Ultra-fast two-qubit ion gate using sequences of resonant pulses

Trapped ions are one of the most accurate platforms for scalable quantum computation. Many ions can be loaded in Paul traps [1, 2], Penning traps [3] or possibly in other scalable architectures [4, 5]. Within these traps, qubits can be stored in long-lived atomic states, which are individually manipulated using lasers or microwaves to implement high-fidelity single-qubit operations and measurements. Finally, using the vibrational states of the ion crystal mediators, it is possible to implement universal multiqubit operations, such as the CNOT gate [6, 7], the Mølmer–Sørensen gate [8, 9], geometric phase gates [10, 11] or Toffoli operations [12, 13]. The actual realization of many of these gates depends on Raman transitions [14–17], with high-fidelity [16, 17] and excellent coherence properties [18]. In practice, fidelity and speed of two-qubit gates are still limiting the depth of actual computations, and prevent the development of scalable fault tolerant computation [19].

Those limitations in fidelity and speed are due to the use of highly detuned lasers, with lengthy control procedures and slow dynamics of the vibrational states. There exist faster gates based on faster and stronger acceleration of the ions [20–23]. Already a strong time-dependent optical lattice may result in high-fidelity gates that are shorter than a trap period [24], but are still constrained by available detuning, power and the Lamb-Dicke limit [25]. Another method is to excite an optical transition using picosecond laser pulses. A properly designed pulse train can create an arbitrarily fast two- or multi-qubit gate [20, 21]. However, as demonstrated in reference [26], it remains a technical challenge to have a strong momentum kick per pulse—a Raman transition might not provide enough momentum—and to switch directions in the pulse laser—which may induce additional sources of error and decoherence.

In this work we study the realization of fast high-fidelity quantum gates using a train of laser pulses that excite a resonant transition [27]. We focus on a simple scenario that only requires pulse-picking from a train of laser pulses with fixed strength and repetition rate. As example, we study a realistic pulsed scheme driving the 4S_1/2 → 4P_3/2 transition in ⁴⁰Ca⁺ [27]. We design the gate protocols with a two-stage global optimization that combines a continuous approximation with a discrete genetic algorithm for fine-tuning the pulse picking. We find many choices of pulses that implement highly entangling gates in a time comparable to the trap frequency, with very weak sensitivity to the pulse arrival time or the temperature of the motional states.

The manuscript is structured as follows: in section 2.1, we revisit the theory for implementing phase gates using spin-dependent kicks [6, 20, 21]. Section 2.2 presents a possible experimental setup and an optimized control protocol based on state-of-the-art kicking and control of trapped ions [27]. The results leading to the implementation of ultra-fast two-qubit gates are discussed in section 3. In section 4 we analyze and quantify the main source of errors in the design of such gates. Finally, we present prospective research lines related to this work in section 5.

2.1. Geometric phase by state-dependent kicks

Consider two ions of mass m in a 1D-harmonic potential of frequency ω, at positions x₁ and x₂. Using the center-of-mass (c) and stretch-mode (s) coordinates, x_c = (x₁ + x₂)/2 and x_s = x₂ − x₁ the free Hamiltonian for this system reads ${H}_{0}=\hslash {\omega }_{\text{c}}{a}_{\text{c}}^{{\dagger}}{a}_{\text{c}}+\hslash {\omega }_{\text{s}}{a}_{\text{s}}^{{\dagger}}{a}_{\text{s}}$. Here ω_c = ω and ${\omega }_{\text{s}}=\omega \sqrt{3}$ and ${a}_{\text{c},\text{s}}^{{\dagger}}\enspace \left({a}_{\text{c},\text{s}}\right)$ are the creation (annihilation) phonon operators for each mode. The ions interact with a laser beam that is resonant with an atomic transition. This interaction is modeled by the effective Hamiltonian

$$\begin{equation}{H}_{1}=\frac{{\Omega}\left(t\right)}{2}\left[{\sigma }_{1}^{{\dagger}}\enspace {\text{e}}^{\text{i}\hslash k{x}_{1}}+{\sigma }_{2}^{{\dagger}}\enspace {\text{e}}^{\text{i}\hslash k{x}_{2}}+\text{H.c.}\right].\end{equation} \tag{ 1 }$$

The pseudospin ladder operator ${\sigma }_{i}^{{\dagger}}$ connects the ground and excited states of the ith ion—in this setup, the 4S_1/2 and 4P_3/2 states of ⁴⁰Ca⁺. The interaction accounts for processes where the ion absorbs or emits a photon, changing its internal state and also modifying its momentum by ±ℏk. The sign of k depends on the direction of the laser and whether the photon is emitted or absorbed. Without loss of generality, we will forego assuming that a single laser simultaneously addresses both ions with a common Rabi frequency Ω(t).

