Driven geometric phase gates with trapped ions

Let us start by describing the system under consideration: a two-ion (N = 2) crystal confined in a linear Paul trap [10]. Under certain conditions [11], such radio-frequency traps provide an effective quadratic confining potential, which is characterized by the so-called axial ω_z, and radial {ω_x,ω_y} trap frequencies. Moreover, when {ω_x,ω_y} ≫ ω_z, the ion equilibrium positions arrange in a string along the trap z-axis. As customary in these cases, when the particles only perform small excursions around their equilibrium positions, their motion can be described in terms of collective vibrational modes, whose quantized excitations lead to the well-known phonons [12]. In the case of ions, the low temperatures provided by laser-cooling techniques justify such a treatment [13], and the vibrations of the crystal are described by

$$\begin{equation} H_{\rm p}=\sum_{\alpha}\sum_{n=1}^{N}\omega_{\alpha,n}a_{n,\alpha}^{\dagger}a_{n,\alpha}^{\phantom{\dagger}}, \end{equation} \tag{ 1 }$$

where a^†_n,α(a _n,α) are the creation (annihilation) phonon operators, ω_α,n is the vibrational frequency of the nth normal mode along the axis α = {x,y,z}, and we set ℏ = 1 throughout this paper. According to equation (1), the vibrations along different axes are decoupled, which allows us to focus our attention on the radial modes along the x-axis. However, we emphasize that our proposal also holds for the other modes. To simplify notation, we write a_n,x → a_n, ω_n,x → ω_n. Note that for a two-ion crystal there are two radial normal modes: the center-of-mass (com) and the zigzag (zz) mode which we also shall denote modes '1' and '2', respectively.

In addition to the vibrational degrees of freedom, our two-ion crystal also has internal (i.e. atomic) degrees of freedom. We consider ion species with a hyperfine structure, such that we can select two levels from the ground-state manifold to form our qubit {|0_i〉,|1_i〉}. This particular choice of qubit has two important properties: (i) spontaneous emission between the qubit states is negligible (i.e. T₁ times are much larger than experimental time scales), and (ii) the typical qubit frequencies ω₀/2π = (E₁ − E₀)/2π ∼ 1–10 GHz allow for microwave, or radio-frequency, radiation to drive directly the qubit transition. To test our scheme with experimental realistic parameters, we shall consider ²⁵Mg⁺ in this work, although we remark that our scheme also works for other ion species (see figure 1). We choose two hyperfine states |0〉 ≡ |F = 3, M = 3〉 and |0〉 ≡ |F' = 2, M' = 2〉, where M and M' are the Zeeman sub-levels of the F and F' levels of the electronic ground state manifold, which are separated by a frequency ω₀/2π ≈ 1.8 GHz.

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By considering the standard Zeeman interaction between the atomic magnetic dipole and an oscillating magnetic field from the microwave source, and using the standard rotating-wave approximation in quantum optics [14], the qubit Hamiltonian can be written as follows:

$$\begin{equation} H_{\rm q}=\sum_{i=1}^N\frac{1}{2}\omega_{0}\sigma_i^z+\frac{1}{2}(\Omega_{\rm d}\sigma_i^+{\rm e}^{-{\rm i}\omega_{\rm d}t}+{\rm H.c.}), \end{equation} \tag{ 2 }$$

where we have introduced the qubit operators σ^z_i = |1_i〉〈1_i| − |0_i〉〈0_i|, σ⁺_i = |1_i〉〈0_i| = (σ⁻_i)^†, and the frequency (Rabi frequency) ω_d(Ω_d) of the microwave driving. Note that the rotating-wave approximation requires ω_d ≈ ω₀, and |Ω_d| ≪ ω₀, which still leaves the possibility of achieving very strong drivings Ω_d/2π ∼ 1–10 MHz, while benefiting from the intensity stability of microwave sources. We also note that the large wavelengths of the microwave traveling waves avoid any qubit–phonon coupling induced by the microwave, which would impose further constraints on the strengths of the drivings. In this work, we exploit the possibility of obtaining such strong microwave drivings to produce fast two-qubit geometric phase gates with robustness to several sources of noise.

As done in previous two-qubit gate schemes [15–21], the main idea is to consider an additional coupling between the qubits and the phonons, and use the latter to mediate interactions between distant qubits in the ion trap. We follow [8], and consider laser-induced qubit–phonon couplings [10], which have already been demonstrated in several laboratories. Therefore, our work can be considered as an instance of a hybrid laser–microwave protocol. For completeness, we remark that for specific qubit choices such qubit–phonon couplings can also be achieved with microwave radiation. It has been shown that in the near field of microwave sources, it is possible to obtain qubit–phonon couplings by producing either static [22, 23], or oscillating magnetic-field gradients [24, 25]. The generality of the scheme presented in the following subsection would allow us to use any of these approaches to achieve qubit–phonon coupling and thus, also to implement the scheme in an all-microwave protocol. Note, however, that we follow none of these approaches [22–25] and therefore, the microwave radiation does not introduce any qubit–phonon coupling.

The qubit–phonon coupling is provided by a two-photon stimulated Raman transition via a third auxiliary level. The transitions to the auxiliary excited state are driven by two far detuned laser beams in a traveling-wave configuration, such that their detuning is much larger than the excited-state decay rate (e.g. for ²⁵Mg⁺ the linewidth of the excited state is Γ/2π ≈ 41.4 MHz, and thus a detuning of the order of Δ/2π ≈ 10–100 GHz would suffice, see figure 1). In this limit, the excited state can be adiabatically eliminated [26], and the qubit–phonon coupling becomes

$$\begin{equation} H_{\rm qp}=\sum \limits_{i} \frac{1}{2}\Omega_{\rm L} \sigma_i^+ {\rm e}^{{\rm i}({\bf k}_{\rm L}\cdot{\bf r}_i- \omega_{\rm L} t)} + {\rm H.c.}, \end{equation} \tag{ 3 }$$

where Ω_L = Ω_L1Ω*_L2/2Δ is the two-photon laser Rabi frequency, and the two-photon wavevector (frequency) k_L =  k_L1 −  k_L2 (ω_L = ω_L1 − ω_L2) are defined in terms of the corresponding parameters of the two laser beams³. By directing k_L along the x-axis, and setting the laser frequencies such that ω_L ≈ ω₀ − ω_n, and the detunings δ_n = ω_L − (ω₀ − ω_n), such that δ_n ≪ ω_n and |Ω_L| ≪ ω_n, a rotating-wave approximation leads to the so-called first red-sideband excitation

$$\begin{equation} H_{\rm qp}=\sum \limits_{i,n} \mathscr{F}_{in} \sigma_i^+ a_n {\rm e}^{-{\rm i} \omega_{\rm L} t} + {\rm H.c.} \end{equation} \tag{ 4 }$$

Here, we have defined the red-sideband coupling strengths

$$\begin{equation} \mathscr{F}_{i1}={\rm i} \frac{|\Omega_{\rm L}| \eta_1}{2\sqrt{2}},\quad \mathscr{F}_{i2}={\rm i} (-1)^i \frac{|\Omega_{\rm L}| \eta_2}{2\sqrt{2}}, \end{equation} \tag{ 5 }$$

where we have set the Raman-beam phase φ_L = 0, and used the Lamb–Dicke parameters $\eta _n={\bf{ k}}_{\mathrm { L}}\cdot {\bf{ e}}_x/\sqrt {2m\omega _n}\ll 1$. We will refer to these couplings generically as forces, since $\mathscr{F}_{in}\sqrt {2m\omega _n}$ corresponds to a force applied on the harmonic oscillator representing the vibrational mode.

As explained below, equations (1), (2) and (4) form the driven single-sideband Hamiltonian

$$\begin{equation} H_{\rm dss}=H_{\rm p}+H_{\rm q}+H_{\rm qp} \end{equation} \tag{ 6 }$$

which can be used to obtain the desired geometric phase gates for strong-enough microwave drivings. At this point, it is worth commenting that strong qubit drivings have also been realized experimentally in combination with a state-dependent force [27]. The driving in this case endows the scheme with protection from dephasing noise, while the resilience to the thermal motion of the ions is provided by the state-dependent force. However, this scheme is prone to phase noise due to fluctuations in the laser-beam paths, or intensity fluctuations of the qubit driving. In a different context, we should also mention that strong-driving-assisted protocols for the generation of entanglement have also been considered theoretically for atoms in cavities [28], where the driving may reduce errors due to thermal population of the cavity modes [29].

