Comparing the performance of practical two-qubit gates for individual ¹⁷¹Yb ions in yttrium orthovanadate

As suggested in Grimm et al. (2021); Kinos et al. (2021a), one can exploit the magnetic dipolar interaction between two Yb ions to perform a controlled-phase gate. In our setup, we consider two nearby Yb ions doped into YVO. For each ion, we define a “passive qubit” by utilizing two of the lowest hyperfine ground state energy levels that share the same electron spin but differ in nuclear spins. We denote these levels as $\ket{\uparrow}$ and $\ket{\downarrow}$, as depicted in Figure 2. These passive qubits have a long coherence time ($31$ ms Kindem et al. (2020)) and therefore can serve as memory qubits. However, for the gate operation, to speed up the slow microwave couplings, mitigate microwave-induced interactions on nearby ions and enhance the interaction strength, the quantum state of passive qubits must be transferred to “active qubits” in the excited state, which possess a significantly different g-tensor compared to the ground state. Grimm et al. (2021); Kinos et al. (2021a) For each Yb, we define active qubit levels using a different electronic spin level. This selection allows the active qubits to interact via the electronic magnetic dipolar interaction, which is proportional to $\mu_{B}$, rather than the nuclear magnetic dipolar interaction, which is proportional to $\mu_{N}$ Grimm et al. (2021). Here, we define the active qubit levels using solid green lines and denote them by $\ket{\uparrow^{\prime}}$ and $\ket{\downarrow^{\prime}}$ as shown in Figure 2. We describe this selection in Section V.1. In the following, we examine the magnetic dipolar interactions between active qubits to design a two-qubit phase gate between two Yb ions.

First, each Yb ion should be initialized in the lowest hyperfine ground state ($\ket{\uparrow}$) through optical pumping. Next, applying a $\pi/2$ microwave pulse can create a superposition of the $\ket{\uparrow}$ and $\ket{\downarrow}$ states. Following this, the population is transferred from the passive qubits to the active qubits by applying four $\pi$ pulses ($P_{1}$) simultaneously (two on each ion) as shown in Figure 2. This results in the transitions $\ket{\uparrow}\mapsto\ket{\uparrow^{\prime}}$ and $\ket{\downarrow}\mapsto\ket{\downarrow^{\prime}}$, placing each Yb ion into a superposition of $\ket{\uparrow^{\prime}}$ and $\ket{\downarrow^{\prime}}$. At this stage, a two-qubit phase gate is performed due to the Ising-type interaction between the active qubits. Specifically, if the control qubit (the first qubit) is in state $\ket{\uparrow^{\prime}}$, the phase of the target qubit (the second qubit) remains unchanged. Conversely, if the control qubit is in state $\ket{\downarrow^{\prime}}$, it induces a phase change in the target qubit. Finally, after a time delay, the quantum states are brought back to the passive qubits by applying another set of $\pi$ pulses ($P_{2}$). A schematic representation of this process is shown in Figure 2.

Note that to perform a CNOT gate between passive qubits, additional single-qubit gates are required in conjunction with the above phase gate Vandersypen and Chuang (2005)

$\displaystyle\mathrm{CNOT^{(1)}}=\left(\sqrt{Z}^{\dagger}\otimes({\sqrt{Z}}\sqrt{X}^{\dagger})\right)U_{1}\left(\mathbb{I}\otimes\sqrt{Y}\right),$

(2)

where $\sqrt{P}$ for a Pauli $P$ is defined as $\sqrt{P}:=\exp(-i\frac{\pi}{4}P)$, and $U_{1}=\exp(i\frac{\pi}{4}(Z\otimes Z))$ is the unitary implemented by the Ising interaction. Single qubit gates can be performed on passive qubits either through microwave fields or with the help of active qubits through optical fields Grimm et al. (2021).

Scattering a single photon from the qubit-cavity system can be used to perform a controlled phase-flip gate between qubits Duan et al. (2005); Lin et al. (2006). This method has been experimentally realized as a locally controlled phase-flip gate with neutral atoms Welte et al. (2018) and has been theoretically explored for non-local controlled phase gates between rare-earth ions in separate microsphere cavities Xiao et al. (2004). While these studies focus on the strong coupling regime, Ref Asadi et al. (2020a) extends the approach to implement a CZ gate between two qubits and provides a fidelity expression applicable to both weak and strong coupling regimes.

This scheme requires the detection of a single photon. Alternatively, it can be adapted for quantum non-demolition (QND) measurements to detect the presence of the photon without measuring or destroying its quantum state by probing the quantum emitter system O’Brien et al. (2016).

