Introduction

High-fidelity entanglement generation among qubits is crucial for quantum information processing1. In most platforms, entangling gates come via direct interactions (e.g. hyperfine fields among nuclear spins) or via a bosonic mediator. Examples of the latter include solid-state qubits coupled to microwave resonators2,3,4,5,6, or trapped ions sharing vibrational modes7. In this scenario, the paradigmatic Mølmer-Sørensen (MS) gate8,9,10,11 and related schemes12,13,14 reach entanglement operations with inherent robustness to uncertainties in the bosonic state.

In recent years, several gate schemes have been developed that operate on the same MS principle but are also robust against other sources of error15. Some examples are frequency or amplitude-modulated gates decoupling from mode decoherence16,17,18,19,20,21, from spectator modes22,23,24,25,26,27,28 or from deviations in the qubit-boson coupling strength29,30,31,32,33. In another vein, dynamical decoupling (DD) is a well established paradigm to protect qubits against decoherence34,35. Some continuous DD techniques have been demonstrated to be suitable for quantum gate implementations36,37,38,39,40, while pulsed DD methods achieve increased robustness employing suited sequences such as XY841,42,43,44,45,46,47,48 or AXY49,50,51. However, the use of pulsed DD to protect oscillator-mediated gates has been mostly limited to dispersive regimes52,53,54,55, and to few spin-echo56,57,58 or rotary-echo37 pulses. Note that the application of several π pulses is desirable for efficient elimination of time-varying noise.

In this regard, Manovitz et al.59 have experimentally shown that the MS gate can be combined with pulse sequences given the ability to tune and turn on-and-off the qubit-boson coupling as many times as the number of DD pulses introduced. Another possibility explored theoretically is to combine an always-on qubit-boson coupling with strong π pulses51,60,61. Although this is possible in certain trapped-ion architectures, turning on-and-off the qubit-boson coupling maybe not practical in other platforms. On the other hand, the use of strong π pulses is experimentally challenging since high-power controls are needed, while these induce crosstalk and hinder the applicability in multimode scenarios.

In this article, we design a DD sequence with low-intensity π pulses—named TQXY16—that achieves faster-than-dispersive entangling gates using static (i.e. non-tunable) qubit-oscillator coupling. Importantly, our gates decouple from dephasing, pulse imperfections, and unwanted finite-pulse effects, leading to high fidelity. Furthermore, we demonstrate the versatility of our protocol by incorporating techniques that lead to additional resilience to decoherence on the bosonic mediator and potential crosstalk effects. Although our method is general, we exemplify its performance in radio-frequency controlled trapped ions demonstrating infidelities within the 10−3 threshold at state-of-the-art experimental conditions, and of 10−4 in near-future setups.

Results

Gate with instantaneous pulses

We consider a system that comprises two qubits and a bosonic mode –with frequencies ω1, ω2 and ν– coupled via longitudinal coupling2,3,4,5,6,7 (here, and throughout the paper, H is H/, meaning all Hamiltonians are given in units of angular frequency),

$${H}_{0}=\nu {a}^{{{{\dagger}}} }a+\eta \nu (a+{a}^{{{{\dagger}}} }){S}_{z}.$$
(1)

Here, a(a) is the creation (annihilation) operator of the bosonic mode, \({S}_{\mu }\equiv {\sigma }_{1}^{\mu }+{\sigma }_{2}^{\mu }\) with μx, y, z are collective qubit operators, and ην is the coupling strength. Also, note that H0 is written in a rotating frame with respect to (w.r.t) the qubit free-energy Hamiltonian \({H}_{q}={\sum }_{\mu }{\omega }_{\mu }{\sigma }_{\mu }^{z}/2\). We assume the usual experimental scenario η 1, thus we stay away from other paradigms that require stronger qubit-boson couplings62,63,64,65,66. H0 contains no driving fields, while in our method we drive the qubits for two main reasons: (i) Accelerate the gate by making the qubits rotate at a frequency close to the bosonic frequency ν, and (ii) Protection of the gate from qubit noise of the form \({\epsilon }_{j}(t){\sigma }_{j}^{z}/2\) leading to dephasing. When driving the qubits, H0 is completed with the term Hd(t) = ∑μ=x,yΩμ(t)Sμ/2. In an interaction picture w.r.t. Hd + νaa we get