2.2.1. Introduction: geometric phase gates with a single sideband

In [8], we have analyzed how the driven single-sideband Hamiltonian H_dss leads to an entangling gate capable of producing two Bell states at particular instants of time:

$$\begin{equation} \begin{array}{ll}\displaystyle |01\rangle\to|\Psi^-\rangle=\frac{1}{\sqrt{2}}(|01\rangle-{\rm i}|10\rangle),\\ \displaystyle |10\rangle\to|\Psi^+\rangle=\frac{1}{\sqrt{2}}(|01\rangle+{\rm i}|10\rangle), \end{array} \end{equation} \tag{ 7 }$$

in the far-detuned regime $|\mathscr{F}_{in}| \ll \delta _n$. In this limit, the weak qubit–phonon coupling (4) is responsible for second-order processes where phonons are virtually created and annihilated. This term alone leads to a flip-flop qubit–qubit interaction H_int∝J^eff_ijσ⁺_iσ⁻_j + H.c. via virtual phonon exchange, where the coupling strength is given by $J^{\rm {eff}}_{ij} = - \sum _n \mathscr{F}_{in} \mathscr{F}_{jn}^*/\delta _n$ with the above introduced forces and detunings $\mathscr{F}_{in}$ and δ_n, respectively. At certain instants of time this interaction can act as an entangling gate. However, we also showed there that the performance of such a gate would be severely limited by thermal and dephasing noise. The main idea put forward in [8] was to exploit a strong resonant microwave driving (2), fulfilling ω_d = ω₀ and $\delta _n\ll \Omega _{\mathrm { d}}\in \mathbb {R}$, as a continuous version [30, 31] of refocusing spin-echo sequences [32, 33], to protect the two-qubit gate from these sources of noise. In this case, the virtual phonon exchange leads to an Ising-type interaction $H_{\mathrm { int}}\propto \skew3\tilde{J}^{\rm {eff}}_{ij} \sigma _i^x\sigma _j^x+{\mathrm { H.c.}}$, where we have introduced σ^x_i = σ⁺_i + σ⁻_i and $\skew3\tilde{J}^{\rm {eff}}_{ij} = J^{\rm {eff}}_{ij}/4$. This qubit–qubit interaction can generate all of the four Bell states:

$$\begin{equation} \begin{array}{llll} \displaystyle |00\rangle\to|\Phi^-\rangle=\frac{1}{\sqrt{2}}(|00\rangle-{\rm i}\,{\rm sgn}(\skew3\tilde{J}^{\rm{eff}}_{12})|11\rangle),\\ \displaystyle |01\rangle\to|\Psi^-\rangle=\frac{1}{\sqrt{2}}(|01\rangle-{\rm i}\,{\rm sgn}(\skew3\tilde{J}^{\rm{eff}}_{12})|10\rangle),\\ \displaystyle |10\rangle\to|\Psi^+\rangle=\frac{1}{\sqrt{2}}(|01\rangle+{\rm i}\,{\rm sgn}(\skew3\tilde{J}^{\rm{eff}}_{12})|10\rangle),\\ \displaystyle |11\rangle\to|\Phi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+{\rm i}\,{\rm sgn}(\skew3\tilde{J}^{\rm{eff}}_{12})|11\rangle). \end{array} \end{equation} \tag{ 8 }$$

Unfortunately, the explored far-detuned regime leads to gates that are more than one order of magnitude slower than state-of-the-art implementations based on other schemes [34]. Therefore, although both the simplicity of the gate and its resilience to different noise sources are interesting advantages, its lower speed presents a considerable drawback.

In this work, we show how one can abandon the far-detuned regime, while still preserving the nice properties of the driven single-sideband gate. Below, we show that by working in the context of the geometric phase gates [17, 19, 35], our driven nearly resonant single-sideband gate: (i) can attain speeds that are one order of magnitude faster than the far-detuned gate [8] for comparable parameters. For the specific parameters considered in this paper (see table 1) we can attain a gate speed of t_g ∼ 63 μs. Gate speeds in the range 10–100 μs can be expected for other parameters, or ion species; (ii) can minimize thermal and dephasing errors down to ε_th, ε_d < 10⁻⁴, which improves by more than one order of magnitude the far-detuned gate [8]; (iii) can withstand fluctuations of the laser phase occurring on time scales longer than the 63 μs, such that ε_ph < 10⁻⁴, which was outlined in [8], but not analyzed carefully; (iv) can also resist relative fluctuation in the intensity of the microwave driving at the 10⁻⁴-level directly with errors ε_I ∼ 10⁻³. Moreover, we show that by adding a secondary driving, the scheme can support stronger intensity fluctuations, while providing smaller gate errors ε_I < 10⁻⁴.

Table 1. Values of trapped-ion setup for the numerical simulation.

ωz/2π (MHz)	ωx/2π (MHz)	δcom/2π (kHz)	ηcom	δzz/2π (kHz)	ηzz	ΩL/2π (kHz)	Ωd/2π (MHz)
1	4	127	0.225	254	0.229	811	7.2

2.2.2. Qualitative analysis: dressed-state interaction picture and rotating-wave approximation

To understand the mechanism underlying the driven nearly resonant single-sideband gate, let us make a small detour, and consider the so-called geometric phase gates by state-dependent forces [17, 19, 35]. By combining the red-sideband (σ⁺_ia_n) term (4) with a blue sideband (σ⁺_ia^†_n) that has an opposite detuning but equal strength and adjusting the laser phases appropriately [17], the qubit–phonon Hamiltonian becomes

$$\begin{equation} \skew3\hat{H}_{\rm qp}=\sum_{i,n}\mathscr{F}_{in}\sigma_i^xa_n\,{\rm e}^{-{\rm i} \delta_nt}+{\rm H.c.}, \end{equation} \tag{ 9 }$$

where the 'hat' refers to the interaction picture with respect H₀ = H_q + H_p, and we have switched off the microwave driving Ω_d = 0. This Hamiltonian (9) can be understood as a pushing force in a direction that depends on the qubit state in the x-basis σ^x_i| ± _i〉_x = ± | ± _i〉_x. Since a single Pauli matrix appears in the above equation, the time evolution under such a state-dependent pushing force reduces to that of a forced quantum harmonic oscillator [36], which can be solved exactly (see e.g. [37]). This leads to the following time-evolution operator:

$$\begin{equation} \skew3\hat{U}(t)={\rm e}^{\sum_{in}\left(\frac{\mathscr{F}_{in}}{\delta_n}({\rm e}^{-{\rm i}\delta_nt}-1)\sigma_i^xa_n-{\rm H.c.}\right)}{\rm e}^{+{\rm i}\sum_{ijn}\frac{\mathscr{F}_{in}\mathscr{F}^*_{jn}}{\delta_n}\left(t-\frac{\sin(\delta_nt)}{\delta_n}\right)\sigma_i^x\sigma_j^x}. \end{equation} \tag{ 10 }$$

The first unitary corresponds to a displacement operator, which is well known in quantum optics [14], with the peculiarity of being state dependent. In phase space, this term induces periodic circular trajectories for the vibrational modes that depend on the collective spin state, and thus leads to qubit–phonon entanglement. When t_g = k_n2π/δ_n, $k_n \in \mathbb {Z}$, the trajectories close and the qubits and the phonons become disentangled, such that the evolution operator becomes

$$\begin{equation} \skew3\hat{U}(t_{\rm g})={\rm e}^{-{\rm i} t_{\rm g}\sum_{ij}J_{ij}^{\rm MS}\sigma_i^x\sigma_j^x},\quad J_{ij}^{\rm MS}=\textstyle{-\sum\limits_n\frac{\mathscr{F}_{in}\mathscr{F}^*_{jn}}{\delta_n}}. \end{equation} \tag{ 11 }$$

This unitary can be easily seen to provide the desired entangled states (8) when t_g(2J^eff₁₂) = π/4. Additionally, if the initial spin states are eigenstates of σ^x₁σ^x₂, the gate gives the table

$$\begin{equation} \begin{array}{llll} \displaystyle |++\rangle_x\to |++\rangle_x,\\ \displaystyle |+-\rangle_x\to{\rm e}^{{\rm i}\frac{\pi}{2}}|+-\rangle_x,\\ \displaystyle |-+\rangle_x\to{\rm e}^{{\rm i}\frac{\pi}{2}}|-+\rangle_x,\\ \displaystyle |--\rangle_x\to |--\rangle_x \end{array} \end{equation} \tag{ 12 }$$

up to an irrelevant global phase. This corresponds to a two-qubit π/2-phase gate which, together with single-qubit rotations, gives a universal set of gates for quantum computation. The fact that the phase–space trajectory is closed at t_g, allows for the interpretation of the π/2-phases as geometric Berry phases determined by the area enclosed by the trajectory [35]. Let us remark two properties of these gates: (i) they do not rely on any far-detuned condition $|\mathscr{F}_{in}|\ll \delta _n$, and can be thus much faster and (ii) the geometric origin of the phase gate underlies its robustness with respect to thermal motion of the ions (i.e. spin-phonon disentanglement occurs when the trajectory closes regardless of the vibrational state).

After this small detour, it becomes easier to understand why a gate based only on a nearly resonant single-sideband cannot lead to a geometric phase gate, and is thus very sensitive to thermal noise. By rewriting the single sideband (4) in the interaction picture with respect to H₀ for Ω_d = 0, we obtain

$$\begin{equation} \skew3\hat{H}_{\rm qp}=\sum \limits_{i,n}\frac{1}{2}\mathscr{F}_{in} (\sigma_i^x+{\rm i}\sigma_i^y) a_n \,{\rm e}^{-{\rm i} \delta_n t} + {\rm H.c.}, \end{equation} \tag{ 13 }$$

where σ^y_i = −iσ⁺_i + iσ⁻_i. Hence, it is clear that the red sideband yields a combination of two state-dependent forces acting on orthogonal bases. This fact forbids an exact solution, such as the one obtained for a single state-dependent force (10), and we inevitably lose the notion of state-dependent trajectories that close independently of the vibrational state. Accordingly, the qubit and phonons get more entangled as the far-detuned condition $|\mathscr{F}_{in}|\ll \delta _n$ is abandoned, and the gate fidelity drops severely for ions in thermal motion.