For this scheme, we employ two of the ground state hyperfine levels of Yb ions as qubit levels ($\ket{\uparrow}$ and $\ket{\downarrow}$), while the first hyperfine level in the excited state serves as an ancillary level ($\ket{\uparrow^{\prime}}$). The interaction between two Yb ions is mediated by scattering a single photon (in the state $\ket{P}$) from a single-sided cavity containing the Yb ions, as illustrated in Figure 3. We tune the $\ket{\uparrow}\mapsto\ket{\uparrow^{\prime}}$ transition of the ions to resonance with each other and with the cavity. If both qubits are in the state $\ket{\downarrow}$, the photon enters the cavity, reflects off its interior, and exits, resulting in a $\pi$-phase shift in the joint ion-photon state. Conversely, if either or both qubits are in the state $\ket{\uparrow}$, the cavity mode is altered, preventing the photon from entering, and the photon is instead reflected off the cavity’s out-coupling mirror. The unitary operator describing this scheme is given with $U_{2}={\exp({i\pi|{\downarrow\downarrow}\rangle\left\langle{\downarrow\downarrow}\right|\otimes\left|P\right\rangle\left\langle P\right|})}$. A CNOT gate is achieved by post-selecting the state of the photon, and conjugated by Hadamard gates on the target qubit $\mathrm{CNOT^{(2)}}=(\mathbb{I}\otimes H)U_{2}(\mathbb{I}\otimes H)$.

This scheme, originally introduced by Barrett and Kok in Barrett and Kok (2005), is a non-deterministic entanglement generation protocol that can be applied in designing quantum network elements. Reiserer et al. (2016); Asadi et al. (2018, 2020b); Ji et al. (2022). This entanglement generation protocol can be adapted into a controlled-Z gate by measuring the photons in the mutually unbiased basis (MUB) instead of the photon number basis Lim et al. (2005).

For this scheme, a three-level system is required, consisting of two qubit levels in the ground state hyperfine manifold, denoted by $\ket{\uparrow}$ and $\ket{\downarrow}$, and an ancillary excited state level, denoted by $\ket{\uparrow^{\prime}}$, as illustrated in Figure 4. Initially, both ions are pumped into the $\ket{\uparrow}$ level. Then, by applying a microwave $\pi/2$ pulse to each ion, a superposition of $\left(\ket{\uparrow}+\ket{\downarrow}\right)/\sqrt{2}$ is created. Next, an optical $\pi$ pulse is applied in resonance with the $\ket{\uparrow}\mapsto\ket{\uparrow^{\prime}}$ transition for each ion. The excited states will eventually decay (with potential enhancement from Purcell effects if the ions are placed in separate cavities), creating entanglement between the qubit state and the photon number. This results in the state $\left(\ket{\uparrow 1}+\ket{\downarrow 0}\right)/\sqrt{2}$, where $1(0)$ indicates the presence (absence) of a photon. The emitted photon(s) are then sent to a beam splitter (BS) positioned midway between the ions, where the which-path information is erased before the photon detection event occurs. Detection of a single photon will project the joint state of the two ions into a maximally entangled state. However, there is a possibility that both ions were excited and emitted photons, but one of the photons was lost during transmission. To rule out this scenario, which would result in a product state rather than a Bell state, a $\pi$ microwave pulse (pulse $P_{2}$ in Figure 4) is applied to the ground state levels right after the first excitation-emission step, causing a spin flip. This is followed by a second round of optical excitation. Detecting two sequential single photons ensures that the remote Yb ions are in an entangled Bell state

$$\left|\psi_{\pm}\right\rangle=\frac{1}{\sqrt{2}}(\left|\uparrow_{\mathrm{Yb}_{1}}\downarrow_{\mathrm{Yb}_{2}}\right\rangle\pm\left|\downarrow_{\mathrm{Yb}_{1}}\uparrow_{\mathrm{Yb}_{2}}\right\rangle),$$

(3)

where $+(-)$ sign indicates the case such that the same (different) detector(s) registered a photon. It can be shown that by altering the measurement basis, this scheme can be transformed into a CZ gate (see Section SIII for the derivation).

Employing the perturbative method, which was described in the previous section, allows us to compute the fidelity of the magnetic dipole-dipole interaction gate as (see Section SII for the derivation)

	$\displaystyle\mathrm{F_{MD}}=$	$\displaystyle 1-{T_{\mathrm{act}}}\left(\frac{{7}}{{8}}({\gamma_{1,\Uparrow^{\prime}}}+{\gamma_{1,\Downarrow^{\prime}}})+\frac{{13}}{{16}}({\gamma_{2}}+{\gamma_{3}})+\frac{1}{{2}}({\gamma_{4}}+{\gamma_{5}})\right)$		(11)
		$\displaystyle-{T_{{\mathop{\rm int}}}}\left({\gamma_{1,\Uparrow^{\prime}}}+{\gamma_{1,\Downarrow^{\prime}}}+\frac{{3}}{{4}}{\gamma_{3}}+\frac{{1}}{{2}}{\gamma_{5}}\right)-\frac{a\left(J_{x}+J_{y}\right)^{2}}{{J_{z}^{2}}},$

