Generally noise-resilient quantum gates for trapped-ions

TL;DR

The paper tackles the challenge of implementing high-fidelity entangling gates for trapped-ion quantum computers in multi-mode, thermally excited environments. It develops a driving-pattern framework that explicitly drives multiple motional sidebands across all modes, using a third-order expansion in the Lamb-Dicke parameter and a Magnus-based solution to tightly suppress unwanted spin-motion terms, yielding a gate with unitary $U = e^{i \phi_T \sum_{j\neq k} \sigma_y^{(j)} \sigma_y^{(k)}}$ while remaining robust to motional heating and detuning errors. Through analytical design and numerical simulations, the authors demonstrate superior fidelity compared to standard MS gates across multi-mode systems, even with significant motional occupation and frequency fluctuations, illustrating improved scalability for large ion chains. The scheme achieves strong resilience to both motional and frequency imperfections, offering practical advantages by functioning with excited motional states and without stringent cooling, thereby advancing the feasibility of scalable trapped-ion quantum computation.

Abstract

We present an entangling gate scheme for trapped-ion chains that achieves high-fidelity operations with excited motional states despite multiple error sources. Our approach incorporates all relevant motional modes and exhibits enhanced robustness against both motional heating effects and detuning errors, critical features for building robust and scalable trapped-ion quantum computers.

Figures (4)

• Figure 1: Energy level diagram for two identical ions and two motional modes. Note that only energy levels involved in the driving process are shown for simplicity. Solid blue (red) sidebands represent transitions with a phonon gain (loss) in motional mode 1. Dashed blue (red) sidebands represent similar processes for motional mode 2.
• Figure 2: Infidelity $1 - F$ of the entangling gate as a function of the initial Fock state occupation $n$ for a 4-qubit (left panel) and 2-qubit (right panel) systems. In the left panel, each data point represents the average infidelity across all qubit pairs, with the shaded area showing the range of values found across different pairs. The infidelity corresponds to the standard MS gate (green), the MS gate applied to all four modes (orange), and the present gate (blue). No motional heating ($\gamma = 0$) is assumed and different values of the coupling strength $\Lambda$ are shown with different markers. In the right panel, the impact of motional heating is evaluated for a fixed spin-motion coupling $\Lambda=0.1$. Decay rates are represented by colors ranging from light green (low) to purple (high) and marker styles differentiate the standard MS gate (crosses) from the proposed robust scheme (circles). In all simulations, the amplitude of the driving fields in the present scheme is limited to match the amplitude of the MS gate, ensuring a fair comparison.
• Figure 3: Combined impact of vibrational and spin frequency errors on the performance of the present entangling scheme (Robust) compared to the standard MS gate in a two-ion system. Each panel shows the ratio of MS gate infidelity to SC gate infidelity $I_{MS}/I_{Robust}$ for a specific initial motional state $\ket{n}$ and decay rate $\gamma$, with fixed coupling strength $\Lambda=0.1$. The $x$-axis represents (in log scale) errors in vibrational frequency $\epsilon_{\nu}$; while the $y$-axis depicts symmetric errors in spin frequencies $\epsilon_{\omega}$. Contour lines reveal actual fidelities achievable by each method. Solid lines enclose regions where infidelities remain below $10^{-4}$, dashed lines correspond to $10^{-3}$ and dotted lines to $10^{-2}$. Green lines represent the MS gate, while white ones represent the robust scheme.
• Figure 4: Infidelity $1-F$ for $N=2$ qubits as a function of the mean occupation $\bar{n}$ of a thermal state (on log scale). Increasing line thickness represents increasing values of the coupling parameter $\Lambda$ (defined in the main text), while the color line distinguishes between the conventional MS approach (orange) and the present robust gate scheme (blue). No motional heating or detuning errors are considered.