If we now switch on a resonant microwave driving ω_d = ω₀, $\Omega _{\mathrm { d}}\in \mathbb {R}$, then the Hamiltonian (13) changes into

$$\begin{equation} \skew3\tilde{H}_{\rm qp}=\sum \limits_{i,n}\! \frac{\mathscr{F}_{in}}{2} (\!\sigma_i^x+{\rm i}\sigma_i^y\cos(\Omega_{\rm d}t)-{\rm i}\sigma_i^z\sin(\Omega_{\rm d}t)\!) a_n \,{\rm e}^{-{\rm i} \delta_n t} + {\rm H.c.} \end{equation} \tag{ 14 }$$

Here, the 'tilde' refers to the following 'dressed-state' interaction picture, namely $\skew3\tilde{H}_{\mathrm { qp}}=\tilde {U}_{\mathrm { d}}(t){H}_{\mathrm { qp}}\tilde {U}^{\dagger }_{\mathrm { d}}(t)$, where

$$\begin{equation} \tilde{U}_{\rm d}(t)={\rm e}^{{\rm i} t\sum_i\textstyle\frac{1}{2}\Omega_{\rm d}\sigma_i^x}\,{\rm e}^{{\rm i} t \sum_i\textstyle\frac{1}{2}\omega_0\sigma_i^z}\,{\rm e}^{{\rm i} t\sum_n\omega_na_n^{\dagger}a_n}. \end{equation} \tag{ 15 }$$

It is now possible to argue that the state-dependent σ^y and σ^z forces rotate very fast for sufficiently strong drivings Ω_d ≫ δ_n, and can be thus neglected in a rotating-wave approximation provided that $|\mathscr{F}_{in}|\ll \Omega _{\mathrm { d}}$. Under this constraint, the driven single-sideband Hamiltonian becomes

$$\begin{equation} \skew3\tilde{H}_{\rm qp}\approx\sum \limits_{i,n}\frac{1}{2} \mathscr{F}_{in} \sigma_i^x a_n\, {\rm e}^{-{\rm i} \delta_n t} + {\rm H.c.} \end{equation} \tag{ 16 }$$

and thus a formal solution such as equation (10) becomes possible again, provided that we make the substitution $\mathscr{F}_{in}\to \textstyle \frac {1}{2} \mathscr{F}_{in}$.

According to this qualitative argument, a strongly driven single-sideband Hamiltonian can also produce the desired geometric phase gate. Therefore, it seems possible to abandon the far-detuned regime $|\mathscr{F}_{in}|\ll \delta _n$ of our previous work [8], and obtain faster gates that are still robust with respect to thermal noise. In the following sections, we will present a quantitative detailed analysis to test the validity of this idea.

2.2.3. Quantitative analysis: Magnus expansion and numerical analysis

As discussed above, the presence of different state-dependent forces in the Hamiltonian (13) avoids an exact solution for the unitary time-evolution operator. However, since we are interested in the strong-driving limit $|\mathscr{F}_{in}|\ll \Omega _{\mathrm { d}}$, we can use the so-called Magnus expansion (ME) [38], truncating it to the desired order to obtain the leading contributions to the dynamics (see appendix A for details). The time-evolution operator in the Schrödinger picture reads

$$\begin{equation} U_{\rm app}(t)=\tilde{U}^{\dagger}_{\rm d}(t)\,{\rm e}^{\Omega(t)},\quad \Omega(t)\approx{\Omega_1(t)+\Omega_2(t)+\mathscr{O}(\xi,\chi)} \end{equation} \tag{ 17 }$$

and ξ = (Ω_Lη_n)²/Ω²_d, χ = (Ω_Lη_n)²/Ω_dδ_n are the small parameters in the ME. In this expression, we have defined the anti-unitary operators containing the different state-dependent displacements

$$\begin{eqnarray} &&\fl \Omega_1(t)= \sum \limits_{i,n} \frac{\mathscr{F}_{in}}{2\delta_n} ( {\rm e}^{-{\rm i} \delta_n t} -1) \sigma_i^x a_n+ \sum \limits_{i,n} \frac{\mathscr{F}_{in}}{4(\Omega_{\rm d}-\delta_n)} ( {\rm e}^{{\rm i} (\Omega_{\rm d} - \delta_n) t} -1) (-{\rm i}\sigma_i^y+\sigma_i^z) a_n -{\rm H.c.} \nonumber\\ &&+ \sum \limits_{i,n} \frac{\mathscr{F}_{in}}{4(\Omega_{\rm d}+\delta_n)} ( {\rm e}^{-{\rm i} (\Omega_{\rm d} + \delta_n) t} -1) ({\rm i}\sigma_i^y+\sigma_i^z) a_n -{\rm H.c.} \end{eqnarray} \tag{ 18 }$$

and the operators leading to the qubit–qubit interactions

$$\begin{eqnarray} &&\fl \Omega_2(t)= {\rm i}\sum_{i,j,n}{\frac{\mathscr{F}_{in}\mathscr{F}_{jn}^*}{4\delta_n}}\sigma_i^x \sigma_j^x\left(t-\frac{\sin\delta_nt}{\delta_n}\right)\nonumber\\ &&+{\rm i} t \sum_{i,n}\!\Delta\Omega_{in}\!\left(\!\!a_n^{\dagger}a_n-\frac{1}{2}\!\right)\sigma_i^x+\sum_{i,n\neq m}(f_{nm}(t)\sigma_i^xa_m^{\dagger}a_n-{\rm H.c.}), \end{eqnarray} \tag{ 19 }$$

where we have introduced the coupling strengths

$$\begin{equation} \Delta\Omega_{in} = -\frac{|\mathscr{F}_{in}|^2}{4} \left( \frac{1}{\Omega_{\rm d} - \delta_n} + \frac{1}{\Omega_{\rm d} + \delta_n} \right) \end{equation} \tag{ 20 }$$

and the following time-dependent functions:

$$\begin{equation} f_{nm}(t) = \frac{\mathscr{F}_{jn}\mathscr{F}_{jm}^*}{8(\delta_n - \delta_m)}\!\! \left( \frac{1}{\Omega_{\rm d} - \delta_m} + \frac{1}{\Omega_{\rm d} + \delta_m}\right)\!\!\big({\rm e}^{-{\rm i}(\delta_n-\delta_m)t} - 1 \big). \end{equation} \tag{ 21 }$$

At this point, it is worth comparing the final expressions (18) and (19) to those corresponding to a state-dependent force in equation (10). The first-order contribution, (18), resembles the state-dependent displacement in (10), but we get in addition the contribution of more state-dependent forces in different bases. From the first line of the second-order contribution (19), we observe also a similarity with the qubit–qubit couplings in (10), but we get additional residual qubit–phonon couplings in the second line of (19). We discuss below how all these additional terms can be minimized, such that we are left with an effective time evolution that is analogous to that of a state-dependent force.

Let us note that equations (18) and (19) correspond to the leading terms in the second-order expansion of U(t), while the complete expression may be found in appendix A. Before analyzing how the geometric phase gates arise from equations (18) and (19), let us check numerically the validity of our derivation by comparing it to the time evolution under the full-driven single-sideband Hamiltonian H_dss (6). To perform the numerics more efficiently, we expressed H_dss in a picture where it becomes time independent, namely

$$\begin{equation} H_{\rm dss}'=\sum_n \delta_{n} a_{n}^{\dagger}a_n+\sum_i\frac{\Omega_{\rm d}}{2}\sigma_i^x+\sum_{i,n}(\mathscr{F}_{in}\sigma_i^+a_n+{\rm H.c.}), \end{equation} \tag{ 22 }$$

where H_dss' = U'(t)H_dss(U'(t))^†, and the unitary is

$$\begin{equation} {U'}(t)={\rm e}^{{\rm i} t\sum_n(\omega_0-\omega_{\rm L})a_n^{\dagger}a_{n}}\,{\rm e}^{{\rm i} t \sum_i\textstyle\frac{1}{2}\omega_0\sigma_i^z}. \end{equation} \tag{ 23 }$$

Accordingly, the time-evolution operator can be written as

$$\begin{equation} U_{\rm exact}(t)=({U'}(t))^{\dagger}\,{\rm e}^{-{\rm i} H_{\rm dss}' t}. \end{equation} \tag{ 24 }$$

For the numerical simulations, we chose realistic parameters for ion-trap experiments, which are summarized in table 1. The outcome of the simulations is shown in figure 2(a), which shows a very good agreement between the ME and the exact time evolution for the qubit dynamics. This supports the validity of our derivations, and allows us to carry on with the description of the driven geometric phase gate.

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The action of the first contribution Ω₁(t) (18) to the time-evolution operator (17) can be understood as follows: for each vibrational mode, three non-commuting state-dependent forces aim at displacing the ions along circular paths in phase space, such that the specific direction of each trajectory depends on the particular eigenstates | ± _x〉,| ± _y〉 and |0/1〉 of the operators σ^x,σ^y and σ^z. For weak drivings, the combination of these forces will deform the circular paths, and lead to phase-space trajectories that are not closed any longer (see figure 3(a)). It is clear from the analytical expression (18) that by applying a sufficiently strong driving $\Omega _{\mathrm { d}}\gg \delta _n,\:|\mathscr{F}_{jn}|$, the forces in the σ^y/z-basis get suppressed, and, to a good approximation, we are left with the desired single state-dependent force in the σ^x-basis with a halved strength $\mathscr{F}_{in}\to \textstyle \frac {1}{2} \mathscr{F}_{in}$ with respect to the standard Mølmer–Sørensen term (9). Accordingly, we should recover the circular phase-space trajectories when the initial state of the qubits is an eigenstate of the σ^x₁σ^x₂ operator. In figure 3(b), we analyze the strong-driving dynamics numerically, and confirm the above prediction. Therefore, the qualitative description of the previous section is put on a firmer ground by the use of the ME.