$$H(t)=\eta \nu (a{e}^{-i\nu t}+{a}^{{{{\dagger}}} }{e}^{i\nu t})\mathop{\sum}\limits_{\mu =x,y,z}{f}_{\mu }(t){S}_{\mu },$$
(2)

where \({\sum }_{\mu = x,y,z}{f}_{\mu }(t){S}_{\mu }={U}_{d}^{{{{\dagger}}} }(t){S}_{z}{U}_{d}(t)\) with \({U}_{d}(t)={{{{{{{\mathcal{T}}}}}}}}\exp [-i\int\nolimits_{0}^{t}{H}_{d}({t}^{{\prime} })d{t}^{{\prime} }]\) being the time-ordered propagator. See supplementary note 1 for additional details.

If driving fields are delivered as instantaneous π pulses (note this requires Ωx,yν during the application of the pulse) spaced τ/2 apart, fx,y(t) can be neglected and fz(t) = 1( − 1) if the number of applied pulses is even (odd), see the grey solid line in Fig. 1(a). For the moment we consider instantaneous pulses, while later we treat the realistic case of non-instantaneous ones. As π pulses are applied periodically, fz(t) takes the form of a function with period τ such that \({f}_{z}(t)=\mathop{\sum }\nolimits_{n = 1}^{\infty }{f}_{n}\cos (n\omega t)\), where ω = 2π/τ and \({f}_{n}=\frac{2}{\tau }\int\nolimits_{0}^{\tau }d{t}^{{\prime} }{f}_{z}({t}^{{\prime} })\cos (n\omega {t}^{{\prime} })\). Hence, under the assumption of instantaneous pulses we get

$$H(t)=\eta \nu \mathop{\sum }\limits_{n=1}^{\infty }{f}_{n}\cos (n\omega t)(a{e}^{-i\nu t}+{a}^{{{{\dagger}}} }{e}^{i\nu t}){S}_{z},$$
(3)

whilst setting an interpulse spacing τ/2 = τk/2 such that ω = ωk ≈ ν/k leads to a resonant qubit-boson interaction via the kth harmonic (from now on τ → τk and ω → ωk, where the subscript k refers to the kth harmonic). As η 1, the terms in Eq. (3) that rotate with frequencies ± ν − nωk (where n ≠ k) and ± ν + nωk can be substituted, using the rotating-wave approximation, by their second-order contribution (here, and in the rest of the paper, second-order stands for second order in η) leading to

$$H(t)\approx \frac{1}{2}\eta \nu {f}_{k}(a{e}^{-i{\xi }_{k}t}+{{{{{{{\rm{H.c.}}}}}}}}){S}_{z}-\frac{1}{2}{\eta }^{2}\nu {J}_{k}{S}_{z}^{2},$$
(4)

where ξk = ν − kωk is the detuning w.r.t. the kth harmonic and \({J}_{k}={f}_{k}^{2}/4+{\sum }_{n\ne k}{f}_{n}^{2}/(1-{n}^{2}/{k}^{2})\) is an effective spin-spin coupling constant that contains contributions from all harmonics. Note that, as η 1 contributions of higher order in η can be neglected. See supplementary note 2 for additional details. The propagator associated to Hamiltonian (3) is

$$U(t)=\exp \{[\alpha (t){a}^{{{{\dagger}}} }-{\alpha }^{* }(t)a]{S}_{z}\}\times \exp [i\theta (t){S}_{z}^{2}]$$
(5)

where \(\alpha (t)=-i\eta \nu \int\nolimits_{0}^{t}d{t}^{{\prime} }{f}_{z}({t}^{{\prime} }){e}^{i\nu {t}^{{\prime} }}\approx -\eta \nu {f}_{k}/(2{\xi }_{k})({e}^{i{\xi }_{k}t}-1)\) and

$$\theta (t)={{{{{{{\rm{Im}}}}}}}}{\int}_{{{{{{{{\mathcal{C}}}}}}}}}\alpha \,d\alpha \approx \frac{{\eta }^{2}{\nu }^{2}{f}_{k}^{2}}{4{\xi }_{k}}\left[t-\frac{\sin ({\xi }_{k}t)}{{\xi }_{k}}\right]+\frac{1}{2}\nu {\eta }^{2}{J}_{k}t,$$
(6)