where $T_{act}=\frac{\pi}{\Omega}$ represents the activation time, the duration needed to transfer the population from passive to active qubits, with $\Omega$ being the Rabi frequency. Here the dissipative parameters affecting the system are considered as optical decay rate $\gamma_{1,\Uparrow^{\prime}}(\gamma_{1,\Downarrow^{\prime}})$ corresponding to the transitions $\Uparrow^{\prime}:\ket{\uparrow^{\prime}}\to\ket{\uparrow}$ ($\Downarrow^{\prime}:\ket{\downarrow^{\prime}}\to\ket{\downarrow}$), ground (excited) state spin decay rate $\gamma_{2(3)}$, and ground (excited) state spin dephase rate $\gamma_{4(5)}$. The interaction time between two active qubits in the excited states is denoted by $T_{int}=\frac{\hbar\pi}{4J_{z}}$ where $J_{z}=\frac{\mu_{0}(\mu_{B}g_{z})^{2}}{8\pi r^{3}}$ is the coupling strength between the qubits, with $\mu_{0}$ being the vacuum permeability, $\mu_{B}$ the Bohr magneton, $g_{z}$ as the principal value of the g-tensor, and $r$ the spatial distance between two Yb ions. In this analysis $J_{i}=\frac{\mu_{0}(\mu_{B}g_{i})^{2}}{16\pi r^{3}}$ (for $i=x,y$) define the transverse components of the Ising-type interaction, where we consider the point symmetry with $g_{x}=g_{y}\equiv g_{\perp}$ and $g_{z}\equiv g_{\|}$. Here, $a$ is a coefficient defined by $a=(32(2-\sqrt{2})-{\pi^{2}})/512$. For $J_{x}=J_{y}\equiv J_{\perp}$ the second order error (denoted by coefficient $a$) is bounded above by $\sim J_{\perp}^{2}/J_{\|}^{2}$ which approves the estimation obtained in Ref. Grimm et al. (2021). In ¹⁷¹Yb³⁺:YVO, the excited state g-tensor components parallel and perpendicular to the crystal symmetry axis (the $c$ axis) are $g_{\|}=2.51$ and $g_{\perp}=1.7$ Kindem et al. (2018). This results in a second-order error contribution to the fidelity of approximately $10^{-3}$, setting a lower bound for the fidelity. This lower bound is relevant in cases where transverse interactions mediate the gate process.

In modelling the system, we neglect certain mechanisms. Below, we describe how these effects scale relative to the two-qubit process.

Following Ref. Grimm et al. (2021), we disregard the nuclear magnetic dipolar interaction, as its strength is $\sim 10^{7}$ times weaker than the electronic magnetic dipolar interaction. This difference arises mostly from the small nuclear magnetic moment ($\mu_{N}$), and the nuclear g factor ($g_{n}=0.987$).

When performing the CNOT gate, some infidelity inevitably arises from the rotating single-qubit gates. Among the single-qubit gates, the Hadamard gate for example can be performed through $\pi/2$ microwave pulses between hyperfine levels which involves nuclear magneton and results in a long gate time of $T_{\mathrm{single}}=\frac{\hbar\pi}{\mu_{N}B}\approx 0.13\mu$s (at $B=500$ mT). Conversely, one can couple nuclear levels indirectly through optical pulses at the cost of adding two consequent optical pulses and advantageous for achieving a faster gate time $T_{\mathrm{single}}=\frac{2\hbar\pi}{\mu_{B}B}\approx 0.14$ ns Grimm et al. (2021).

Ref. Grimm et al. (2021) discusses that the intrinsic optical decoherence rate causes the infidelity $\sim T_{\mathrm{single}}/T_{2o}$, with $T_{2o}$ being the optical coherence time. Plugging in $T_{2o}=91\,\mu\mathrm{s}$ (at $B=500$ mT) Kindem et al. (2018), we get an infidelity of about $\epsilon_{\mathrm{single}}\simeq 1.5\times 10^{-6}$. Therefore, in these analyses, we assume that the fidelity of the CNOT gate is primarily determined by the two-qubit gate processes. However, we include these fast excitations in our analysis to capture their effect on the two-qubit activation process (see the second term in Equation 11).

Moreover, the error corresponding to off-resonant excitations due to optical $\pi$ pulses (involved in both single-qubit and activation operations) cause infidelity of $\sim\exp(-\frac{\pi}{2}\left(\frac{\Delta_{s}}{\Omega}\right)^{2})$ with $\Delta_{s}$ being the hyperfine splitting Grimm et al. (2021). To avoid any off-resonance excitation, the Rabi frequency of the activation pulses should be much smaller than the smallest hyperfine splitting, and hence, choosing $\Delta_{s}\geq 10\Omega$ yields an insignificant infidelity of $\sim 10^{-68}$. For example for a splitting of $\sim 100$ MHz, a Rabi frequency of at most 10 MHz is required. Employing this Rabi frequency to apply four simultaneous $\pi$-pulses results in an activation process to take approximately $T_{act}=\frac{\pi}{\Omega}\sim 0.31\mu$s.