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The ME also allows us to improve further the geometric character of the gate. By setting the correct values to the detunings and intensities, it is possible to cancel exactly the qubit–phonon entanglement introduced by all three state-dependent displacements in (18), and the residual qubit–phonon entanglement introduced in the second term of the second line of equation (19). In order to achieve this, we need to adjust the Hamiltonian parameters such that, at the gate time t_g, the following constraints are fulfilled:

$$\begin{equation} \begin{array}{ll} \displaystyle t_{\rm g}= r \frac{2 \pi}{\delta_1}, \quad \delta_2=k\delta_1, \quad \Omega_{\rm d}=p\delta_1, \quad \: r,k,p\in \mathbb{Z}, \\ \displaystyle \frac{r}{k} \in \mathbb{Z} \quad {\rm and} \quad |p|> |r|+|k|. \end{array} \end{equation} \tag{ 25 }$$

In other words, we need the detuning of the zz mode δ₂ and the microwave Rabi frequency Ω_d to be an integer multiple of the com mode detuning δ₁. Physically, the constraints can be understood in the following way. By fulfilling equation (25) the com mode performs r closed loops in phase space during one gate cycle while the zz mode performs r/k loops in phase space. Thus, in order to attain the geometric character of the gate r/k must be an integer for then the phase-space trajectories of both normal modes close simultaneously at t_g. The constraints on p, however, lack a clear physical interpretation. They are motivated by the results of the ME.

For the above conditions, Ω₁(t_g) = 0 vanishes exactly. Moreover, we reduce considerably the qubit–phonon entanglement due to the second-order term Ω₂(t_g) = −it_gH_dss, where we have introduced the Hamiltonian

$$\begin{equation} H_{\rm dss}=\sum_{i,j}J_{ij}^{\rm dss}\sigma_i^x \sigma_j^x-\sum_{i,n}\Delta\Omega_{in}\left(a_n^{\dagger}a_n-\frac{1}{2}\right)\sigma_i^x \end{equation} \tag{ 26 }$$

and the following coupling strengths:

$$\begin{equation} J_{ij}^{\rm dss}=-\sum_n \frac{1}{4\delta_n}\mathscr{F}_{in} \mathscr{F}_{jn}^{*}. \end{equation} \tag{ 27 }$$

Note that a closer inspection of all the additional terms in the ME (see appendix A) shows that they all cancel exactly under these constraints. Therefore, all the dynamics for the gate is exactly described by Hamiltonian (26).

Finally, in the limit of very strong drivings Ω_d ≫ δ_n, the residual qubit–phonon coupling is minimized as ΔΩ_in (20) decreases with increasing driving. Moreover, one can even cancel exactly the contribution of the second term in equation (26) by introducing a spin-echo pulse [8] or, equivalently, a phase reversal [9] of the microwave driving at half the expected gate time. As an additional benefit, this procedure guarantees the closure of phase-space trajectories in the case of an imperfect detuning if the forces in the σ^y/z bases can be neglected in a rotating-wave approximation. This has been demonstrated in the recent experiment [9]. Therefore, the full time evolution can be written approximately as

$$\begin{equation} U_{\rm app}(t_{\rm g})=\tilde{U}^{\dagger}_{\rm d}(t_{\rm g}){\rm e}^{-{\rm i} H_{\rm dss}t_{\rm g}},\quad H_{\rm dss}=\sum_{i,j}J^{\rm{dss}}_{ij}\sigma_i^x\sigma_j^x. \end{equation} \tag{ 28 }$$

From this expression, we can extract the gate time to be (2J^dss₁₂)t_g = π/4, which will give rise to the entangling table (8), or to the phase-gate table (12), depending on the chosen basis for the initial states. In figure 2(b), we check the validity of this description by computing the fidelity of generating the Bell state |Ψ⁺〉 at the expected gate time, starting from the initial qubit state |10〉. As seen from the figure, the agreement between the full ME and the exact dynamics is very good, and we obtain the desired entangled state at t_g with very high fidelities. It can also be observed in the figure that the fidelity exhibits oscillations. These can be understood from equation (10) which we realize to a good approximation with our choice of the driving strength. The state |10〉 couples to the state-dependent forces on both modes and each of the modes is displaced in phase space. In this process entanglement between the internal and motional states is created and the fidelity of producing the Bell state drops. Whenever a trajectory is closed the entanglement between qubit and motional states vanishes and the fidelity exhibits a local maximum. The oscillations observed in the fidelity can then be understood as the beatnote of the two state-dependent forces, each acting on a different mode.

Note that we have not made any assumption on the ratio between the sideband strengths and the laser detunings $|\mathscr{F}_{in}|/\delta _n$, and thus we do not require to work in the far-detuned regime $|\mathscr{F}_{in}|/\delta _n\ll 1$ of our previous proposal [8]. According to the expression of the gate coupling strengths J^dss_ij, this means that we can increase the gate speed considerably by working closer to the resonance (i.e. t_g ≈ 63 μs). The conditions (25) also imply that the phase-space trajectories of the vibrational modes are closed, providing a geometric character to the two-qubit gate, and its resilience to the thermal noise.

In the following sections, we will analyze the speed and noise robustness of this driven nearly resonant single-sideband gate. However, let us first address how the constraints (25) can be met in practice. The detunings for the com and the zz modes are given by

$$\begin{equation} \delta_{1} = \omega_{\rm L} - (\omega_0 -\omega_{1} ),\quad \delta_{2} = \omega_{\rm L} - (\omega_0 -\omega_{2} ). \end{equation} \tag{ 29 }$$

By setting δ₁ = cω₁, such that c is a constant, and demanding that δ₂ = kδ₁ with $k\in \mathbb {Z}$, we obtain

$$\begin{equation} c = \frac{\xi -1 }{k -1}, \end{equation} \tag{ 30 }$$

where ξ = ω₂/ω₁ < 1 is the ratio of the normal-mode frequencies. To make the gate as fast as possible t_g = 2π/(|c|ω₁), the modulus of c should be as big as possible, and we thus choose k = 2, so that |c| = 1 − ξ. According to these considerations, the gate time in units of the trap frequency is t_gω_x = 2π/(1 − ξ), which is minimized by minimizing ξ (i.e. large difference of the com and the zz modes). This occurs when approaching the linear-to-zigzag instability.

Finally, the laser intensities in J^dss₁₂ have to be adjusted such that we can fulfill the following condition:

$$\begin{equation} t_{\rm g}=\frac{\pi}{8 J^{\rm{dss}}_{12}} = r \frac{2 \pi}{\delta_1}, \end{equation} \tag{ 31 }$$

where $r\in \mathbb {Z}$ is an integer that determines how many loops in phase space the com mode performs. On the one hand, r should be as small as possible to maximize the gate speed. On the other hand, it turns out that the smaller r is, the more fragile the gate becomes with respect to dephasing noise (see sections below). Therefore, we optimize the value of r to find a compromise between gate speed and gate fidelity.

Inserting expression (27) in the above equation, we obtain

$$\begin{equation} |\Omega_{\rm L}|=\frac{|\delta_1|}{\eta_1 \sqrt{\frac{r}{2} \left(1-\frac{1}{2 \xi} \right)}} \end{equation} \tag{ 32 }$$

which fixes the laser intensities. Finally, we note that the microwave driving strength can be modified in the experiment

$$\begin{equation} \Omega_{\rm d}=p\delta_1 \end{equation} \tag{ 33 }$$

with $p\in \mathbb {Z}$ being a very large integer to meet the strong-driving condition Ω_d ≫ δ_n. With these expressions, the parameters of the setup are fixed in terms of two integers (r,p). In the following section, we optimize the choice of these integers to maximize simultaneously the gate fidelity and speed. We explore how such fidelities approach the FT in the presence of different sources of noise as the microwave driving strength is increased.

One of the primary sources of gate infidelities in early trapped-ion proposals [15] was the noise introduced by residual thermal motion of the ions after a stage of resolved sideband cooling. In order to consider such thermal noise, we consider an initial state $\rho _0=| {10}\rangle \langle {10}| \otimes \rho _{\mathrm { th}}(\bar {n}_1,\bar {n}_2)$ for a thermal phonon state $\rho _{\mathrm { th}}(\bar {n}_1,\bar {n}_2)=\rho _{\mathrm { th}}(\bar {n}_1)\otimes \rho _{\mathrm { th}}(\bar {n}_2)$ characterized by the mean phonon numbers $\{\bar {n}_1,\bar {n}_2\}$ for each mode. Note that due to the small difference in frequency of the normal modes in x-direction we have $\bar {n}_1 \approx \bar {n}_2$. According to the table (8), the target state after the gate is $\rho (t_{\mathrm { g}})=| {\Psi ^+}\rangle \langle {\Psi ^+}| \otimes \rho _{\mathrm { th}}(\bar {n}_1,\bar {n}_2)$, where $|\Psi ^+\rangle =(|01\rangle +{\mathrm { i}}|10\rangle )/\sqrt {2}$.

