate between the superconducting qubit and spin, $R_{\textrm{sc-e}}$, can be expressed as

where $\eta_{\textrm{sc}}^{\textrm{mw}}$ ($\eta_{\textrm{e}}^{\textrm{mw}}$) is the probability of successful microwave emission or absorption operation by the superconductor (spin). Here, we can confirm the generation of entanglement with the heralding techniques mentioned above. Figure 3 shows $R_{\textrm{sc-e}}$ as a function of $\eta_{\textrm{e}}^{\textrm{mw}}$. Although we calculate $R_{\textrm{sc-e}}$ for $\eta_{\textrm{e}}^{\textrm{mw}}$ = 0.9, 1.0, it does not seriously affect $R_{\textrm{sc-e}}$. Assuming the entanglement generation rate between remote color centers to be 10 kHz based on the discussion in the previous section, $R_{\textrm{sc-e}}>10$ kHz do not seriously degrade the whole entanglement generation speed between the superconducting qubits. Thus, $\eta_{\textrm{e}}^{\textrm{mw}}=0.1$, corresponding to $R_{\textrm{sc-e}}\sim 50$ kHz, could be sufficient to generate entanglement between the superconducting qubit and the spin. Even $\eta_{\textrm{e}}^{\textrm{mw}}\sim 0.02$, corresponding to $R_{\textrm{sc-e}}\sim 10$ kHz, could work if the decrease in the whole entanglement generation rate to 5 kHz is acceptable. Although the interaction between a microwave photon and single spin still needs proof-of-concept experiments, our numerical simulation indicates that $\eta_{\textrm{e}}^{\textrm{mw}}\sim 0.1$ is feasible considering the current technologies and strain sensitivity of a specific color center (see Appendix C).

After the successful entanglement generation between the superconducting qubit and the spin in nodes A and B, BSM is performed between the electron and nuclear spins in each node (Fig. 4). Depending on the BSM result, the quantum state of the superconducting qubits in nodes A and B is projected to the states, $(1/\sqrt{2})(\ket{g}_{\textrm{sc}}^{\textrm{A}}\ket{g}_{\textrm{sc}}^{\textrm{B}}\pm\ket{e}_{\textrm{sc}}^{\textrm{A}}\ket{e}_{\textrm{sc}}^{\textrm{B}}$) or $(1/\sqrt{2})(\ket{g}_{\textrm{sc}}^{\textrm{A}}\ket{e}_{\textrm{sc}}^{\textrm{B}}\pm\ket{e}_{\textrm{sc}}^{\textrm{A}}\ket{g}_{\textrm{sc}}^{\textrm{B}}$). The overall gate sequence of the entanglement generation is shown in Fig. 4. Also, in Table 1, we summarize the parameters needed to control and read out the quantum states based on the above discussions and references shown in the caption. Here, we consider three situations: (A) moderate hyperfine coupling and weak microwave drive power, (B) strong hyperfine coupling and moderate microwave drive power, and (C) strong hyperfine coupling and strong microwave drive power. Based on these parameters, the total gate operation time for BSM is 20-30 $\mu$s, as can be seen from Fig. 4 and Table 1. To efficiently generate entanglement, the time for the BSM must be shorter than the coherence time of the superconducting qubits. State-of-the-art superconducting qubits have sufficient performance, since their coherence time reaches 100 $\mu$s [1] or even $\sim 1$ ms [36], which is longer than the BSM time. However, a much quicker BSM is still needed to generate the entanglement with high fidelity.

Considering the success probability of entanglement generation between the remote spins and that between the superconducting qubit and the spin, the average entanglement generation rate between the superconducting qubits, $R_{\textrm{sc-sc}}^{\textrm{mem}}$, depends on $R_{\textrm{e-e}}$ and $R_{\textrm{sc-e}}$ as

where $\tau^{\textrm{init}}$ is the time to initialize the electron and the nuclear spins, $\tau_{\textrm{n}}^{\textrm{SWAP}}$ is the time to swap the state of the electron spin with that of the nuclear spin, $\tau_{\textrm{sc-e}}$ is the average time to generate entanglement between the electron spin and the superconducting qubit, and $\tau^{\textrm{BSM}}$ is the time to perform BSM for the electron and nuclear spins. $\tau_{\textrm{sc-e}}$ can be calculated by

In Table 1, assuming the highest efficiency, $\eta_{\textrm{e}}^{\textrm{opt}}=1$ and $\eta_{\textrm{sc}}^{\textrm{mw}}=\eta_{\textrm{e}}^{\textrm{mw}}=1$, we estimate $R_{\textrm{sc-sc}}^{\textrm{mem}}$ to be 10-20 kHz at maximum.

If we can use multiple quantum spin memories [21] as shown in Fig.5, we can prepare the remote entanglement between electron spins during the BSM of another spin memory. Based on the assumption shown in Table 1, a system with an additional parallel memory is sufficient to maintain the entanglement between the remote spins and to supply it to the superconducting qubits. The overhead time to generate electron spin-spin entanglement is again eliminated, resulting in the entanglement attempt rate of $\sim$40 kHz at maximum, which is limited by the speed of the BSM of the electron and nuclear spins.

We estimate the entanglement generation rate between remote superconducting qubits using the quantum spin memory. Here, we compare the performance of the memory-based scheme to those of other protocols without memory.

We proposed a protocol for entanglement generation between remote superconducting qubits using solid-state spin quantum memories, which can be used as interfaces for both microwave and optical photons. Based on the current state-of-the-art technologies, we estimated the maximum entanglement generation rate to be 10-40 kHz. The proposed memory-based scheme’s entanglement generation rate can exceed that of the direct conversion (DC) scheme as long as the microwave-to-optical conversion efficiency of DC is below 0.1. Moreover, the memory-based scheme can reduce the amount of heat generated below the mixing chamber compared to that generated by the entangled-photon scheme, which uses a strong optical pump to generate entangled-photon pairs. For quantum communication between remote superconducting qubits, the reduction of heat is a central issue considering that operation occurs inside a dilution refrigerator. This suggests the usefulness of introducing the quantum memory.

This work was supported by the Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (20H05661, 20K20441); by a Japan Science and Technology Agency (JST) CREST grant (JPMJCR1773); and by a JST Moonshot R$\&$D grant (JPMJMS2062). We also acknowledge the Ministry of Internal Affairs and Communications (MIC) for funding, research and development for the construction of a global quantum cryptography network (JPMI00316).

H. Kurokawa conceived the main protocol and performed the calculation. M. Y. and Y. S. contributed the design of partial protocol. H. Kosaka supervised the project.