where \({{{{{{{\mathcal{C}}}}}}}}\) is the phase-space trajectory followed by α(t). Note that, if the gate time is chosen as tg = 2π/ξk, α(tg) ≈ 0 at the end of the gate, making the gate insensitive to the bosonic state. To satisfy condition θ(tg) = π/8, we choose τk such that \({\xi }_{k}=2\eta \nu \{\sqrt{{f}_{k}^{2}+4{\eta }^{2}{J}_{k}^{2}}+2\eta {J}_{k}\}\) for Jk > 0. After a time tg the propagator U(t) approximates to \(\exp (i\frac{\pi }{8}{S}_{z}^{2})\). For two qubits, this is equivalent (up to a global qubit rotation) to the CPHASE gate, and transforms the state \(\left\vert ++\right\rangle\) into the Bell state \(\left\vert {\Phi }^{+}\right\rangle =\frac{1}{\sqrt{2}}(\left\vert ++\right\rangle +i\left\vert --\right\rangle )\). It is noteworthy that the choice of ξk (thus τk) is, in general, not trivial, as both fk and Jk depend on τk. However, in the cases discussed here, this dependance does not hold, making the choice of ξk direct. See supplementary note 3 for analytic expressions for fk and Jk.

Fig. 1: Entangling gates with instantaneous π pulses.
figure 1

a Rabi frequency Ω(t) and modulation function fz(t) during a period τk for k = 5. For comparison, we plot \(\cos (k{\omega }_{k}t)\) in green. b Phase-space trajectory of α(t) during the application of a pulse sequence with k = 1 and k = 5 in blue and green, respectively.

For instantaneous π pulses one finds \({f}_{k}={f}_{k}^{{{{{{{{\rm{ins}}}}}}}}}=\frac{4}{k\pi }\sin (\frac{k\pi }{2})\). Thus, if resonance is achieved via a low harmonic, e.g. k = 1, the gate time is \({t}_{{{{{{{{\rm{g}}}}}}}}}^{k = 1}\approx {\pi }^{2}/4\eta \nu\), a factor π/2 longer than the original MS gate. On the other hand, for sufficiently large harmonics the gate is mostly governed by the dispersive term \(\frac{1}{2}{\eta }^{2}\nu {J}_{k\to \infty }{S}_{z}^{2}\approx {\eta }^{2}\nu {S}_{z}^{2}\) in Eq. (4), leading to \({t}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{d}}}}}}}}}=\pi /8{\eta }^{2}\nu\). We define faster-than-dispersive gates as those that satisfy \({t}_{{{{{{{{\rm{g}}}}}}}}}^{k}/{t}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{d}}}}}}}}} < 1\). For example, in the case η = 0.01 and k = 1 we find a faster-than-dispersive gate with \({t}_{{{{{{{{\rm{g}}}}}}}}}^{k = 1}/{t}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{d}}}}}}}}}\approx 1/16\). This is, the gate is 16 times faster than the dispersive one. In Fig. 1(b), we show α(t) for η = 0.03, with k = 1 and 5.

It is noteworthy that the analysis conducted above is valid for Nq qubits homogeneously coupled to the same bosonic mode, i.e. \({S}_{\mu }\to \mathop{\sum }\nolimits_{j = 1}^{{N}_{q}}{\sigma }_{j}^{\mu }\). For two qubits with inhomogeneous coupling, i.e. \({S}_{z}\to {S}_{z}=({\eta }_{1}{\sigma }_{1}^{z}+{\eta }_{2}{\sigma }_{2}^{z})/\eta\) where ηj 1, the method also yields to a CPHASE gate, however, in this case, the correct expression for the detuning ξk is that in which every η is substituted by \(\sqrt{{\eta }_{1}{\eta }_{2}}\). For larger η, the rotating-wave approximation is not justified and terms neglected from Eq. (3) to Eq. (4) will lead to significant residual qubit-boson entanglement at the end of the gate. See supplementary note 4 for additional details.