Based on the ME evolution operator (17), it is possible to derive a closed expression for the gate fidelity

$$\begin{equation} \mathscr{F}_{|\Psi^+\rangle}=\mathop{\rm Tr}\{ |\Psi^+\rangle \langle \Psi^+| \otimes \mathbbm{1}_{{\rm ph}} U_{\rm M}(t_{\rm g})\rho_0 U_{\rm M}^{\dagger}(t_{\rm g})\} \end{equation} \tag{ 34 }$$

in the strong-driving limit Ω_d ≫ δ_n (i.e. neglecting all terms that are suppressed with 1/Ω_d). After some algebra, the fidelity can be expressed as a sum of expectation values of the displacement operator

$$\begin{equation} \langle D(\alpha_n)\rangle={\rm Tr}_{\rm ph}\{{\rm e}^{\alpha_n a_n^{\dagger}-\alpha_n^* a_n}\rho_{\rm th}(\bar{n}_n)\}={\rm e}^{-|\alpha_n|^2\left(\bar{n}_n+\textstyle\frac{1}{2}\right)}. \end{equation} \tag{ 35 }$$

Provided that the gate time fulfills t_gΩ_d = 4πp where $p\in \mathbb {Z}$, the fidelity of the entangling operation reads as follows:

$$\begin{equation} \mathscr{F}_{|\Psi^+\rangle}=\frac{1}{4}+\frac{1}{2}\,{\rm e}^{-\sum_n\kappa_n(1-\cos(\delta_n t_{\rm g}))}+\frac{1}{8}\sum\limits_n\,{\rm e}^{-4 \kappa_n(1-\cos(\delta_n t_{\rm g}))}, \end{equation} \tag{ 36 }$$

where $\kappa _n=\frac {|\mathscr{F}_{1n}|^2}{\delta _n^2}(2\bar {n}_n+1)$. From this expression, it becomes clear that the fidelity would decrease exponentially with the mean number of thermal phonons, unless the phase-space trajectories are closed δ_nt_g = 2π. Under such condition, the gate fidelity is maximized $\mathscr{F}_{|\Psi ^+\rangle }(t_{\mathrm { g}}=2\pi /\delta _n)=1$ in the limit of Ω_d → ∞. This expression not only unveils the importance of the geometric character of the gate, but will also be useful in understanding the effect of other sources of noise.

In order to check how the fidelity approaches unity as the microwave driving Ω_d is increased, we study numerically the exact time evolution (24). To avoid timing errors due to the fast oscillations induced by the strong microwave driving, we add the above-mentioned refocusing spin-echo pulse at half the expected gate time, such that the total time evolution is $U_{\mathrm { full}} (t_{\mathrm { g}},0)= U_{\mathrm { exact}}(t_{\mathrm { g}},\frac {t_{\mathrm { g}}}{2})(\sigma ^z_1 \sigma _2^z) U_{\mathrm { exact}}(\frac {t_{\mathrm { g}}}{2},0)$. We then compute numerically the fidelity of producing the desired Bell state

$$\begin{equation} \mathscr{F}_{|\Psi^+\rangle} = \mathop{\rm Tr}\{ |\Psi^+\rangle \langle \Psi^+| \otimes \mathbbm{1}_{{\rm ph}} U_{\rm full}(t_{\rm g},0) \rho_0 U_{\rm full}^{\dagger} (t_{\rm g},0)\}. \end{equation} \tag{ 37 }$$

In order to integrate the full Hamiltonian numerically, we truncate each vibrational Hilbert space to a maximum number of phonons of n_max = 25 per mode, which is sufficiently high so that no appreciable error is introduced.

In figure 4, we represent the numerical results for the gate error $\varepsilon =1-\mathscr{F}$ for various microwave driving strengths. By setting r = 8 in equation (31), we obtain a gate time of t_g ≈ 63 μs, which improves the gate speed of our previous work [8] by one order of magnitude. Besides, the results shown in figure 4, show that the gate errors surpass the FT ε_FT2 ∼ 10⁻⁴ for sufficiently strong drivings. This figure illustrates clearly the benefit of this nearly resonant single-sideband gate in comparison to our previous scheme [8], since the gate error even for the largest thermal phonon numbers $\bar {n}_1=1$ is reduced by several orders of magnitude.

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Another source of gate infidelities for several trapped-ion qubits is the noise caused by fluctuating magnetic fields, or laser intensities, which lead to a shift in the resonance frequency of the qubit through either the Zeeman shift, or an uncompensated ac-Stark shift (see appendix B). On the time scales considered, these fluctuations can be modeled as

$$\begin{equation} H_{\rm fluc} (t)=\sum_i \frac{1}{2} \Delta \omega_0(t) \sigma_i^z, \end{equation} \tag{ 38 }$$

where Δω₀(t) is a specific Markov process known as an Ornstein–Uhlenbeck (O–U) process, and we have assumed that the fluctuating source acts globally on the two ions (e.g. global fluctuating magnetic field).

The O–U process, which is defined by a relaxation time τ and a diffusion constant c [39], is characterized by the following identities:

$$\begin{equation} \overline{\Delta\omega_0(t)}=0,\quad \overline{\Delta\omega_0(t)\Delta\omega_0(t-s)}={\frac{c\tau}{2}}{\rm e}^{-\frac{s}{\tau}}, \end{equation} \tag{ 39 }$$

where the 'bar' refers to the stochastic average, and we assume a Markovian limit where the time scales of interest greatly exceed the correlation time. The time evolution of the density matrix in the so-called Born–Markov approximation

$$\begin{equation} \dot{\rho}=-\int_0^{\infty}{\rm d}s\overline{[H_{\rm fluc}(t),[H_{\rm fluc}(t-s),\rho(t)]]} \end{equation} \tag{ 40 }$$

can be expressed in the following Lindblad form:

$$\begin{equation} \dot{\rho}=\mathscr{L}_{\rm d}({\rho})=\Gamma_{\rm d}\left(S_z\rho S_z-\frac{1}{2} S_z^2\rho-\frac{1}{2}\rho S_z^2\right),\quad \Gamma_{\rm d}=\frac{c\tau^2}{4}, \end{equation} \tag{ 41 }$$

where we have introduced the collective operator $S_{\alpha }=\sum _i\sigma _i^{\alpha }$ for α = z. By means of the adjoint master equation [40], one finds that $\sigma _i^x(t)={\mathrm { e}}^{\mathscr{L}^{\dagger }_{\mathrm { d}}t}\sigma _i^x(0)=\sigma _i^x(0){\mathrm { e}}^{-t c\tau ^2/2}$, which allows us to define the decoherence time as T₂ = 2/cτ². This expression, together with the condition of short noise correlation times, fulfilled by setting τ = 0.1T₂, allows us to set the noise-model parameters for the study of a variety of decoherence times T₂∈[15,40] μs. We note that these values are overly pessimist, and would only occur for a poor shielding of the magnetic fields, or a bad stabilization of the lasers.

As we show below, even for these poor decoherence times, the strong microwave driving suppresses the noise, and reestablishes a good coherent behavior. The qualitative argument is again to move to the dressed-state interaction picture (15), where $\skew3\tilde{H}_{\mathrm { fluc}} (t)=\tilde {U}_{\mathrm { d}}(t){H}_{\mathrm { fluc}} (t)\tilde {U}_{\mathrm { d}}^{\dagger }(t)$ becomes

$$\begin{equation} \skew3\tilde{H}_{\rm fluc} (t)= \sum \limits_{i} \frac{1}{2}\Delta \omega_0(t) \left( \cos(\Omega_{\rm d} t) \sigma_i^z + \sin(\Omega_{\rm d} t) \sigma_i^y \right). \end{equation} \tag{ 42 }$$

For a sufficiently strong driving, the noisy terms rotate very fast, and we can neglect them in a rotating-wave approximation.

In order to put this argument on a firmer footing, let us use again the Born–Markov approximation (40). We obtain a Lindlad-type master equation $\dot {\tilde {\rho }}=\tilde {\mathscr{L}}_{\mathrm { d}}(\tilde {\rho })$, where the new dephasing super-operator is

$$\begin{equation} \tilde{\mathscr{L}}_{\rm d}(\tilde{\rho})=-{\rm i}\left[\frac{\Delta\Omega_{\rm d}}{2}S_x,\tilde{\rho}\right]+\frac{\tilde{\Gamma}_{\rm d}}{2}\!\!\sum_{\alpha=z,y}\left(S_{\alpha}\tilde{\rho} S_{\alpha}-\frac{1}{2} S_{\alpha}^2\tilde{\rho}-\frac{1}{2}\tilde{\rho} S_{\alpha}^2\right) \end{equation} \tag{ 43 }$$

and we have assumed that Ω_dT₂ ≫ 1. Here, we have defined a dressed-state energy shift and a renormalized dephasing rate:

$$\begin{equation} \Delta\Omega_{\rm d}=\frac{\Omega_{\rm d}\tau}{4T_2(1+(\Omega_{\rm d}\tau)^2)},\quad \tilde{\Gamma}_{\rm d}=\frac{\Gamma_{\rm d}}{(1+(\Omega_{\rm d}\tau)^2)}. \end{equation} \tag{ 44 }$$

The adjoint master equation associated with such Liouvillian (43) yields the coherences decay $\sigma _i^x(t)=\sigma _i^x(0){\mathrm { e}}^{-2\tilde {\Gamma }_{\mathrm { d}}t}$. Accordingly, we obtain a renormalized decoherence time

$$\begin{equation} \tilde{T}_2=T_2(1+(\Omega_{\rm d}\tau)^2) \end{equation} \tag{ 45 }$$

which increases quadratically with the driving strength [41].