Our gate protocols [20] alternate free evolution H = H₀, where the laser is switched off (Ω = 0), with a very fast, pulsed interaction kicking the ion. As shown in figure 1(b), we assume pairs of pulses coming from counter-propagating directions such the first pulse activates the selected transition and the second coherently de-excites the ion back to the lower energy configuration. The Rabi frequency Ω(t) and the duration of each pulse δt satisfy ${\int }_{0}^{\delta t}{\Omega}\left(\tau \right)\mathrm{d}\tau =\pi$ and δt ≪ 2π/ω. The pulses kick the ions, accelerating them along the same direction. In between each pair of kicks, the ions oscillate freely in the trap. The combination of both effects can be modeled analytically. The evolution operator for N pulses is $\mathcal{U}={\mathcal{U}}_{\text{c}}{\mathcal{U}}_{\text{s}}$ with ${\mathcal{U}}_{\text{c},\text{s}}={\prod }_{n=1}^{N}{U}_{\text{c},\text{s}}\left({t}_{n},z\right)$ and

$$\begin{equation}\begin{aligned}\hfill {U}_{\text{c}}\left({t}_{n},z\right)& ={\text{e}}^{\text{i}{\alpha }_{\text{c}}\left({\sigma }_{1}^{z}+{\sigma }_{2}^{z}\right)\left({a}_{\text{c}}+{a}_{\text{c}}^{{\dagger}}\right)}\enspace {\text{e}}^{\text{i}{\omega }_{\text{c}}{t}_{n}{a}_{\text{c}}^{{\dagger}}{a}_{\text{c}}}\hfill \\ \hfill {U}_{\text{s}}\left({t}_{n},z\right)& ={\text{e}}^{\text{i}{\alpha }_{\text{s}}\left({\sigma }_{1}^{z}-{\sigma }_{2}^{z}\right)\left({a}_{\text{s}}+{a}_{\text{s}}^{{\dagger}}\right)}\enspace {\text{e}}^{\text{i}{\omega }_{\text{s}}{t}_{n}{a}_{\text{s}}^{{\dagger}}{a}_{\text{s}}}.\hfill \end{aligned}\end{equation} \tag{ 2 }$$

The amplitudes α_c = ηz/2^3/2 and α_s = α_c/3^1/4 depend on the Lamb-Dicke parameter $\eta =\sqrt{\frac{\hslash }{2m\omega }}k$. The sign z = ±1 indicates the net orientation of the combined kick. It depends on the relative order of pulses within each pair: z = +1 if the first pulse comes from the left and the second from the right, z = −1 in the opposite case. In the setup from figure 1(b), the sign z is fixed to one of these values throughout the experiment.

Zoom In Zoom Out Reset image size

A kicking sequence with N pulses displaces the Fock operators a_c,s by a complex number A_c,s that depends on the collective state of the ions

$$\begin{equation}\begin{aligned}\hfill {a}_{c}& \to \hfill & \hfill {a}_{\text{c}}+{A}_{\text{c}}={a}_{\text{c}}+\text{i}\left({\sigma }_{1}^{z}+{\sigma }_{2}^{z}\right){\alpha }_{\text{c}}\sum _{n=1}^{N}{\text{e}}^{-\text{i}{\omega }_{\text{c}}{t}_{n}}\\ \hfill {a}_{\text{s}}& \to \hfill & \hfill {a}_{\text{s}}+{A}_{\text{s}}={a}_{\text{s}}+\text{i}\left({\sigma }_{1}^{z}-{\sigma }_{2}^{z}\right){\alpha }_{\text{s}}\sum _{n=1}^{N}{\text{e}}^{-\text{i}{\omega }_{\text{s}}{t}_{n}}.\end{aligned}\end{equation} \tag{ 3 }$$