Gate with low-intensity pulses

In what follows, we discuss the realistic case of non-instantaneous pulses. For standard top-hat pulses the Fourier coefficient fk that quantifies the strength of the qubit-boson interaction reads (see supplementary note 3 for the derivation)

$${f}_{k}^{{{{{{{{\rm{th}}}}}}}}}\approx \frac{{f}_{k}^{{{{{{{{\rm{ins}}}}}}}}}}{1-{\nu }^{2}/{\Omega }^{2}}\cos \left(\frac{\pi \nu }{2\Omega }\right).$$
(7)

Notice that for low-intensity pulses—defined as those holding Ω < ν—the value of fk decays with (Ω/ν)2. As a result, achieving faster-than-dispersive gates is no longer possible. Note that fk directly relates to the non-dispersive contribution in θ(t), and, through condition θ(tg) = π/8, to the gate time tg.

To solve this problem and optimise the strength of the qubit-boson interaction, we propose to modulate the Rabi frequency during the execution of each π pulse. Specifically, we pose the following ansatz for fz(t)

$${f}_{z}(t)=\cos [\pi (t-{t}_{i})/{t}_{\pi }]+\beta (t)\sin [k{\omega }_{k}(t-{t}_{m})],$$
(8)

where tπ is the π pulse duration, and ti and tm = ti + tπ/2 are the initial and central points of the pulse. Note that the Rabi frequency is then given by \(\Omega (t)=-\frac{\partial {f}_{z}(t)}{\partial t}\times {[1-{f}_{z}^{2}(t)]}^{-1/2}\). For the envelope function β(t), we propose

$$\beta (t)=\frac{d}{\pi kb}\sin (\pi k/2)\left[{{{{{{{\rm{erf}}}}}}}}\left(\frac{t-{t}_{l}}{c{t}_{\pi }}\right)-{{{{{{{\rm{erf}}}}}}}}\left(\frac{t-{t}_{r}}{c{t}_{\pi }}\right)\right],$$
(9)

where tr = tm + btπ and tl = tm − btπ. The free parameters b and c serve to control the width of the envelope function β(t), while d is proportional to its amplitude. Suitable values for b, c and d for the first harmonics are shown in Table 2 (Methods).

From now on, we assume tπ = τk/2, i.e., the pulse extends over a whole period τk/2, leading to solutions with the lowest intensities. As a result of our pulse design with suitable b and c, the value for the Fourier coefficient fk is given by \({f}_{k}^{{{{{{{{\rm{m}}}}}}}}}=-4d/\pi k\) where d can take values from 0 to \(| {d}_{\max }| > 1\). See supplementary note 3 for the derivation. Since now fk depends on d, this serves to control the strength of the interaction, thus the duration of the gate tg. Also, d relates to the amplitude of the pulse, thus to the maximum value of the Rabi frequency Ωpp. Typically, we look for large values of d, bounded by \({d}_{\max }\) or by the experimentally available Ωpp.

Now we describe the recipe to design faster-than-dispersive gates using low-intensity pulses. First we choose a value for the harmonic k. Larger k allow for lower pulse intensities at the price of longer gates. Second, we use Eq. (8) to generate the modulation function fz(t) and the Rabi frequency Ω(t) for different values of d, and calculate both the gate time tg and \({\Omega }_{{{{{{{{\rm{pp}}}}}}}}}=\max [| \Omega (t)| ]\). We note that the obtained Ω(t) can lead to pulses along arbitrary axes (e.g. X or Y). In particular, for reasons described later, we target gates formed by concatenating blocks of 16 pulses. For that, the gate time tg must be 8Nτk, where N is an integer number. This translates into the condition \((\nu -{\xi }_{k})/8k| {\xi }_{k}| \in {\mathbb{N}}\) (note tg = 8Nτk, while tg = 2π/ξk and τk = 2πk/(ν − ξk)). The final step is to select the values of d for which this last condition is satisfied. As a result, we obtain all possible gates within the harmonic k as well as the corresponding values for Ωpp.

Figure 2(b) shows values of tg and Ωpp obtained following the previous prescription for η = 0.005 and k = 7, 9. Notice that there are plenty of solutions giving faster-than-dispersive gates, i.e. \({t}_{{{{{{{{\rm{g}}}}}}}}}/{t}_{{{{{{{{\rm{g}}}}}}}}}^{{{{{{{{\rm{d}}}}}}}}} < 1\), using low-intensity pulses with values of Ωpp well below the frequency ν.