If the splitting is small, achieving fast excitations can be challenging. In that case, one should examine how close the ions have to be to obtain fidelities above $90\%$. Therefore, there is a trade-off between splitting and separation of ions. In Figure 5, we demonstrate fidelity as a function of Rabi frequency and ions’ separation illustrating the variation in laser power corresponding to the changing distance between two ions. The dipolar interaction strength falls off with $1/r^{3}$, making the distance between two Yb ions a crucial factor in this scheme. As Figure 5 confirms, the distance cannot be arbitrarily large, as the fidelity of the two-qubit process decreases with increased separation. Two nearby ions can be found by implanting each ion at precise locations within the crystal, achieving a separation of just a few nanometers between them Kornher et al. (2016).

To limit the decay process, we choose two of the lowest ground state hyperfine levels with long enough coherence time as the passive qubit levels. Furthermore, in ¹⁷¹Yb³⁺:YVO system with a nonzero perpendicular component of the g-tensor $(g_{\perp}\neq 0)$, choosing opposite nuclear levels within each electronic doublet reduces spin flip-flop errors during the two-qubit gate process Grimm et al. (2021). This approach offers two options for selecting active qubits within both different electron spin and nuclear spins (e.g., both solid or dashed green lines in Figure 2).

The optical decay rate for an ensemble of ¹⁷¹Yb³⁺: YVO is $\gamma_{1,\Downarrow^{\prime}}\simeq\gamma_{1,\Uparrow^{\prime}}\equiv\gamma_{1}=2\pi\times 596$ Hz Kindem et al. (2018). We estimate the ground state spin dephasing rate with relation $\gamma_{4}=1/{T_{2s,g}}-\gamma_{2}/2\simeq 2\pi\times 3.6$ Hz, where $T_{2s,g}=6.6$ ms (at $B=440$ mT) is the ground state spin coherence time Kindem et al. (2018). For single ${}^{171}\text{Yb}^{3+}$ ions coupled to a YVO photonic crystal cavity, the ground state spin coherence time is extended to $31$ ms using Carr-Purcell Meiboom-Gill (CPMG) decoupling sequences and the ground state spin decay rate and is measured to be $\gamma_{2}=2\pi\times 2.95$ Hz Kindem et al. (2020). Similarly, we estimate the excited state spin dephasing rate with relation $\gamma_{5}=1/T_{2s,e}-\gamma_{3}/2=2\pi\times 4.5$ KHz, where $T_{2s,e}$ is the excited state spin coherence time, measured to be $35\mu$s at low magnetic fields Bartholomew et al. (2020). We assume that the excited state spin decay rate ($\gamma_{3}$) is the same as the ground state ($\gamma_{2}$), as there is no available information on this to the best of our knowledge. Therefore, the fidelities of 0.98 and 0.92 can be achieved for distances $r=5$ nm, and $r=10$ nm respectively. The overall gate time is defined to be $T_{g,MD}=2T_{act}+T_{int}$, where the coefficient $2$ is considered for activation and deactivation pulses. The corresponding gate times for distances $r=5$ nm, and $r=10$ nm are $1\mu$s, and $3.68\mu$s respectively.

In this scheme, photon loss due to spontaneous emission, scattering, detector efficiency and other deficiencies make the gate operation probabilistic. However, they do not cause errors as long as the photon count is properly recorded Duan et al. (2005). Therefore, we assume that the photon is successfully detected and focus on the remaining sources of error that impact the fidelity. The fidelity of the photon-scattering gate scheme in high cooperativity regime $C\gg 1$ is computed in Asadi et al. (2020a) to be

	$\displaystyle\mathrm{F_{PS}}=$	$\displaystyle 1-\frac{5}{2C}-\frac{(\delta_{p}^{2}+\sigma_{p}^{2})}{4\gamma_{1}^{2}C^{2}}\left(11-80\alpha^{2}+192\alpha^{4}\right)$		(12)
		$\displaystyle-\frac{\left(\delta_{Yb_{1}}\!-\delta_{Yb_{2}}\right)^{2}}{2\gamma_{1}^{2}C}-\Gamma T_{g,PS},$

where $\delta_{p}$ is the cavity-photon detuning, $\delta_{Yb_{n}}$ denotes the detuning of the cavity frequency from the optical transition of the $n^{\text{th}}$ system, $\sigma_{p}$ is the spectral standard deviation of the incident photon that is modelled assuming a Gaussian wave packet for the photon, $\Gamma$ represents the effective decoherence rate, and $T_{g,PS}$ is the gate time. The parameter $\alpha=g/\kappa$ defines the ratio between the cavity-ion coupling rate $g$, and the cavity decay rate $\kappa$ and the cavity cooperativity is $C=4g^{2}/\kappa\gamma_{1}$.