In phase space (⟨x_c,s⟩, ⟨p_c,s⟩), the normal modes follow polygonal orbits (cf figure 2). The edges of the polygon all have uniform length ∼α_c,s and the angles between edges are determined by the arrival times of the kicks ω_ct_n. A different frequency ω'_c rotates the same amount if the pulse is allocated at t'_n = t_nω_c/ω'_c leading to the same angle between consecutive edges. A perfect gate restores the motional state of the ion A_c = A_s = 0, bringing them back to their original oscillator trajectories

$$\begin{equation}\sum _{n=1}^{N}{\text{e}}^{\text{i}\omega {t}_{n}}=\sum _{n=1}^{N}{\text{e}}^{\text{i}\sqrt{3}\omega {t}_{n}}=0,\end{equation} \tag{ 4 }$$

and closing the orbits. Under these conditions, after a time T the evolution operator becomes [20]

$$\begin{equation}\mathcal{U}\left(\phi ,T\right)={\text{e}}^{-\text{i}\phi {\sigma }_{1}^{z}{\sigma }_{2}^{z}}\enspace {\text{e}}^{\text{i}{\omega }_{\text{c}}T{a}_{\text{c}}^{{\dagger}}{a}_{\text{c}}}\enspace {\text{e}}^{\text{i}{\omega }_{\text{s}}T{a}_{\text{s}}^{{\dagger}}{a}_{\text{s}}}.\end{equation} \tag{ 5 }$$

This is equivalent to free evolution in the trap, combined with a global phase ϕ that does not depend on the motional state,

$$\begin{align}\hfill \phi & ={\alpha }_{\text{c}}^{2}\sum _{j=2}^{N}\sum _{k=1}^{j-1}\left[\frac{\mathrm{sin}\left(\sqrt{3}\omega \left({t}_{j}-{t}_{k}\right)\right)}{\sqrt{3}}-\mathrm{sin}\left(\omega \left({t}_{j}-{t}_{k}\right)\right)\right]\hfill \\ \hfill & =:{\alpha }_{\text{c}}^{2}\varphi .\hfill \end{align} \tag{ 6 }$$

When equation (4) holds and the total phase satisfies

$$\begin{equation}\phi =\pi /4+2n\pi \quad n\in \mathbb{Z},\end{equation} \tag{ 7 }$$

the combined evolution implements a control-phase gate on the internal state of the ions. The set of equations that determines the operation of the gate are solved in two steps. First, calculating the allocation positions x_n = ωt_n that fulfill equation (4) and determines the value φ, see equation (6). The determination of x_n leads to equivalent different pulse sequences where the new pulse arrival times are re-scaled t'_n = t_nω/ω' but the value of φ holds. Second, we change the trapping frequency and scale the arrival times to calibrate the Lamb-Dicke parameter and the amplitude α_c, to implement a control-phase gate (7). The new trapping frequencies are given by

$$\begin{equation}\omega =\frac{\hslash {k}^{2}\varphi }{16m\left(\pi /4+2n\pi \right)}.\end{equation} \tag{ 8 }$$

Given an optimal pulse sequence and closed phase-space trajectory (4), there are several trapping frequencies (8) that implement the same gate. These frequencies are found to overshoot the accumulated phase, exceeding the minimum value π/4 by an integer multiple n of 2π. As we will see later, a clever choice of overshoot factors n allows us to fine tune the frequency within experimentally reasonable bounds.

Zoom In Zoom Out Reset image size

The control scheme presented above relies on the interaction Hamiltonian (1), characteristic in trapped ions setups. Such interaction is independent of the considered species. However, the selected resonant transition must be able to displace the ions and accumulate a phase. As the displacements in phase space are proportional to the wavenumber k the scheme is suitable for optical transitions, but less energetic transitions such microwave controls do not provide enough momentum to the ion to accumulate the requested phase.

2.2. Experimental setup and parameters

2.3. Design and optimization of a discrete control

Section 2.1 established that a control-phase gate can be implemented by a sequence of pulse pairs that satisfies equations (4) and (7). In this work we address the design of the pulse sequence as two consecutive tasks: (i) find a set of pulse arrival times ${\left\{{t}_{n}\right\}}_{n=1}^{N}$ that meet conditions (4), (ii) fine tune the trapping frequency ω so that the total acquired phase is compatible with the implementation of a control-phase gate (7).

