Radial Fast Entangling Gates Under Micromotion in Trapped-Ion Quantum Computers

TL;DR

This work reframes RF micromotion from a detrimental effect into a controllable resource for fast, high-fidelity entangling gates in trapped-ion systems. By leveraging state-dependent kicks on the radial modes of a two-ion crystal and optimizing through the Generalised Pulse Group framework, the authors demonstrate sub-trap-period gate operation with fidelities approaching unity under favorable micromotion, while also analyzing robustness to SDK errors, timing/frequency noise, and finite laser repetition rates. The results establish the feasibility of MHz-rate, high-fidelity two-qubit gates that do not require cooling to the Lamb-Dicke regime, with clear routes toward scalable architectures in 2D/3D ion crystals and mixed-species implementations. These findings have practical implications for accelerating trapped-ion quantum computation and inform experimental efforts to harness micromotion as a design resource rather than a nuisance.

Abstract

Micromotion in radio-frequency ion traps is generally considered detrimental for quantum logic gates, and is typically minimized in state-of-the-art experiments. However, as a deterministic effect, it can be incorporated into quantum control frameworks aimed at designing high-fidelity quantum logic controls. In this work, we demonstrate that micromotion can be beneficial to the design of fast gates utilizing the radial modes of a two-ion crystal, particularly in the sub-trap-period regime where high-fidelity control sequences are identified with operation times ranging from hundreds of nanoseconds to microseconds. Through analysis of select fast gate solutions, we uncover the physical origin of micromotion enhancement and further study the induced gate error under experimental noises and control imperfections. This analysis establishes the feasibility of realising high-fidelity entangling gates in hundreds of nanoseconds using the micromotion-sensitive radial modes of trapped-ion crystals.

Figures (6)

• Figure 1: Impulsive fast two-qubit gates using the transverse (radial) modes of a two-ion crystal. (a) i. Ions are subjected to SDKs using focussed optical beams (shown in red) transverse to the null of the RF trap (dashed line). ii. Addressing both ions with SDKs excites the in-phase (common-motional) mode for same-spin two-qubit states, and the out-of-phase (stretch) mode for different-spin two-qubit states. (b) First-order stability region for the trapping parameters $(a,q)$ along the $x$-axis (blue shaded region) and the $y$-axis (red shaded region). The overlap between the two regions corresponds to stable trapping in all dimensions (hatched region). Trapping parameter values along the $x$-axis chosen to maintain the same trapping potential in the secular limit indicated with dots (diamonds) for an RF drive of $40\omega_0$ ($20\omega_0$), where $\omega_0$ is the secular frequency of the RF trap. (c) i. Example dynamics of fast gates of duration $\approx 0.7 (2\pi/\omega_0)$ for regimes of low-amplitude micromotion ($q_x=0.01$) and high-amplitude micromotion ($q_x=0.5$). ii. Arrival and magnitude of SDKs within the RF cycle in the high-micromotion regime, $q_x=0.5$, illustrating state-dependent impulses occur at various points in the RF cycle. iii. Histogram of the RF phases at the time of each SDK ($t_k$), i.e. $\phi^{(k)} = \phi_{\rm RF}+\Omega_{\rm RF}t_k$.
• Figure 2: Micromotion enhancement of radial fast gates in a RF trap with relative mode splitting $\chi=-1.4\times 10^{-2}$ and RF drive frequency i. $\Omega_{\rm RF}=20\omega_0$ and ii. $\Omega_{\rm RF}=40\omega_0$. (a) Infidelities of generalised fast gate solutions as a function of gate time in trap periods. For fast gates with gate times between $0.6 \le \omega_0 t_{\rm g} / (2 \pi) \le 1.0$ trap periods, gate errors are suppressed in environments with larger micromotion amplitudes by up to two orders of magnitude. (b) Number of SDKs per gate ($\mathcal{N}$) for gate solutions with stage-averaged fidelities exceeding 99.9%, with horizontal dashed lines indicating the minimum numer of SDKs to implement a maximally entangling gate in less than two trapping periods
• Figure 3: Analysis of gate solutions from Fig. \ref{['fig:MMEnhancement_InfRR']}ii. illustrating micromotion enhancement in gates with sub-trap-period duration $\omega_0 t_{\rm g} /(2 \pi) \approx 0.7$. (a) Presents the motional dynamics in terms of the dimensionless position and momentum of each mode, and the difference between the accumulated two-qubit phase $\Theta$ and its target value, $\Delta \Phi = \Theta - \pi/4$. The centre-of-mass mode couples solely to same-spin two-qubit states $\{\ket{\uparrow\uparrow},\ket{\downarrow\downarrow}\}$, and the breathing/stretch mode couples solely to the anti-aligned spin states $\{\ket{\downarrow\uparrow},\ket{\uparrow\downarrow}\}$. (b) i. The vectors $\mu^{(s)}_0(t_{\rm g},t_j)$ and $\mu^{(c)}_0(t_{\rm g},t_j)$ are plotted for all SDK timings $\{t_j\}$, and can be interpreted as high-frequency corrections to the time-dependent ion positions as compared to the secular limit (indicated by the dashed grey lines). ii. Similarly, the vectors $\kappa^{(s)}_0(t_{\rm g},t_j)$ and $\kappa^{(c)}_0(t_{\rm g},t_j)$ are the micromotion corrections to the time-dependent momentum of the ions, c.f Eq. \ref{['eq:ConditionEquations']}.
• Figure 4: Infidelities of fast gates evaluated with a finite repetition rate for relative RF frequencies of $\Omega_{\rm{RF}}/\omega_0 =$ (i) $20$ and (ii) $40$. (a) Fast gates of duration $\omega_0 t_{\rm g} / (2 \pi) = 1.0$ trap periods that are optimised and evaluated with a finite repetition rate $f_{\rm rep}$. For micromotion amplitudes of $q_x \ge 0.3$, $2\pi f_{\rm{rep}}/\omega_0=$ 100 -- 300 is sufficient to achieve 99.9%-fidelity fast gates for most environments. (b) Fast gate solutions optimized for a target duration $\omega_0 t_{\rm g} / (2 \pi) = 3.0$ trap periods.
• Figure 5: The effect of SDK pulse area errors on the fast gate mechanism. (a) Pulse area errors lead to imperfect population inversion from each $\pi$-pulse, which for an initial spin-state $\ket{\downarrow}$ can be visualised as diffusion of the state at the pole of the single-qubit Bloch sphere. Engineering a $\pi$-phase shift on the counter-propagating $\pi$-pulse -- akin to an 'echo' of the Bloch vector -- cancels population inversion errors, though motional state errors are not suppressed. (b) Degredation of gate fidelity as a function of $\pi$-pulse errors ($\epsilon_\pi$), with the vertical dashed line corresponding to state-of-the-art SDK population transfer errors of $\epsilon_\pi\approx 0.007$Johnson2017b. (b.ii) Shows the effect of pulse area imperfections in the worst-case scenario where population errors are not suppressed, with the gate fidelity degrading for larger imperfections according to Eq. \ref{['eq:Fidelity_PopErrors']}. (b.ii) Gate infidelity in the case where population errors are suppressed, estimated as the ensemble average over $10^4$ Monte-Carlo samples and a maximum of $m_{\rm max}=4$ errors per SDK sequence. (c) Reconstructed probability distribution function from the Monte-Carlo simulation for an SDK population error of $0.007$. Vertical lines indicate the state-averaged gate error in the absence of SDK errors.