We now address qualitatively the noise effects on the fidelity of the entangling operation in the strong-driving limit. Note that both |10〉 and |Ψ⁺〉 lie in the 'zero-magnetization' subspace, which can be easily checked to be a decoherence-free subspace of the original dephasing (38). Therefore, the effects of the dephasing noise must be tested for another Bell state, such as $|\Phi ^-\rangle =(|00\rangle - {\rm i}\ |11\rangle )/\sqrt {2}$, which is generated from |00〉 (see equation (8)). We identify two possible effects. (i) On the one hand, the qubit coherences are degraded on a timescale given by $\tilde {T}_2=T_2(1+(\Omega _{\mathrm { d}}\tau )^2)$. (ii) On the other hand, the state-dependent forces for the leading term in (17) are damped according to $\mathscr{F}_{in}\to \mathscr{F}_{in}{\mathrm { e}}^{-t/\tilde {T}_2}$ (i.e. recall $\sigma _i^x(t)=\sigma _i^x (0){\mathrm { e}}^{-2\tilde {\Gamma }_{\mathrm { d}}t}$). This avoids the perfect closure of the phase-space trajectories, and thus decreases the gate fidelity.

To validate these predictions, we integrate the full Hamiltonian (22) incorporating the noise term (38) and once again a refocusing spin-echo pulse at half the gate time. In appendix B, we describe in detail the conditions under which the stochastic noise process can be numerically propagated in time, and incorporated into the full time evolution. In figure 5, we show the numerical results for the gate error as a function of the microwave driving strength, where we have set $\bar {n}_1=\bar {n}_2=0$ to distinguish clearly between the effects of thermal and dephasing noise. This figure shows neatly how the dephasing error is considerably suppressed by increasing the driving strength. Errors well below the FT can be achieved again for sufficiently strong drivings. Additionally, this figure also shows the advantage of the near-resonant gates with respect to the far-detuned ones [8], since the error is reduced by several orders of magnitude.

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Another possible source of noise for trapped-ion gates are fluctuations in the laser phases at the position of the ions. Such fluctuations become especially dangerous for the geometric phase gates based on two non-copropagating Raman laser beams, which are more sensitive to laser path fluctuations. This noise can be modeled by substituting the sideband couplings $\mathscr{F}_{in}\to \mathscr{F}_{in}\,{\mathrm { e}}^{{\mathrm { i}}\Delta \varphi (t)}$, where Δφ(t) is again a stochastic variable. Therefore, the full Hamiltonian (22) becomes

$$\begin{equation} H_{\rm dss}'=\sum_n\delta_na_n^{\dagger}a_n+\sum_i\frac{\Omega_{\rm d}}{2}\sigma_i^x+\sum_{i,n}(\mathscr{F}_{in}\,{\rm e}^{{\rm i}\Delta\varphi(t)}\sigma_i^+a_n+{\rm H.c.}). \end{equation} \tag{ 46 }$$

However, in this case we model the noise as another normal stochastic process, namely a Wiener process Δφ∈[0,2π]. This stochastic process is fully characterized by a single diffusion constant c and its initial value

$$\begin{equation} \overline{\Delta\varphi(t)}=\Delta\varphi(0)=0,\quad \overline{\Delta\varphi(t)\Delta\varphi(t)}=c t. \end{equation} \tag{ 47 }$$

As phase fluctuations typically occur on a much longer time scale as compared to the typical gate times [42], they contribute with slow phase drifts. Therefore, we simulated N_n = 10³ subsequent realizations of the gate where the phase drifts were modeled as a Wiener process with

$$\begin{equation} c=\frac{(\zeta_{\rm p}\pi)^2}{10^3 t_{\mathrm{g}}},\quad \zeta_{\rm p}\in[0.01,0.1], \end{equation} \tag{ 48 }$$

where the parameter ζ_p determines how large the phase drifts are. A more detailed discussion of the process, and the motivation for the choice of the diffusion constant, can be found in appendix B.

Let us note that our analytical study (17) predicts that the gate is insensitive to such slow phase drifts. In this regime, we can set the phase to be constant Δφ(t) = Δφ₀ during the gate interval. Accordingly, we can use the analytic expression in equations (18) and (19) after substituting $\mathscr{F}_{in}\to \mathscr{F}_{in}\,{\mathrm { e}}^{{\mathrm { i}}\Delta \varphi _0}$, where Δφ₀∈[0,2π] is a time-independent random variable. From these expressions, one can see that: (i) the geometric character of the gate, and thus its robustness to thermal noise, is not altered by any constant value of Δφ₀ (i.e. the fulfillment of the constraints (25), such that Ω₁(t_g) = 0, is independent of Δφ₀); (ii) the effective qubit–qubit interactions (27), which determine the gate time, $J_{ij}^{\mathrm { dss}}=\sum _{n}(\mathscr{F}_{in}\,{\mathrm { e}}^{{\mathrm { i}}\Delta \varphi _0})(\mathscr{F}_{jn}\,{\mathrm { e}}^{{\mathrm { i}}\Delta \varphi _0})^*/4\delta _n=\sum _{n}\mathscr{F}_{in}\mathscr{F}_{jn}^*/4\delta _n$ are independent of the phase value. Therefore, we conclude that for strong-enough drivings and sufficiently slow phase drifts, the gate should be robust against phase noise.

In figure 6, we present our numerical results for the resilience of the gate to slow drifts in the phases of the lasers. From these results, we can conclude that for moderate drivings, and phases with drifts as big as Δφ ≈ 0.1π over 10³ gate realizations, the gate errors still lie well below the stringent FT ε_2q < ε_FT2. We thus confirm that the proposed gate is indeed robust against a realistic phase noise, where the phase drifts for the time scales of interest are well below Δφ ≈ 0.1π (see the discussion in appendix B).

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Let us now discuss the impact of possible fluctuations of the microwave intensity on the gate fidelities. This noise is modeled by substituting the microwave Rabi frequency Ω_d → Ω_d(t) = Ω_d + ΔΩ_d(t), where ΔΩ_d(t) is a stochastic process representing the microwave intensity fluctuations. Hence, the full Hamiltonian (22) should be substituted for

$$\begin{equation} H_{\rm dss}'=\sum_n\delta_na_n^{\dagger}a_n+\sum_i\frac{\Omega_{\rm d}(t)}{2}\sigma_i^x+\sum_{in}(\mathscr{F}_{i,n}\sigma_i^+a_n+{\rm H.c.}). \end{equation} \tag{ 49 }$$

A reasonable choice for the intensity fluctuations is to consider an O–U process, and set its mean and variance to

$$\begin{equation} \overline{\Delta\Omega_{\rm d}(t)}=0,\quad \overline{\Delta\Omega_{\rm d}(t)\Delta\Omega_{\rm d}(t)}={\frac{c\tau}{2}}=\zeta_{\rm I}^2\Omega_{\rm d}^2 \end{equation} \tag{ 50 }$$

such that ζ_I fixes the relative intensity fluctuations, which we vary in the range ζ_I ≈ 10⁻⁴. In addition, we assume that the correlation time of the intensity fluctuations is longer than the gate time, and set it to τ = 1 ms.

In figure 7, we integrate numerically the Hamiltonian with a fluctuating microwave intensity (49) and a refocusing spin-echo pulse at half the gate time. As seen in this figure, for intensity stabilization within the ζ_I ≈ 10⁻⁴ regime, and intermediate drivings Ω_d/2π∈[5,10] MHz, the gate error is still well below the higher threshold ε_FT1 , and approaches the FT ε_FT2 for optimal drivings and the smallest noise ζ_I = 0.7 × 10⁻⁴. In any case, however, the full advantage of the strong-driving limit shown in figures 4, 5 and 6, where ε_2q < ε_FT2 is attained for sufficiently strong microwave intensities, is lost due to the associated increase of the intensity fluctuations. Hence, it would be desirable to modify the scheme such that it becomes more robust to intensity fluctuations, while preserving its resilience to thermal, dephasing and phase noise. This is accomplished in the following section.

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Let us point out that the specific form of the curve in figure 7 depends on our choice of parameters. Still, it also describes qualitatively the behavior of the gate under driving-intensity noise for other parameters. Let us recall that the microwave driving suppresses the unwanted contributions of the forces in the σ^y and σ^z bases, while we minimize the detrimental effect of the spin-dependent force in σ^x by closing the phase-space trajectories. Thus, we expect the gate error to decrease with increasing driving. However, by increasing the driving strength, the noise will eventually become some percentage of the effective qubit–qubit coupling J^dss₁₂. Since the noise acts in the same basis as the gate, it will eventually start to deteriorate the gate performance outweighing the benefits of the driving. Hence, we expect a minimum error at a certain driving after which the error increases with stronger drivings. Now, if we make faster gates, the qubit–qubit coupling J^dss₁₂ is larger, and we expect the minimum error to appear for a stronger driving. This behavior is confirmed by figure 11, where we carried out a numerical simulation of a faster gate with equal noise parameters, and found that the error minimum is shifted to larger driving strengths. The same considerations can be applied for slower gates, where the minimum would arise at slower drivings.