The first task decides our pulse picking strategy. To measure the goodness of the sequences fulfilling (4) we define the gate error that accounts for the ability to restore the motional state of the ions. Note that the total Hamiltonian H = H₀ + H_I is quadratic in the phonon operators for both the c- and s-modes. Therefore, a Gaussian state remains Gaussian after the whole kicking sequence. The lack of overlap between the initial state located at the origin (0, 0) and the final state produced by the pulse sequence located at (|A_c|, |A_s|) defines which in the small displacement limit, i.e. |A_c|, |A_s| → 0, takes the form

$$\begin{equation}{\epsilon}=\vert {A}_{\text{c}}{\vert }^{2}+\vert {A}_{\text{s}}{\vert }^{2},\end{equation} \tag{ 9 }$$

that implicitly depends on the amount of applied pulses.

The minimization of implies solving a combinatorial optimization problem, where the times t_n = k_n × t_R + t₁ are spaced by integer multiples of the laser pulse period t_R. In phase space, equation (4) ensures closed polygonal trajectories (cf figure 2), with angles between edges proportional to ω_c,s(t_n+1 − t_n) and edge lengths proportional to α_c,s. The area enclosed by the polygons determines the geometric phase ϕ. By adjusting the trap frequency ω_c, we tune the kick strengths α_c = α_c(ω), scaling the whole trajectory in phase space. This allows us to fine tune the accumulated phase (7) to the desired value, modulo an irrelevant integer n.

The design of the pulse sequence is a hard combinatorial optimization problem, where we pick N pulses out of a much longer train. To avoid the exponential complexity in this search, we find good approximate solutions using a two-stage method. The first stage is a regular minimization of the gate error over a set of N continuous arrival times ${t}_{n}\in \mathbb{R}$. We apply a standard algorithm to minimize the gate error (9) over a set of N variables, using K_seed random initial seeds $\overrightarrow {t}\equiv \left\{{t}_{1},{t}_{2},\dots ,{t}_{N}\right\}$ of ordered times t_n+1 > t_n and t_N ⩽ 2π/ω. We select a subset of K_opt controls maximizing the phase ϕ, rejecting slow solutions T > 2 × 2π/ω. In the second stage of this process, we introduce the finite repetition of the laser. We round the K_opt continuous solutions to the nearest laser pulses, which are spaced by a multiple of t_R = 1/R. These discrete protocols introduce a possible timing error ξ = |t_n − nt_R|. Instead of just a local minimization of each ξ, we minimize the global error (9) with a genetic algorithm that fine tunes the pulse allocation.

A genetic algorithm [28, 29] is a discrete optimizer that builds on the concept of natural selection, where solutions are iteratively improved using biologically inspired operations such as selection, crossover and mutation. In each iteration, a set or population is composed by several candidate solutions (called individuals) and are evolved towards better solutions based on a fitness function—the cost function to be optimized. On each iteration or generation, the algorithm selects a subset of individuals that maximize the fitness. These so called parents merge and mutate, giving rise to new solutions, the offspring that form the next generation. This process of selection and reproduction is repeated until the fitness reaches the desired optimal value, selected by a user-defined tolerance, or until the maximum number of generations is reached.

To bring our problem into this form, we take the N continuous times t_n and find out the M_max closest pulses within the discrete sequence created by the laser (cf figure 3(a)). We then encode a solution as a chromosome with N × M_max genes. Each gene is a bit that becomes 1 when the corresponding pulse is selected or 0 otherwise (cf figure 3(b)). Taking the optimal solution provided by the regular minimization we generate a seed. Our initial population is formed by K_ind individuals that results from rounding to nearest and randomly distributing M active genes within the M_max discrete positions at each ${\left\{{t}_{n}\right\}}_{n=1}^{N}$. As result each individual has N × M active genes, indicating that we have N groups of M pulses around the times t_n. From this pool, we select a subset of K_p parents as the individuals exhibiting the best value of the fitness function (9). Parents mate in pairs and each offspring receives part of its chromosome from the first parent and the rest from the second. In our algorithm this proportion is 50/50 made at the middle of each parent chromosome, see figure 3(b). If an offspring improves the fitness function it joins the parents to constitute the new population for the next generation. If not, a mutation is produced on the offspring creating random variations in the chromosome. To preserve the total number of N × M pulses, we randomly swap the values of two genes from a M_max sequence placed around one of the times t_n, see figure 3(b). These mutants join the new population, irrespective of their value of the fitness function, and the whole process is repeated. This workflow, sketched in figure 3, is repeated over K_ite generations. At the end, we select the individual that produces the best value of the fitness function, thereby minimizing the error equation (9).