While the analytical expression for gate fidelity has been previously derived, it does not account for optical pure dephasing and spin decoherence effects. To address these limitations, we approach the problem using a fully numerical method based on input-output theory (see Section SIV). However, in our numerical simulation, we consider the photon to have a Lorentzian spectrum rather than the Gaussian spectrum used in the analytic expression since otherwise a large time-dependent master equation must be solved that substantially increases simulation time and restricts the parameter range that we can explore. Thus, although the qualitative behaviour and scaling properties are expected to be the same as in Equation 12, quantitative comparisons require caution.

In Figure 6, we display our fidelity estimations as a function of cavity parameters ($g,\kappa$) using the numerical simulation where we consider a photon with a Lorentzian-shaped spectrum. We also use this numerical simulation to validate properties of the previous analytical approximation derived in Equation 12. Our numerical simulation reveals that this analytic expression can be modified to capture the degrading effect of optical pure dephasing when $\alpha\ll 1$ by replacing the cavity cooperativity with an effective cavity cooperativity $C_{eff}=C/(1+w(T_{1o}/T_{2o}-1))$, where $w\simeq 0.7$, $T_{1o}$ is the optical lifetime, and $T_{2o}$ is the optical coherence time. Additionally, we find that $\Gamma\propto 1/T_{2s,g}$, where $T_{2s,g}$ represents the ground state spin coherence time, offers a reliable estimate of the error caused by spin decoherence. The exact proportionality coefficient, however, depends on the shape of the photon. When using the same definition of gate time as in Asadi et al. (2020a) along with $\sigma_{p}=1/T_{1p}$ where $T_{1p}$ is the scattered photon lifetime, we find $\Gamma\simeq 1/T_{2s,g}$ quantitatively agrees with the numerical simulation.

The numerical simulation also leads us to identify a previously-overlooked alteration of the lower-bound error scaling when spin decoherence is non-negligible. Assuming $\delta_{p}=0$ and the ions are in perfect resonance with each other and with the cavity, from Equation 12, we find that in the absence of spin dephasing ($\Gamma=0$), it is always possible to reach the minimum error of $1-\mathrm{F_{PS,max}}\propto 1/C$ by scattering a very narrow photon ($\sigma_{p}\ll 1$). However, the gate time is proportional to the time it takes to scatter the photon ($T_{g,PS}\propto 1/\sigma_{p}$). Thus, as $C$ is increased to improve fidelity, it is necessary to increase $T_{g,PS}$ and so if $\Gamma\neq 0$ this causes the last term in Equation 12 to increase. This imposes a trade-off between having an error from the spectral broadening (i.e., $\sigma_{p}^{2}$ term) or from the spin dephasing (i.e., $\Gamma T_{g,PS}$ term). In Ref. Asadi et al. (2020a), this trade-off was explored but assumed to always be of a second-order correction to the seemingly dominant $1/C$ term when the gate time was optimized. This is, in fact, not the case.

For a given cavity cooperativity and spin dephasing rate $\Gamma$, there is an optimal bandwidth of the photon that minimizes the error. This optimal bandwidth can be computed by setting the third and the last terms in Equation 12 to be equal. Solving for the bandwidth implies that it must scale as $\sigma_{p}\propto C^{2/3}$ and hence $\Gamma T_{g,PS}\propto 1/C^{2/3}$. Thus, if either $\Gamma$ or $C$ is large enough, this scheme will have an error scaling of $1/C^{2/3}$ rather than $1/C$, which is significantly worse for large $C$.

In addition to setting the photon bandwidth, it is necessary to define a photon truncation time that defines the total gate time (denoted by $T_{g,PS}$). If one stops measuring the photon too soon, there will be low efficiency. On the other hand, if one waits too long, spin decoherence will be exacerbated. For the value of $T_{2s,g}$ expected, we select this additional truncation time to be proportional to the time-scale of the scattered photon and such that it has a minimal impact on the fidelity for the range of cooperativity explored, leading to $T_{g,PS}=7T_{1p}$ which implies only 0.1% loss.

In our simulations, we also assume that the gate operation is sufficiently fast for the ions to effectively avoid experiencing spectral diffusion during this period. However, even for single emitters, spectral diffusion of the optical transition can reduce the Ramsey coherence time Kindem et al. (2020). Any remaining spectral diffusion could, in principle, be handled using active feedback or special decoupling techniques. Therefore, we focus on the upper bound fidelity imposed by the measured $T_{2o}$ value only. Although the spectral diffusion can be overcome with additional filtering or time binning, if unmitigated, this effect must be incorporated into the fidelity calculation. Assuming that spectral diffusion primarily determines the detuning between the ions’ optical transitions during the gate time, one can approximately account for this by including the time-dependent spectral diffusion function in the fidelity equations via the $\delta_{Yb_{1}}\!-\delta_{Yb_{2}}$ term.