Inspired by recent results for prolonging single-qubit coherence times by means of continuous drivings [43], we present a modification of the driven single-sideband Hamiltonian (6), which will allow us to obtain two-qubit gates that are also robust to intensity fluctuations of the microwave driving. We complement the qubit Hamiltonian (2) with a secondary microwave driving, such that H_q → H_q,2, where

$$\begin{equation} H_{\rm q,2}=\sum_{i=1}^N\frac{1}{2}\omega_{0}\sigma_i^z+\frac{1}{2}(\Omega_{\rm d}(t)\sigma_i^+{\rm e}^{-{\rm i}\omega_{\rm d}t}+\tilde{\Omega}_{\rm d}(t)\sigma_i^+{\rm e}^{-{\rm i}\tilde{\omega}_{\rm d}t}+{\rm H.c.}) \end{equation} \tag{ 51 }$$

and the secondary driving is characterized by a fluctuating Rabi frequency $\tilde {\Omega }_{\mathrm { d}}(t)=\tilde {\Omega }_{\mathrm { d}}+\Delta \tilde {\Omega }_{\mathrm { d}}(t)$, and a frequency $\tilde {\omega }_{\mathrm { d}}$, fulfilling $\tilde {\omega }_{\mathrm { d }}\approx \omega _0$, and $\tilde {\Omega }_{\mathrm { d}}\ll \omega _0$, where we have assumed again that $\tilde {\Omega }_{\mathrm { d}}\in \mathbb {R}$ and $\Delta \tilde {\Omega }_{\mathrm { d}}(t)$ is an O–U process. This yields the doubly driven single-sideband Hamiltonian

$$\begin{equation} H_{\rm 2dss}=H_{\rm q,2}+H_{\rm p}+H_{\rm qp}, \end{equation} \tag{ 52 }$$

where the phonon H_p and qubit–phonon H_qp terms correspond to equations (1) and (4) described above.

We show below that this doubly driven model leads to a geometric phase gate with an additional property: it is robust to intensity fluctuations of the first microwave driving ΔΩ_d(t), while being sensitive to intensity fluctuations of the secondary driving $\Delta \tilde {\Omega }_{\mathrm { d}}(t)$. This already tells us that the secondary driving should be much weaker $\tilde {\Omega }_{\mathrm { d}}\ll {\Omega }_{\mathrm { d}}$, such that the impact of its intensity fluctuations on the gate performance is considerably weaker. However, in order to find a more detailed parameter regime for the high-fidelity gates, we need to explore equation (52) analytically.

Based on the experience gained from the analytical study of the single-driving geometric phase gates (see section 2.2), we move to the 'dressed-state' interaction picture with respect to the first driving (15), which is again assumed to be on resonance with the qubit ω_d = ω₀, and $\Omega _{\mathrm { d}}\in \mathbb {R}$. By focusing first on the qubit part of the Hamiltonian, which parallels the discussion in [43], we obtain the following expression:

$$\begin{equation} \fl \skew3\tilde{H}_{\rm q}=\sum_i\frac{\Delta\Omega_{\rm d}(t)}{2}\sigma_i^x +\sum_i \left( \frac{\tilde{\Omega}_{\rm d}(t)}{4}(\!\sigma_i^x+{\rm i}\sigma_i^y\cos(\Omega_{\rm d}t)-{\rm i}\sigma_i^z\sin(\Omega_{\rm d}t)\!) {\rm e}^{-{\rm i} \tilde{\delta}_{\rm d} t} + {\rm H.c.} \right), \end{equation} \tag{ 53 }$$

where we have introduced the detuning of the secondary driving $\tilde {\delta }_{\mathrm { d}}=\tilde {\omega }_{\mathrm { d}}-\omega _0$. If this detuning is set correctly, the secondary driving will decouple the qubit from the intensity fluctuations in the first driving, just as the dephasing noise is minimized by the first driving (see equation (42) in section 3.2). In particular, this is achieved for $\tilde {\delta _{\mathrm { d}}}=\Omega _{\mathrm { d}}$, such that a rotating-wave approximation for $\tilde {\Omega }_{\mathrm { d}}\ll 4 \Omega _{\mathrm { d}}$ leads to

$$\begin{equation} \skew3\tilde{H}_{\rm q}=\frac{1}{2}\sum\limits_i\Delta\Omega_{\rm d}(t)\sigma_i^x-\frac{1}{4}\sum\limits_i (\tilde{\Omega}_{\rm d}+\Delta\tilde{\Omega}_{\rm d}(t))\sigma_i^z. \end{equation} \tag{ 54 }$$

If we now move to the dressed-state interaction picture of the secondary driving, which we shall call the double dressed-state interaction picture

$$\begin{equation} \skew3\tilde{\skew3\tilde{U}}_{\rm d}(t)={\rm e}^{-{\rm i} t\sum_i\frac{1}{4}\tilde{\Omega}_{\rm d}\sigma_i^z}\,{\rm e}^{{\rm i} t\sum_i\frac{1}{2}\Omega_{\rm d}\sigma_i^x}\,{\rm e}^{{\rm i} t \sum_i\frac{1}{2}\omega_0\sigma_i^z}\,{\rm e}^{{\rm i} t\sum_n\omega_na_n^{\dagger}a_n} \end{equation} \tag{ 55 }$$

the qubit Hamiltonian can be rewritten as

$$\begin{equation} \fl \skew3\tilde{\skew3\tilde{H}}_{\rm q}=\frac{1}{2}\sum_i\Delta\Omega_{\rm d}(t)\left(\cos\left(\frac{1}{2}\tilde{\Omega}_{\rm d}t\right)\sigma_i^x+\sin\left(\frac{1}{2}\tilde{\Omega}_{\rm d}t\right)\sigma_i^y\right)-\frac{1}{4}\sum_i \Delta\tilde{\Omega}_{\rm d}(t)\sigma_i^z. \end{equation} \tag{ 56 }$$

In this picture, it becomes clear that the noisy terms due to the first driving become rapidly rotating even for a weaker second driving provided that $\zeta _{\mathrm { I}}\Omega _{\mathrm { d}}\ll \tilde {\Omega }_{\mathrm { d}}\ll 4\Omega _{\mathrm { d}}$. Hence, we can neglect them in a rotating-wave approximation, such that the qubit is only sensitive to the fluctuations of the second driving $\tilde {\skew3\tilde{H}}_{\mathrm { q}}\approx -\frac {1}{4}\sum _i \Delta \tilde {\Omega }_{\mathrm { d}}(t)\sigma _i^z$, which have a weaker effect.

So far, we have only treated the qubit part of the doubly driven single-sideband Hamiltonian (52). The crucial point to address now is whether the two strong drivings can be combined with the qubit–phonon couplings responsible for the entangling gate. This question is by no means trivial, since the secondary driving acting in the σ^z-basis (54) will make the σ^x state-dependent force (16) rotate very fast, and can thus inhibit the required qubit–phonon couplings. To find the correct parameter regime, let us express the qubit–phonon Hamiltonian in the double dressed-state interaction picture

$$\begin{equation} \skew3\tilde{\skew3\tilde{H}}_{\rm qp}\!=\!\sum \limits_{i,n}\! \frac{\mathscr{F}_{in}}{2} \big(\!\sigma_i^xf_x(t)+\sigma_i^yf_y(t)-{\rm i}\sigma_i^zf_z(t)\!\big) a_n \,{\rm e}^{-{\rm i} \tilde{\delta}_n t} + {\rm H.c.}, \end{equation} \tag{ 57 }$$

where we have introduced the following time dependences:

$$\begin{eqnarray} f_x(t)&=&\cos(\textstyle\frac{1}{2}\tilde{\Omega}_{\rm d}t)-{\rm i}\sin(\textstyle\frac{1}{2}\tilde{\Omega}_{\rm d}t)\cos({\Omega}_{\rm d}t),\nonumber\\ f_y(t)&=&\sin(\textstyle\frac{1}{2}\tilde{\Omega}_{\rm d}t)+{\rm i}\cos(\textstyle\frac{1}{2}\tilde{\Omega}_{\rm d}t)\cos({\Omega}_{\rm d}t),\\ f_z(t)&=&\sin({\Omega}_{\rm d}t).\nonumber \end{eqnarray} \tag{ 58 }$$

We now study all the possibilities. (i) If the laser-induced detunings fulfill $\tilde\delta _n\ll \tilde {\Omega }_{\mathrm { d}}\ll \Omega _{\mathrm { d}}$, then all the terms in (57) become rapidly rotating, and the qubit–phonon coupling is inhibited by the double driving. (ii) If $\tilde\delta _n\approx \tilde{\Omega}_{{\rm d}/2}\ll \tilde {\Omega }_{\mathrm {d}}$, such that $\textstyle \frac {1}{2}|\mathscr{F}_{in}|\ll \tilde {\Omega }_{\mathrm { d}}\ll \Omega _{\mathrm { d}}$, then only the σ^z force can be neglected in a rotating-wave approximation. Unfortunately, the σ^x and σ^y forces then contribute equally, and we would lose the geometric character of the entangling gate (i.e. it will be very sensitive to thermal motion of the ions). (iii) Finally, if $\tilde {\Omega }_{\mathrm { d}}\ll \tilde\delta _n\approx \Omega _{\mathrm { d}}$, such that $\textstyle \frac {1}{2}|\mathscr{F}_{in}|\ll \tilde {\Omega }_{\mathrm { d}}$, Ω_d, then both σ^x/y forces become rapidly rotating and can be neglected, but the σ^z force preserves its nearly resonant character. To achieve this condition, instead of setting the laser frequency close to the bare sideband ω_L ≈ ω₀ − ω_n (see section 2.1), we need to adjust it close to the dressed sideband resonance

$$\begin{equation} \omega_{\rm L}\approx\omega_0+\Omega_{\rm d}-\omega_n,\quad \tilde{\delta}_n=\omega_{\rm L}-(\omega_0+\Omega_{\rm d}-\omega_n). \end{equation} \tag{ 59 }$$

In this regime, we obtain a single state-dependent force

$$\begin{equation} \skew3\tilde{\skew3\tilde{H}}_{\rm qp}\!\approx-\!\sum \limits_{i,n}\! \frac{\mathscr{F}_{in}}{4} \sigma_i^z a_n\, {\rm e}^{-{\rm i} \tilde{\delta}_n t} + {\rm H.c.} \end{equation} \tag{ 60 }$$

which will lead to a geometric phase gate in a different basis. We also note that the condition for the laser Rabi frequency is modified in this case to $|\Omega _{\mathrm { L}}|\ll 8 (\omega _n-|\tilde {\Omega }_{\mathrm { d}}/2|)$.