Zoom In Zoom Out Reset image size

For the two-stage optimization strategy the regular minimization of equation (9) represents the main computational cost that grows with N as the number of unknown variables t_n to be optimized becomes larger. Once the solutions fulfilling the commensurability equation (4) are obtained and fed into the genetic algorithm, the discrete minimization in contrast does not involve significant computational resources. The optimizer only needs to evaluate equation (9) for different sequences that are obtained by replacing part of the genes 0 ↔ 1 using the crossover and mutation processes explained above. Based on the randomness in the generation of new solutions, for our simulations, the algorithm needs a computational time $\mathcal{O}\left(\text{secs}\right)$ to provide a minimization improvement of the gate error that range from 0, where the initial seed corresponding to a local minimization of ξ represents the optimal solution, to cases where the genetic algorithm improves 2–3 orders of magnitudes the solution.

As mentioned above, our simulations consider a scenario where the direction of the kicks is fixed. This happens when a single pulse picker is connected to an interferometric setup, creating pairs of pulses all arriving with the same relative delay [cf figure 1]—e.g. the left pulse always excites the ion and the right pulse immediately de-excites it, setting z = +1. Scenarios where both the relative direction and the Lamb-Dicke parameter are tuned have been considered before [20, 22, 30, 31] leading to different degrees of controllability and thus to different gate times. Here we will show that, despite our experimentally-motivated constraints [27], it is possible to implement control-phase gates in a time shorter than the trap period T < 2π/ω.

Before illustrating the final protocols, figure 4 shows the intermediate results obtained when solving the commensurability equation (4) with continuous variables ${\left\{{t}_{n}\right\}}_{n=1}^{N}$. Note how for a fixed number of pulses N there exist multiple schemes that restore the motional state of the ions and implement a control-phase gate. Out of those combinations we select those that maximize the ratio $\varphi =\vert \phi /{\alpha }_{\text{c}}^{2}\vert$, and feed them to the genetic algorithm to create discrete pulse sequences. Note that the two-qubit phase depends on α_c and therefore on the trap frequency ω_c. The preselection of continuous protocols with large φ provides a broader choice of pulse sequences and frequencies (8) that satisfy both the experimental restriction ω ∈ [ω_min, ω_max] and the phase relation equation (7), with either n = 0 or n ≠ 0 (overshooting).

Zoom In Zoom Out Reset image size

A numerical fit shows that for the sequences maximizing the accumulated phase, phase grows with the number of pulses in the discrete protocol as |φ| ∝ N^0.6, (cf figure 5(a)), while the duration of the gate remains below T ≲ 1.055 × 2π/ω and is close to the sequences minimizing the gate time T [cf figures 4(a), 5(b), and 7(a)]. When solving equation (4) for x_n this fit avoids the re-scaling of the arrival times t_n since it allows to fine tune in advance the frequency (8) leading to a control-phase gate as φ is known. The error introduced by the finite repetition rate is also negligible, figure 5(c) shows the theoretical error for one protocol consisting of N = 6 pulses. The global optimization of performed with the genetic algorithm (blue-solid) provides pulses sequences that minimize the gate error several orders of magnitude compared to the local minimization of ξ that rounds the continuous solutions allocating the pulses to the nearest discrete time nt_R compatible with the source repetition rate (red-dashed). A laser with a repetition rate R ≳ 1 GHz already produces an ultra-fast two-qubit gate with fidelity above 99.999%.

Zoom In Zoom Out Reset image size

As shown in figure 5(b), a short sequence with N = 4 pulses produces very fast gates T < 2π/ω, but with a small acquired phase. We may increase the accumulated φ, concentrating M pulses around each of the N kicking times [cf figure 3(a)]. This maintains the shape of the orbits, scaling the edges by a factor of M [cf figure 5(a)]. As shown in figures 5(a) and (b), the duration of the gate is preserved and the accumulated phase grows with the area as φ ∝ M². Note that, since the phase increases in discrete steps, we still need to fine tune the trap frequency to match the desired control-phase gate. Figures 6(a) and (b) show that this is possible for realistic trapping frequencies [27], using different multiplication factors M. Due to the existence of multiple compatible frequencies in the range ω ∈ 2π × [78 kHz, 2 MHz] depending on the overshooting factor n, in figure 6(a) we show the frequencies (8) that implement a control-phase gate and which are closest to a specific reference frequency ω ∼ 2π × 0.82 MHz [27] chosen as an example. As φ grows with both N and M, figure 6(b) shows that the specific frequency is achievable compensating the phase with a large overshooting factor n.