In his scheme, similar to the previous approach, we employ two hyperfine levels in the ground state as qubit levels. However, unlike the previous scheme, only a single hyperfine level in the excited state is required to serve as an auxiliary state. After qubit initialization, the transitions $\ket{\uparrow}\mapsto\ket{\uparrow^{\prime}}$ of each ion should be brought into resonance with one another and with the cavity mode. A magnetic field gradient can tune the optical transitions of the ions into resonance with each other. On the other hand, to bring the cavity-ion system into resonance, the piezoelectric effect can be employed to adjust the cavity frequency relative to the transition frequency of the ions within it Goswami et al. (2018). When bringing the ions into resonance with each other, it should be noted that the laser field can couple both hyperfine qubit levels to the excited state. To avoid this this undesired coupling, the transition $\ket{\downarrow}\mapsto\ket{\uparrow^{\prime}}$ should be detuned far from the cavity resonance.

Photonic cavities have been fabricated in YVO crystals using ion beam milling, achieving a cavity-Yb ion coupling rate of $g=2\pi\times 23$ MHz and a cavity decay rate of $\kappa=2\pi\times 30.7$ GHz Kindem et al. (2020). For single Yb ions coupled to this cavity a reduced optical lifetime of $2.27\mu$s have been measured Kindem et al. (2020). However, the fabrication process poses a significant risk of crystal damage, making it a challenging method. An alternative approach involves designing a hybrid photonic crystal cavity system, where the cavity is fabricated separately and later transferred onto the crystal. For instance, a Gallium Arsenide (GaAs) photonic crystal cavity has been developed, demonstrating a reduced optical lifetime of $4.2\mu$s corresponding to the Purcell factor of 179 for strongly coupled single Yb ions coupled to hybrid GaAs-YVO system Wu et al. (2023).

In this scheme, the fidelity is primarily restricted by the optical quality of the source. Specifically, when transferring to an MUB basis, we assume all single-qubit gates and the beam splitter to be ideal. Therefore, we consider the fidelity of the two-qubit gate to be limited by the two-qubit operation.

This scheme requires the detection of two consecutive single photons. Assuming the detection time ($T_{d}$) is longer than the optical lifetime ($T_{d}>T_{1o}$) the fidelity is given by Wein et al. (2020); Asadi et al. (2020b)

$$\mathrm{F_{PI}}=\frac{1}{2}\left(1+\frac{\gamma_{1}^{2}}{(\gamma_{1}+2\gamma^{*})^{2}+\Delta_{\omega}^{2}}\right),$$

(13)

where $\gamma_{1}$ represents the optical decay rate from the auxiliary excited state $\ket{\uparrow^{\prime}}$ to the ground-state qubit level $\ket{\uparrow}$ (assuming the decay rate from $\ket{\uparrow^{\prime}}\mapsto\ket{\downarrow}$ is negligible), $\gamma^{*}$ is the optical pure dephasing rate, and $\Delta_{\omega}$ is the detuning between optical transitions of two Yb ions. We estimate the optical pure dephasing rate with relation $\gamma^{*}=1/T_{2o}-\gamma_{1}/2=2\pi\times 1.4$ kHz, where $T_{2o}=91\mu$s is the optical coherence time of Yb ions in YVO crystal bulk Kindem et al. (2018). This results in a fidelity of $\mathrm{F_{PI}}=0.51$.

This reduced fidelity is attributed to the Yb ion optical pure dephasing rate captured by $T_{2o}$, which is influenced by both spin decoherence and spectral fluctuations of the optical transition that occur faster than the optical lifetime. However, knowing that the ground state spin coherence time $T_{2s}=31$ ms Kindem et al. (2020) is already two orders of magnitude larger than the unenhanced optical lifetime $T_{1o}=267\mu\text{s}$ Kindem et al. (2018), the contribution of the ground state spin dephasing to the optical decoherence is negligible. Furthermore, any amount of Purcell enhancement needed to overcome the optical dephasing rate will further reduce the impact of the ground state spin decoherence and hence it can be safely neglected in the analysis.