Let us now support the above discussion by integrating numerically the dynamics induced by the doubly driven single-sideband Hamiltonian in equations (57) and (58), such that the laser is now tuned to the regime (59). From the above discussion, we know that by applying a sufficiently strong secondary driving $\tilde {\Omega }_{\mathrm { d}}\gg |\mathscr{F}_{in}|$, we are left with a single state-dependent force in the σ^z-basis with a halved strength $\textstyle \frac {1}{2}\mathscr{F}_{in}\to \textstyle \frac {1}{4} \mathscr{F}_{in}$ with respect to the previous σ^x-force (16). In this regime, we should recover circular phase-space trajectories when the initial state of the qubits is an eigenstate of the σ^z₁σ^z₂ operator. In figure 8, we analyze the strong-driving dynamics numerically. For weak secondary drivings (figure 8(a)), we observe how the different phase-space trajectories are not closed. Conversely, for strong secondary drivings (figure 8(b)), we recover circular orbits that close exactly at the expected gate time. These numerical results support the validity of our arguments leading to equation (60).

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Starting from the state-dependent force in the σ^z-basis (60), we can apply once more the ME to get the following approximation to the time-evolution operator:

$$\begin{equation} U_{\rm z}(t_{\rm g})\approx\skew3\tilde{\skew3\tilde{U}}^{\dagger}_{\rm d}(t){\rm e}^{-{\rm i} H_{\rm ddss}t_{\rm g}},\quad H_{\rm ddss}=\sum_{i,j}J^{\rm{ddss}}_{ij}\sigma_i^z\sigma_j^z, \end{equation} \tag{ 61 }$$

where the coupling strengths now are given by

$$\begin{equation} J_{ij}^{\rm ddss}=-\sum_n \frac{1}{16\tilde{\delta_n}}\mathscr{F}_{in} \mathscr{F}_{jn}^{*}. \end{equation} \tag{ 62 }$$

Note that, in order to obtain this time evolution, it is necessary to neglect all other contributions coming from the ME. This is possible by considering the strong-driving limit $\textstyle \frac {1}{2}|\mathscr{F}_{in}|\ll \tilde {\Omega }_{\mathrm { d}}\ll 4\Omega _{\mathrm { d}}$, where additional conditions similar to the single-driving case (25) are also fulfilled

$$\begin{equation} t_{\rm g}= r \frac{2 \pi}{\tilde{\delta}_1}, \quad \tilde{\delta}_2=k \tilde{\delta}_1, \quad \Omega_{\rm d}=p \tilde{\delta}_1,\quad \tilde{\Omega}_{\rm d}=q \tilde{\delta}_1, \quad r,k,p,q\in \mathbb{Z}. \end{equation} \tag{ 63 }$$

These conditions allow us to close exactly the phase-space trajectories (figure 8(b)) or, equivalently, to minimize the residual qubit–phonon couplings that would make the gate sensitive to the thermal motion of the ions. We have found that these conditions can be met by imposing

$$\begin{equation} |\Omega_{\rm L}|=\frac{2|\tilde{\delta}_1|}{\eta_1 \sqrt{\frac{r}{2}\left(1-\frac{1}{2 \xi} \right)}} \end{equation} \tag{ 64 }$$

in analogy to our older choice (32) for the single-driving gates. We use the same values for the detunings as in table 1 with $\delta _{1/2} \to \tilde {\delta }_{1/2}$, but now set the parameter r = 32 to optimize the gate fidelities, at the price of diminishing the gate speed. Note that the unitary evolution (61) leads to the following table for t_g = π/(8J^ddss₁₂):

$$\begin{equation} \begin{array}{llll} \displaystyle |++\rangle\to|\tilde{\Phi}^-\rangle=\frac{1}{\sqrt{2}}(|++\rangle-{\rm i}\,{\rm sgn}(J^{\rm{ddss}}_{12})|--\rangle),\\ \displaystyle |+-\rangle\to|\tilde{\Psi}^-\rangle=\frac{1}{\sqrt{2}}(|+-\rangle-{\rm i}\,{\rm sgn}(J^{\rm{ddss}}_{12})|-+\rangle),\\ \displaystyle |-+\rangle\to|\tilde{\Psi}^+\rangle=\frac{1}{\sqrt{2}}(|+-\rangle+{\rm i}\,{\rm sgn}(J^{\rm{ddss}}_{12})|-+\rangle),\\ \displaystyle |--\rangle\to|\tilde{\Phi}^+\rangle=\frac{1}{\sqrt{2}}(|++\rangle+{\rm i}\,{\rm sgn}(J^{\rm{ddss}}_{12})|--\rangle). \end{array} \end{equation} \tag{ 65 }$$

In figure 9(b), we represent the fidelity for the unitary generation of the entangled state $|\tilde {\Psi }^-\rangle$ from the initially unentangled state | + −〉. It becomes clear from this figure, that the fidelities that can be achieved are again close to 100%.

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However, the fast and small-amplitude oscillations in the fidelity set an upper limit to the achieved fidelities, which is below the desired FTs. In order to improve the gate fidelities, we introduce a simple spin-echo refocusing pulse, such that the complete time-evolution operator is $U_{\mathrm { full}} (t_{\mathrm { g}},0)= U_{\mathrm { z,noise}}(t_{\mathrm { g}},\frac {t_{\mathrm { g}}}{2})(\sigma ^y_1 \sigma _2^y) U_{\mathrm { z,noise}}(\frac {t_{\mathrm { g}}}{2},0)$. In this expression, we have also included the intensity fluctuations of the first driving, such that U_z,noise(t₂,t₁) is the time-evolution operator induced by the noisy Hamiltonian

$$\begin{eqnarray} &&\fl H_{\rm z,noise}=\frac{1}{2}\sum_i\Delta\Omega_{\rm d}(t)\left(\cos\left(\frac{1}{2}\tilde{\Omega}_{\rm d}t\right)\sigma_i^x+\sin\left(\frac{1}{2}\tilde{\Omega}_{\rm d}t\right)\sigma_i^y\right)\nonumber\\ &&+\sum \limits_{i,n}\! \frac{\mathscr{F}_{in}}{2} \big(\!\sigma_i^xf_x(t)+\sigma_i^yf_y(t)-{\rm i}\sigma_i^zf_z(t)\!\big) a_n \,{\rm e}^{-{\rm i} \tilde\delta_n t} + {\rm H.c.}, \end{eqnarray} \tag{ 66 }$$

where all the parameters have already been introduced above. Note that due to the refocusing pulse, the target entangled state obtained from | + −〉 is no longer $|\tilde {\Psi }^-\rangle$, but rather $|\tilde {\Psi }^+\rangle$.

In figure 10, we represent the achieved error of this doubly driven geometric phase gate for the generation of the entangled state $|\tilde {\Psi }^+\rangle$ in the presence of noise on the first driving. We note that with our choice of parameters for the simulation displayed in figure 10 we do not fulfill $|\tilde {\Omega }_{\mathrm { d}}|\ll |\Omega _{\mathrm { d}}|$ but the analytic calculation showed that it is enough to fulfill $|\tilde {\Omega }_{\mathrm { d}}|\ll 4|\Omega _{\mathrm { d}}|$ to obtain the doubly driven gate in the σ^z basis. The second, more relaxed constraint is met here.

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This numerical result displays the superior performance of this new gate, and its resilience to the intensity fluctuations of the first microwave driving allowing for ε_2q < ε_FT2. At this point, it might be worth emphasizing the benefit of the secondary driving. Noting that the scheme also works for clock states, one might ask whether the impact of the intensity fluctuations of the microwave driving could be reduced by making faster gates. In fact, assuming a clock state, we can increase the gate speed by decreasing the parameter r from equation (25) without compromising the error due to the dephasing noise. In figure 11, the error for a gate with r = 2 in the presence of relative intensity noise ζ_I = 0.7 × 10⁻⁴ on the microwave driving is shown. As can be seen in the figure, the error reaches ε_2q ≈ 1 × 10⁻⁴, while the error achieved with the secondary driving is lower ε_2q ≈ 5 × 10⁻⁵.

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We remark that intensity fluctuations of the secondary driving have not been considered. Such fluctuations will eventually be the limiting factor for the gate fidelity. However, since the secondary driving is weaker than the first one, their effect will be smaller. Moreover, due to the excellent fidelity obtained, it may also be possible to reduce the second driving further, making the gate more insensitive to its fluctuations. Another possibility would be to introduce a tertiary microwave driving, which would be now detuned by the Rabi frequency of the secondary one $\tilde {\tilde {\omega }}_{\mathrm { d}}-\omega _0=\tilde {\Omega }_{\mathrm { d}}$. This driving would act as a decoupling mechanism from intensity fluctuations of the second driving. We believe it should be possible to modify the sideband resonance (59), such that we obtain a triply driven geometric phase gate in a different basis (possibly the σ^y basis) that enjoys a further robustness against all these sources of noise. Such concatenated schemes can be followed until the desired fidelities are achieved.