Zoom In Zoom Out Reset image size

This resonant pulse scheme is intrinsically limited to kicks along the same direction. In the rotating reference frame, two consecutive segments (kicks) are separated an angle that depends on the waiting time and the direction of the kicks. In our scheme, the angle always grows in the same direction, due to kicks having the same orientation. Consequently, both the c- and s-modes describe closed polyhedral orbits—with edges that have the same length—only when the pulse sequence takes about a trap period, T ∼ 2π/ω = T_ω, see equation (4). If we preserve this approximate duration for sequences with large number of pulses N, then angles decrease and the asymptotic orbits (N ≫ 1) approximate distorted ellipses that have similar shapes, but grow in area with N, see figure 7. As we showed, sequences with small numbers of pulses can improve over this asymptotic behavior, producing gates in shorter times T < T_ω, independently of the concatenated amount of M pulses around t_n.

Zoom In Zoom Out Reset image size

We could even improve further if we can control the magnitude of the acceleration of each kick, or the orientation. For example, if we can switch the order in which the pulses arrive, the momentum kick can take different signs ±1 at each kick time, producing sequences where the gate time decreases with the number of pulses N. In this context, the angles are free to evolve and gates T ≪ T_ω are feasible. Additionally, if we provide even more controllability, for example tilting the laser with respect to the axial trapping axis [20] leads to even faster gates. This extra-controllability is still an experimental challenge and may induce other sources of error.

We have presented a route for the implementation of ultra-fast T < 2π/ω quantum gates using a train of laser pulses that are resonant with the transition frequency of a trapped ion. In these protocols, the motional state of the ion is almost perfectly restored with a high-fidelity ∼ 10⁻⁹–10⁻⁷ using source generators with a constant repetition rate R ∼ 5 GHz. When implementing these protocols, actual experiments will suffer from imperfections in the control of the ion, due to spontaneous emission during the time that the ion remains in the excited state 4P_3/2 (i.e., during pulses and waiting time), and due to intensity fluctuations in the pulses.

A trivial model to quantify the spontaneous emission errors, giving an upper bound on them, is to write a density matrix

$$\begin{equation}\rho =\left(1-{P}_{\text{err}}\right)\vert \psi \rangle \langle \psi \vert +{P}_{\text{err}}\vert g\rangle \langle g\vert \end{equation} \tag{ 10 }$$

where |g⟩ is a fictitious state accumulating the probability that an error took place. The fidelity is given by P_errF₀, where F₀ is the fidelity of the gate implemented by ideal kicks. In this model, P_err feeds from spontaneous emission effects: we assume that whenever the emission takes place, the experiment must be repeated. The probability that the ion is in an excited state |e⟩ is

$$\begin{equation}\frac{\mathrm{d}{P}_{\text{ok}}}{\mathrm{d}t}=-\gamma \vert \langle e\vert \psi \left(t\right)\rangle {\vert }^{2}\left(1-{P}_{\text{err}}\right),\end{equation} \tag{ 11 }$$

with P_ok + P_err = 1. The decay rate γ = 1/t_γ is inversely proportional to the lifetime t_γ. The solution to this problem is

$$\begin{equation}\begin{aligned}\hfill {{\epsilon}}_{\gamma }& =1-{P}_{\text{ok}}\left(T\right),\hfill \\ \hfill {P}_{\text{ok}}\left(T\right)& =\mathrm{exp}\left(-\gamma {\int }_{0}^{T}\vert \langle e\vert \psi \left(t\right)\rangle {\vert }^{2}\mathrm{d}t\right){P}_{\text{ok}}\left(0\right).\hfill \end{aligned}\end{equation} \tag{ 12 }$$

In a very crude scenario, we upper bound the error probability, assuming that the ion is in the excited state from the beginning of the exciting pulse, to the end of the following, that is T_e ≃ δt + τ,

$$\begin{equation}{{\epsilon}}_{\gamma }=1-\mathrm{exp}\left(-\gamma {T}_{\text{e}}\right)\simeq \gamma {T}_{\text{e}}.\end{equation} \tag{ 13 }$$