Fig. 2: Amplitude modulated π pulses.
figure 2

a Ω(t) for a single XY8 block. The whole sequence here is a concatenation of 2N blocks. b Gate time tg as a function of Ωpp, for values of d between 0 and \({d}_{\max }\). Values satisfying tg = 8Nτk are represented by square and round markers for k = 9 and k = 7, respectively. c Ω(t) and fz(t) for cases N = 5 with d = 1.888 (blue) and N = 10 with d = 0.908 (red) of the 9th harmonic. d Gate phase θ(t) for N = 5 (dashed blue line), N = 10 (solid red line), and the dispersive case (dotted line).

As an example, we choose two solutions within the 9th harmonic, where τk extends over approximately nine oscillator periods. In Fig. 2c the shapes of Ω(t) and fz(t) are displayed for cases N = 5 and 10. Notice that Ω(t) achieves a larger amplitude when N = 5. As a consequence, it generates a faster gate. This is shown in Fig. 2d, where the two-qubit gate phase θ(t) related to the N = 5 gate reaches the target value π/8 faster than the N = 10 gate or the dispersive gate.

The reason for choosing the gate time as an integer multiple of 8τk has to do with an efficient decoupling from finite-pulse effects produced by the terms fx,y(t) neglected in Eq. (3). In the same way, the XY8 ≡ XYXYYXYX pulse structure assures cancelation of σz type noise, as well as of Rabi frequency fluctuations. In Fig. 2a the Rabi frequency is plotted (Ωx(t) when ‘X’; Ωy(t) when ‘Y’) for an XY8 block.

To understand the elimination of finite-pulse effects, we calculate the second-order Hamiltonian of Eq. (2) after an XYXY block leading to (see supplementary note 5 for the derivation)

$${H}_{{{{{{{{\rm{XYXY}}}}}}}}}=-\frac{1}{2}{\eta }^{2}\nu \left\{{J}_{k}^{\perp }({S}_{x}^{2}+{S}_{y}^{2})-{B}_{k}({a}^{{{{\dagger}}} }a){S}_{z}\right\}.$$
(10)

If our gate contains only XYXY blocks, HXYXY adds to Hamiltonian (4) spoiling a high-fidelity performance. To overcome this problem, we use a two-step strategy. Firstly, we concatenate XYXY and YXYX blocks (which form a XY8 block) such that the term Bk(aa)Sz gets refocused. Note that in the presence of bosonic decoherence, this term will induce qubit dephasing. Secondly, we cancel the remaining term \({J}_{k}^{\perp }({S}_{x}^{2}+{S}_{y}^{2})\) by driving the two qubits with opposite phases every second XY8 block. This is, when rotating by an angle π the phase of the second qubit’s driving \({H}_{d}^{(+)}={\sum }_{\mu = x,y}{\Omega }_{\mu }(t){S}_{\mu }^{(+)}/2\) becomes \({H}_{d}^{(-)}={\sum }_{\mu = x,y}{\Omega }_{\mu }(t){S}_{\mu }^{(-)}/2\) instead, where \({S}_{\mu }^{(\pm )}\equiv {\sigma }_{1}^{\mu }\pm {\sigma }_{2}^{\mu }\). This changes the sign of the \({\sigma }_{1}^{u}{\sigma }_{2}^{\mu }\) terms with u, μ {x, y}, leading to refocusing of the term \({J}_{k}^{\perp }({S}_{x}^{2}+{S}_{y}^{2})\) after every pair of XY8 blocks.

Note that the second step requires the ability to address each qubit individually, and assumes that Nq = 2, i.e. Sμ is given by the sum of two qubit operators. In the absence of individual addressing, one can incorporate the term \({J}_{k}^{\perp }({S}_{x}^{2}+{S}_{y}^{2})\) into the gate, but then the operation applied is not equivalent to the CPHASE gate. For a discussion regarding this alternative gate, as well as the extension to the multiqubit case, see supplementary note 6.

Summarising, our two-qubit gates are generated by nesting TQXY16 ≡ XY8(+)XY8(−) blocks, where TQXY16 stands for ‘two-qubit’ XY16, while XY8(±) implies qubits driven in phase or in anti-phase as discussed in the previous paragraph, while, importantly, each π pulse is implemented according to the designs for fz(t) and β(t) presented in Eqs. (8), (9).