While this gate scheme does not inherently require a cavity to be performed, below, we discuss how employing a cavity can enhance the fidelity of the gate between two Yb ions. In case of implying a cavity Equation 13 is modified to

$$\mathrm{F^{\prime}_{PI}}=\frac{1}{2}\left(1+\frac{\gamma_{1}^{{}^{\prime}2}}{(\gamma_{1}^{\prime}+2\gamma^{*})^{2}+\Delta_{\omega}^{2}}\right),$$

(14)

where $\gamma_{1}^{\prime}=\gamma_{1r}F_{p}+\gamma_{1}$ defines the Purcell-enhanced optical decay rate of the ion in the presence of the cavity, $\gamma_{1}=\gamma_{1r}+\gamma_{1nr}$ is Yb ion decay rate with radiative ($\gamma_{1r}$) and non-radiative decay rate ($\gamma_{1nr}$) parts. The Purcell factor in both strong coupling and bad-cavity regimes is defined as Wein (2021)

$$F_{p}=\frac{R\kappa}{\gamma_{r}(\kappa+R)},$$

(15)

with $\kappa$ being the cavity decay rate, and

$$R=\frac{4g^{2}(\kappa+\gamma_{1}+2\gamma^{*})}{(\kappa+\gamma_{1}+2\gamma^{*})^{2}+4\Delta_{\omega}^{2}}.$$

(16)

We demonstrate how the fidelity of this scheme varies with different cavity parameters in Figure 7. Additionally, we determine the necessary Purcell factor for achieving a specific fidelity threshold.

Assuming resonant excitation results in photons with a Lorentzian spectral shape, and since this scheme takes two emissions, we consider the gate time for this scheme twice the gate time definition in the photon scattering gate $T_{g,PI}=14T_{1p}^{\prime}$. Therefore, we allow at most $7T_{1p}$ for each emission time bin. However, it should be noted that here $T_{1p}^{\prime}$ denotes the Purcell-enhanced lifetime of the photon set by the emitter while in photon scattering scheme, the photon originates from an external source and not the ion. Furthermore, this gate time neglects the time taken to perform the necessary spin-flip operation between the emission events. However, as discussed in Section V.1 we assume that single-qubit operations can be implemented much faster than the time it takes to emit two photons.

In this scheme, as in the photon scattering scheme, qubit levels are defined using two of the lowest hyperfine levels in the ground state, along with an ancillary level in the excited state.

Let us start by examining the fidelity for the first-order approximation $\rho_{I}^{(1)}$ in Equation S7. This gives

$\displaystyle\begin{split}1-F&=-\frac{i\delta}{\hbar}\int_{0}^{T_{g}}\langle\psi(0)|\,[H_{e,I}(t^{\prime}),\ket{\psi(0)}\bra{\psi(0)}]\,\ket{\psi(0)}\\ &\qquad+\int_{0}^{T_{g}}\bra{\psi(0)}\mathcal{D}_{I}\left[\ket{\psi(0)}\bra{\psi(0)}\right]\ket{\psi(0)}\\ &\qquad+O\left((|\delta|+\sum_{k}|\gamma_{k}|)^{2}\right).\end{split}$

(S10)

Note that $\langle\psi(0)|[H_{e,I}(t^{\prime}),\ket{\psi(0)}\bra{\psi(0)}]\ket{\psi(0)}=0$, which means that the first-order contribution of $\delta$ (i.e., reversible errors) to infidelity is zero. However, accounting for first-order irreversible errors, we have

$\displaystyle\begin{split}&\bra{\psi(0)}\mathcal{D}_{I}\left[\ket{\psi(0)}\bra{\psi(0)}\right]\ket{\psi(0)}=\\ &\qquad\qquad\sum_{k}\gamma_{k}\langle\psi(0)|L_{k,I}|\psi(0)\rangle\langle\psi(0)|L^{\dagger}_{k,I}|\psi(0)\rangle\\ &\qquad\qquad-\frac{1}{2}\bra{\psi(0)}\{L_{k,I}^{\dagger}L_{k,I},\ket{\psi(0)}\bra{\psi(0)}\}\ket{\psi(0)},\end{split}$

(S11)

which follows directly from Equation S3. Note that

$\displaystyle\begin{split}\langle\psi(0)|L_{k,I}\ket{\psi(0)}&=\bra{\psi(0)}U^{\dagger}(t)L_{k}U(t)\ket{\psi(0)}\\ &=\langle L_{k}(t)\rangle,\end{split}$

(S12)

as we are adapting the notation $\langle\mathcal{O}(t)\rangle:=\bra{\psi(t)}\mathcal{O}\ket{\psi(t)}$. Using a similar argument for the second term on the right-hand side of Equation S11 we get

$\displaystyle 1-F=\epsilon_{L}^{(1)}+O\left((|\delta|+\sum_{k}|\gamma_{k}|)^{2}\right),$

(S13)

with

$$\epsilon_{L}^{(1)}=\sum_{k}\int_{0}^{T_{g}}\gamma_{k}\left(\langle L_{k}^{\dagger}(t^{\prime})L_{k}(t^{\prime})\rangle-|\langle L_{k}(t^{\prime})\rangle|^{2}\right)\,\mathrm{d}t^{\prime},$$

which is identical to Equation 8. We proceed to the second term correction by computing $\bra{\psi(0)}\rho^{(2)}_{I}-\rho^{(1)}_{I}\ket{\psi(0)}$. Note that $\epsilon_{L}^{(1)}+\bra{\psi(0)}\rho^{(2)}_{I}-\rho^{(1)}_{I}\ket{\psi(0)}$ gives us the second order approximation to $F$. However, as we are merely interested in the first non-zero perturbation terms, we will keep only the terms including $\delta^{2}$ and $\delta\cdot\gamma_{k}$ as the first-order contribution of $\delta$ is zero. We have