In the experimental setup from figure 1(b), the waiting time τ between counter-propagating pulses is controlled by the relative length of the optical paths. The minimum separation is given by the pulse duration, τ ≳ δt to avoid interference. In our system, the excited state 4P_3/2 has a lifetime t_γ = 6.9 ns and T_e ≃ 1 ps [27, 32] leading to errors ${{\epsilon}}_{\gamma }\sim \mathcal{O}\left(\delta t/{t}_{\gamma }\right)\simeq 1.4{\times}1{0}^{-4}$. For a sequence containing N kicks, the infidelity of the gate is approximately ${{\epsilon}}_{\gamma }^{\text{gate}}=1-{\left(1-{{\epsilon}}_{\gamma }\right)}^{N}\sim \mathcal{O}\left(N\delta t/{t}_{\gamma }\right)$.

We can also quantify the errors _A due to fluctuations in the π−pulses. For a general pulse shape $\theta {=\int }_{0}^{\delta t}{\Omega}\left(\tau \right)\mathrm{d}\tau$ the unitary generated by the interaction Hamiltonian (1) is

$$\begin{equation}{\hat{U}}_{k}=\left(c-\text{i}s{\hat{\sigma }}_{1}^{x}\enspace {\text{e}}^{\text{i}k{x}_{1}{\hat{\sigma }}_{1}^{z}}\right)\left(c-\text{i}s{\hat{\sigma }}_{2}^{x}\enspace {\text{e}}^{\text{i}k{x}_{2}{\hat{\sigma }}_{2}^{z}}\right)\end{equation} \tag{ 14 }$$

with c = cos(θ/2) and s = sin(θ/2). A perfect π−pulse, i.e. θ = π, generates the unitary ${\hat{U}}_{\text{kick}}=-{\hat{\sigma }}_{1}^{x}{\hat{\sigma }}_{2}^{x}\enspace {\text{e}}^{\text{i}k\left({x}_{1}{\hat{\sigma }}_{1}^{z}+{x}_{2}{\hat{\sigma }}_{2}^{2}\right)}$. In order to quantify the errors due to area fluctuations when combining two counter-propagating pulses ${\hat{U}}_{\text{pair}}={\hat{U}}_{k}{\hat{U}}_{-k}$ we consider small fluctuations $\pi +{\Delta}\theta {=\int }_{0}^{\delta t}{\Omega}\left(\tau \right)\mathrm{d}\tau$ (with Δθ → 0) in the pulse area. Retaining the first order terms in Δθ an imperfect pair of counter-propagating pulses generates the transformation

$$\begin{equation}{\hat{U}}_{\text{pair}}=\left(1-{\Delta}{\theta }^{2}/2\right){\hat{U}}_{0}-{\Delta}\theta {\hat{U}}_{e}^{1}-{\Delta}{\theta }^{2}{\hat{U}}_{e}^{2}+\mathcal{O}\left({\Delta}{\theta }^{3}\right)\end{equation} \tag{ 15 }$$

with ${\hat{U}}_{0}={\text{e}}^{-2\text{i}k\left({x}_{1}{\hat{\sigma }}_{1}^{z}+{x}_{2}{\hat{\sigma }}_{2}^{z}\right)}$ the optimal unitary generated by two perfect counter-propagating ${\hat{U}}_{\text{kick}}$ pulses, and ${\hat{U}}_{e}^{1}=\text{i}\left({\sigma }_{1}^{x}\enspace \mathrm{cos}\left(k{x}_{1}\right){\text{e}}^{-2\text{i}k{\hat{\sigma }}_{2}^{z}{x}_{2}}+{\sigma }_{2}^{x}\enspace \mathrm{cos}\left(k{x}_{2}\right){\text{e}}^{-2\text{i}k{\hat{\sigma }}_{2}^{z}{x}_{1}}\right)$ and ${\hat{U}}_{e}^{2}=\mathrm{cos}\left(k{x}_{1}\right)\mathrm{cos}\left(k{x}_{2}\right){\hat{\sigma }}_{1}^{x}{\hat{\sigma }}_{2}^{x}+\left({\text{e}}^{\text{i}k{x}_{1}{\hat{\sigma }}_{1}^{z}}+{\text{e}}^{\text{i}k{x}_{z}{\hat{\sigma }}_{2}^{z}}\right)/4$ accounting for unrestored and incorrect motional dynamics. The total unitary of a gate can be approximated by the product of N pairs