Trapped-ion implementation and numerical results

We benchmark our method by simulating its performance in a pair of trapped ions in a static magnetic field gradient7. In this scenario, qubit frequencies ωμ take values around (2π) × 10 GHz, ν = (2π) × 220 kHz is the frequency of the centre-of-mass vibrational mode, \(\eta ={\gamma }_{e}{g}_{B}/8\nu \sqrt{\hslash /M\nu }\) is an effective Lamb-Dicke factor where γe = (2π) × 2.8 MHz/G, gB is the magnetic field gradient, and M is the ion mass. The two-ion system has a second vibrational mode ‘b’ with its corresponding qubit-boson coupling. Thus, Hamiltonian (1) is replaced by H0 + H2M, where \({H}_{2{{{{{{{\rm{M}}}}}}}}}=\sqrt{3}\nu {b}^{{{{\dagger}}} }b-{3}^{-1/4}\eta \nu (b+{b}^{{{{\dagger}}} }){S}_{z}^{(-)}\). The addition of H2M changes the dispersive coupling in Eq. (4) as \({J}_{k}\to {J}_{k}-1/3\mathop{\sum }\nolimits_{n = 1}^{\infty }{f}_{n}^{2}/(1-{n}^{2}/3{k}^{2})\), which must be taken into account when following the prescription to calculate the valid gates. This step can be done for an arbitrary amount of spectator modes, given that the mode frequencies νm fulfil the condition ηfnνm − nωk for all odd n.

Although we simulate the performance of the gate with the two-mode Hamiltonian \({H}_{f}={H}_{d}^{(\pm )}+{H}_{0}+{H}_{2{{{{{{{\rm{M}}}}}}}}}\) (see column \(\Delta {{{{{{{{\mathcal{I}}}}}}}}}_{2M}\) in Table 1), due to computational limitations we use the single-mode Hamiltonian \({H}_{s}={H}_{d}^{(\pm )}+{H}_{0}+{H}_{2{{{{{{{\rm{M}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}\) instead, where \({H}_{2{{{{{{{\rm{M}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}=\frac{1}{3}\nu {\eta }^{2}r{S}_{z}^{2}\) is the second-order contribution of H2M. See supplementary note 7 for additional details. Here, \({H}_{d}^{(\pm )}\) stands for \({H}_{d}^{(+)}\) (\({H}_{d}^{(-)}\)) every first (second) half of a TQXY16 block.

Table 1 Error budget: Column XY8 and TQXY16 show the infidelities after an evolution with Hamiltonians \({H}_{d}^{(+)}+{H}_{0}+{H}_{2{{{{{{{\rm{M}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}\) and Hs, respectively.

We investigate two regimes: (i) η = 0.005 (gB = 19.16 T/m), which is the state-of-the-art of current experiments18,54, and (ii) η = 0.04 (gB = 153.2 T/m), which can be reached in near future setups39.

In regime (i), we consider three different gates, all within the 9th harmonic. The first gate (G1), with a duration tg = 1.64 ms, appears after five TQXY16 blocks with pulse length tπ = 20.5 μs reaching Ωpp = (2π) × 124 kHz. The second gate (G2) with gate time tg = 3.28 ms uses ten TQXY16 blocks with pulse length tπ = 20.5 μs reaching Ωpp = (2π) × 77.8 kHz. The third gate (G3), with the gate-time tg = 3.94 ms and Ωpp = (2π) × 78.69 kHz, uses twelve blocks, each with a different pulse length and detuning, while it incorporates a technique to mitigate errors due to mode decoherence, see supplementary note 8. In regime (ii) we consider a gate within the 5th harmonic (G4). This gate occurs after two TQXY16 blocks where tg = 368 μs, tπ = 11.5 μs, and Ωpp = (2π) × 80.9 kHz. For further details regarding pulse parameters, see supplementary note 7.

The performance of the four gates in the presence of distinct error sources is shown in Table 1. Each simulated experiment starts from the state \(\vert {+}_{x}{+}_{y}\rangle\) and targets the Bell-state \(\vert {\tilde{\Phi }}^{+}\rangle =\frac{1}{\sqrt{2}}(\vert {+}_{x}{+}_{y}\rangle +i\vert {-}_{x}{-}_{y}\rangle )\), while in all cases we consider an initial motional thermal state with \(\bar{n}=1\)54. Other initial states result in similar values for fidelity.