$\displaystyle\begin{split}\rho^{(2)}_{I}-\rho^{(1)}_{I}=&-\frac{\delta^{2}}{\hbar^{2}}\int_{0}^{T_{g}}\int_{0}^{t}[H_{e,I}(t^{\prime}),[H_{e,I}(t^{\prime\prime}),\rho_{0}]]\,\mathrm{d}t^{\prime}\,\mathrm{d}t^{\prime\prime}\\ &-\frac{i\epsilon}{\hbar}\bigg{(}\int_{0}^{T_{g}}\int_{0}^{t}[H_{e,I}(t^{\prime}),\mathcal{D}_{I}(t^{\prime\prime})[\rho(0)]]\,\mathrm{d}t^{\prime}\,\mathrm{d}t^{\prime\prime}\\ &-\int_{0}^{T_{g}}\int_{0}^{t}\mathcal{D}_{I}(t^{\prime})[[H_{e,I}(t^{\prime}),\rho_{0}]]\,\mathrm{d}t^{\prime}\,\mathrm{d}t^{\prime\prime}\bigg{)}\\ &+O((\sum_{k}|\gamma_{k}|)^{2}).\end{split}$

(S14)

By a straightforward calculation, we get

$\displaystyle\begin{split}\bra{\psi_{0}}[H_{e,I}(t^{\prime}),[H_{e,I}(t^{\prime\prime}),\rho_{0}]]&\ket{\psi_{0}}=\\ &\real(\langle H_{e}(t^{\prime})H_{e}(t^{\prime\prime})\rangle\\ &-\langle H_{e}(t^{\prime})\rangle\langle H_{e}(t^{\prime\prime})\rangle\rangle\bigg{)},\end{split}$

(S15)

which corresponds to $\epsilon^{(2)}_{HH}$ as in Equation 10. Similarly, we get

$\displaystyle\begin{split}-i\bra{\psi_{0}}[H_{e,I}(t^{\prime}),\mathcal{D}_{I}[\rho_{0}]]&\ket{\psi_{0}}=\\ &\imaginary(\sum_{k}\gamma_{k}\langle H_{e}(t^{\prime})L_{k}(t^{\prime\prime})\rangle\langle L_{k}(t^{\prime\prime})\rangle\\ &-2\langle H_{e}(t^{\prime})L_{k}(t^{\prime\prime})\rangle\langle L_{k}^{\dagger}(t^{\prime\prime})\rangle\bigg{)},\end{split}$

(S16)

and

$\displaystyle\begin{split}-i\bra{\psi_{0}}\mathcal{D}_{I}(t^{\prime})[[H_{e,I}(t^{\prime}),\rho_{0}]]&\ket{\psi_{0}}=\\ &\imaginary(\langle L_{k}^{\dagger}(t^{\prime})L_{k}(t^{\prime})H_{e}(t^{\prime\prime})\rangle\\ &-2\langle L_{k}(t^{\prime})H_{e}(t^{\prime\prime})\rangle\langle L_{k}^{\dagger}(t^{\prime})\rangle\bigg{)},\end{split}$

(S17)

which gives

$\displaystyle\begin{split}\epsilon_{LH}^{(2)}&=\sum_{k}\frac{\gamma_{k}\delta}{\hbar}\int_{t^{\prime}=0}^{T_{g}}\int_{t^{\prime\prime}=0}^{t^{\prime}}\mathrm{Im}\bigg{(}\langle H_{e}(t^{\prime})L_{k}^{\dagger}(t^{\prime\prime})L_{k}(t^{\prime\prime})\rangle\\ &\qquad\qquad-2\langle H_{e}(t^{\prime})L_{k}(t^{\prime\prime})\rangle\langle L_{k}^{\dagger}(t^{\prime\prime})\rangle\bigg{)}\,\mathrm{d}t^{\prime\prime}\,\mathrm{d}t^{\prime}\\ &+\sum_{k}\frac{\gamma_{k}\delta}{\hbar}\int_{t^{\prime}=0}^{T_{g}}\int_{t^{\prime\prime}=0}^{t^{\prime}}\mathrm{Im}\bigg{(}\langle L_{k}^{\dagger}(t^{\prime})L_{k}(t^{\prime})H_{e}(t^{\prime\prime})\rangle\\ &\quad-2\langle L_{k}(t^{\prime})H_{e}(t^{\prime\prime})\rangle\langle L_{k}^{\dagger}(t^{\prime})\rangle\bigg{)}\,\mathrm{d}t^{\prime\prime}\,\mathrm{d}t^{\prime}.\end{split}$

(S18)