$$\begin{equation}{\hat{U}}_{\text{gate}}\approx \left(1-{\Delta}{\theta }^{2}N/2\right){\hat{U}}_{N}-N{\Delta}\theta {\hat{U}}_{\text{err}}\end{equation} \tag{ 16 }$$

with ${\hat{U}}_{N}={\hat{U}}_{0}^{N}$ and collecting all the errant dynamics in ${\hat{U}}_{\text{err}}$ that it is assumed orthogonal to the ideal unitary ${\hat{U}}_{0}$. This is a conservative approximation that neglects terms that result in an incorrect motional state, but includes those that correctly restore the internal state [31]. For any initial state |ψ⟩ of the computational basis we can compare the dynamics of the optimal gate ${\hat{U}}_{\text{opt}}$ with the one generated by ${\hat{U}}_{\text{gate}}$. To this end we estimate the fidelity

$$\begin{equation}F=\vert \langle \psi \vert {\hat{U}}_{\text{opt}}^{{\dagger}}{\hat{U}}_{\text{gate}}\vert \psi \rangle {\vert }^{2}=\left(1-N{{\epsilon}}_{\text{A}}+{N}^{2}{{\epsilon}}_{\text{A}}^{2}/4\right){F}_{0},\end{equation} \tag{ 17 }$$

with _A = Δθ². The magnitude of the fluctuations _A depends on the specific characteristics of the laser pulses. In real setups with picosecond pulses [27, 33] these fluctuations are found to induce errors of around _A ∝ ΔI/I ∼ 10⁻³. However, these intensity fluctuations can be reduced experimentally, using methods such as adiabatic rapid passages with chirped laser pulses [34–36].

Our analysis shows that it is possible to engineer ultra-fast gates T < 2π/ω, using pulse picking strategies for an experimentally relevant setup [27, 32]. Current two-qubit Mølmer–Sørensen gate operations require a duration of $\bar{T}\sim 40\enspace \mu \text{s}$ for entangling two qubits at a trapping frequency ω ≃ 2π × 1.4 MHz [19]. Compared to these numbers, our scheme can provide a speedup factor $\bar{T}/T{ >}50$, for a conservative gate duration T ∼ 2π/ω.

Our investigation leaves some open questions, to be addressed in later works. The first one concerns the robustness of the protocol with respect to intensity fluctuations and spontaneous emission. Both problems may be overcome if we use STIRAP techniques [37–39], to induce excitation between the 4S_1/2 and a metastable state, such as 3D_5/2 or 3D_3/2. Experimentally, ⁴⁰Ca⁺ ions have been robustly manipulated using such techniques [40–42]. For our proposal, we could detune the pulsed laser exciting the 4S_1/2 → 4P_3/2 transition and combine it with another pulse connecting the 4P_3/2 ↔ 3D_5/2 states. These improvements can be supplemented with pulse shaping techniques [43–45], to minimize the AC Stark-shifts and dephasing associated with high-intensity pulses.

A second, more pressing question, concerns the parallelizability and scalability of our pulsed schemes. Recent works have addressed theoretically [21, 46, 47] and demonstrated experimentally [48, 49] the simultaneous implementation of arbitrary two-qubit gates among a subset or all pairs of K ions in a trap. We can use our two-step protocol to perform this task with significant speed ups. As in this work, the first step is a continuous optimization of the desired gate operation, subject to the now 2K dynamical constraints [21]. The resulting pulsed protocol is fine tuned with our genetic algorithm, to match the repetition rate of the laser. The process has an increased optimization cost, but the multi-qubit gates do not seem to take longer than the two-qubit ones [21].

Current ion trap quantum computers are able to run programs with up to several hundred one and two-qubit operations [50]. We expect that these methods and subsequent improvements ion trap quantum computers will be able to improve at least one, if not two orders of magnitude, leading to an increased quantum volume in NISQ devices. Moreover, the estimated ideal gate fidelities are compatible with existing error thresholds [51], which makes these methods a promising alternative for implementing fault-tolerant computation schemes [19].

We acknowledge support from Project PGC2018-094792-B-I00 (MCIU/AEI/FEDER,UE), CSIC Research Platform PTI-001, and CAM/FEDER Project No. S2018/TCS-4342 (QUITEMAD-CM). Authors also acknowledge support by the Institut für Quanteninformation GmbH.