In the 2nd and 3rd columns of Table 1, we show the gate error \({{{{{{{\mathcal{I}}}}}}}}=1-{{{{{{{\mathcal{F}}}}}}}}\) obtained by concatenating XY8 or TQXY16 blocks, respectively. Here, \({{{{{{{\mathcal{F}}}}}}}}=\langle {\tilde{\Phi }}^{+}\vert \rho \vert {\tilde{\Phi }}^{+}\rangle /\sqrt{{{{{{{{\rm{Tr}}}}}}}}({\rho }^{2})}\)67, where ρ is the final state after tracing out the bosonic states. Notice that TQXY16 blocks achieve a clearly superior performance due to efficient decoupling from finite pulse effects. For these, \({{{{{{{{\mathcal{I}}}}}}}}}_{{{{{{{{\rm{TQXY}}}}}}}}16}\le 1{0}^{-6}\) for all gates except G4, where finite the residual qubit-boson entanglement limits the error to approximately 10−5. In the fourth column we evaluate the effect of the second mode b by numerically simulating the two-mode Hamiltonian Hf (initialising the second mode b in a thermal state with \(\bar{n}=1\)), which results in \({{{{{{{{\mathcal{I}}}}}}}}}_{2{{{{{{{\rm{M}}}}}}}}}\). The infidelities relative to the previous case (i.e. \(\Delta {{{{{{{{\mathcal{I}}}}}}}}}_{2{{{{{{{\rm{M}}}}}}}}}={{{{{{{{\mathcal{I}}}}}}}}}_{2{{{{{{{\rm{M}}}}}}}}}-{{{{{{{{\mathcal{I}}}}}}}}}_{{{{{{{{\rm{TQXY}}}}}}}}16}\)) are given in the ‘\(\Delta {{{{{{{{\mathcal{I}}}}}}}}}_{2{{{{{{{\rm{M}}}}}}}}}\)’ column of Table 1. Again, the effect of the second mode is relevant only for G4, which contributes 3.8 × 10−5 to the total error. Importantly, this demonstrates that our gate is compatible with the presence of spectator modes.

To investigate the effect of crosstalk, we add the term \({H}_{c}^{(\pm )}={\sum }_{\mu = x,y}\frac{{\Omega }_{\mu }(t)}{2}({\sigma }_{2}^{-}{e}^{-i\Delta \omega t}\pm {\sigma }_{1}^{-}{e}^{i\Delta \omega t}+{{{{{{{\rm{H.c.}}}}}}}})\) to Hs, where Δω/(2π) = (ω2 − ω1)/(2π) = 2.54 and 20.34 MHz for regimes (i) and (ii), respectively. The results are given in the ‘\(\Delta {{{{{{{{\mathcal{I}}}}}}}}}_{{{{{{{{\rm{CT}}}}}}}}}\)’ column of Table 1. In contrast to the effect of the spectator mode, crosstalk is most harmless with the G4 gate. This is expected, as G4 operates with a larger qubit detuning Δω than the rest, while using a similar Rabi frequency. To reduce the impact of crosstalk, we combine our pulses with sin2-shaped ramps at the beginning and end of each pulse, see supplementary note 7, and optimise the length of the ramp using numerical simulations. The resulting infidelities are shown in the ‘\(\Delta {{{{{{{{\mathcal{I}}}}}}}}}_{{{{{{{{\rm{CT}}}}}}}}* }\)’ column. Note that the sin2 ramp reduces the value of \({{{{{{{\mathcal{I}}}}}}}}\) by at least an order of magnitude in most cases.

Robustness w.r.t. common errors such as dephasing over qubits due to static shifts ωj → ωj ± δω, Rabi-frequency shifts (i.e. Ω(t) → (1 ± δΩ) Ω(t)), and shifts on the mode frequency, ν → (1 ± δν) ν, is shown in Fig. 3a–c, where the infidelity is plotted versus the degree of uncertainty. In columns 7–9 of Table 1, we display the relative infidelities for a dephasing time \({T}_{2}^{* }\approx 500\mu{{{{{\rm{s}}}}}}\) 54, a Rabi-frequency shift δΩ = 5 × 10−3, and a mode-frequency shift of δν = 10−568. For further details, see ‘Methods’. Furthermore, in Fig. 3d we plot the infidelity versus \(\dot{\bar{n}}\), while in column \(\dot{\bar{n}}\), we show the relative infidelities for G1-3 and G4 for mode heating rates \(\dot{\bar{n}}=35\) and 100 ph/s, respectively. For further details, see Supplementary note 7. Figure 3e shows the phase-space trajectory of α(t) for all gates G1-4.

Fig. 3: Sensitivity to errors.
figure 3

a Gate error \({{{{{{{\mathcal{I}}}}}}}}\) versus \(1/{T}_{2}^{* }\) for the gates G1 (blue), G2 (red), G3 (purple) and G4 (green). In (b) and (c), the error is shown for Rabi frequency shifts Ω(t) → (1 ± δΩ)Ω(t) and in the mode frequency ν → (1 ± δν)ν. d Error under heating for different rates \(\dot{\bar{n}}\). e Trajectory of α(t) for gates G1 (circular, blue), G2 (circular, red), G3 (non-circular, purple), and G4 (circular, green).

Table 1 shows that mode heating is the main source of error for gates in regime (i). This, along with dephasing and crosstalk, limits the fidelity of these gates to the 10−3 regime. Despite its longer gate duration, G3 achieves better performance in terms of motion-induced errors than G1 and G2, proving the validity of the mode decoherence protecting technique described in supplementary note 8. Finally, table 1 shows that G4 is the most robust w.r.t. experimental imperfections. This is reasonable since it uses a larger η and is an order of magnitude faster than the other gates. In particular, we find that G4 achieves infidelities on the 10−4 regime, mainly limited by residual qubit-boson entanglement caused by off-resonant harmonics and the spectator mode. Note that the influence of this error has been taken into account in supplementary note 3, where we also discuss the potential effects of micromotion.

Discussion

We have presented a DD sequence (TQXY16) based on the delivery of low-intensity π pulses that achieve faster-than-dispersive two-qubit gates. Without the need of any numerical optimisation, we have designed entangling gates which are robust to fluctuations in qubit frequencies and control fields, as well as to finite-pulse effects hindering a high-fidelity performance. In addition, we have demonstrated the versatility of our protocol to adopt forms that provide increased robustness against crosstalk and mode decoherence.

Our scheme is best suited for systems i) using longitudinal qubit-boson coupling with η 1, ii) where dephasing is the main source of qubit decoherence, and iii) where the Rabi frequencies Ω(t) are of the order (or far below) the mode frequencies ν. This is the case, e.g., for spin qubits coupled to microwave cavities5,6. In ref. 6, η ~ 10−2 and Ω/ν ~ 0.1. Superconducting qubit architectures exploiting longitudinal qubit-boson coupling has also been proposed3,4. Our method is also well suited for these systems when working with small η.

Finally, we tested the performance of our protocol in trapped ions coupled via static magnetic field gradients, where conditions (i), (ii), and (iii) are perfectly satisfied. Compared to existing multi-level schemes18,39, our method has the advantage of using only two levels, which lowers the experimental requirements. Compared to previous pulsed DD methods51, our method has the advantage of using realistic pulse intensities. Using detailed numerical simulations, we have obtained infidelities within the 10−3 threshold at state-of-the-art conditions, and in the 10−4 regime in near-future setups.

Methods

In Table 2 we show suitable values of b and c given the harmonic k. Also, we show the maximum value of d for which a physical pulse (i.e, fz(t) ≤ 1) can still be generated with the ansatz given in Eqs. (8), (9). For gates G1, G2 and G4, the selected values of d are 1.915, 0.933, and −0.321, respectively. The list of detunings used in gate G3 is \({\overrightarrow{\xi }}_{k}=(2\pi )\times [1.24,0.31,0.64,0.09,0.55,0.05,0.54,0.06,0.57,0.14,0.73,0.80]\) kHz.

Table 2 Suitable pulse parameters.

Our numerical simulations for dephasing consider an additional ± δω/2Sz term in Hs, where \(\delta \omega =\sqrt{2}/{T}_{2}^{* }\). In all three cases, each point is the average error obtained by a positive (e.g. ωj → ωj + δω) and a negative (e.g. ωj → ωj − δω) displacement.