Realization and Calibration of Continuously Parameterized Two-Qubit Gates on a Trapped-Ion Quantum Processor

TL;DR

The paper addresses the challenge of realizing continuously parameterized two-qubit Mølmer-Sørensen gates on a trapped-ion processor by developing a practical calibration framework that links hardware-driven amplitude scaling to a controllable entangling angle $\theta$. It introduces Gaussian pulse shaping, frequency-mode balancing, and a detailed treatment of fourth-order light shifts, complemented by a dynamic frame-rotation scheme to cancel residual phase errors, and a robust ZZ($\theta$) implementation via wrapper gates. Key contributions include precise AOM-saturation parameters $a_{\rm sat}$ and $\Xi$, empirical light-shift cancellation through optimized tone ratios $\zeta_{br}$, and a dual-frame approach to manage phases across gate sequences, enabling high-fidelity arbitrary-angle entangling operations. The work has broad impact by enabling more efficient circuit compilation and deeper quantum algorithms on trapped-ion hardware, with implications for both current and future quantum processors that rely on continuous entangling-angle gates.

Abstract

Continuously parameterized two-qubit gates are a key feature of state-of-the-art trapped-ion quantum processors as they have favorable error scalings and show distinct improvements in circuit performance over more restricted maximally entangling gatesets. In this work, we provide a comprehensive and pedagogical discussion on how to practically implement these continuously parameterized Mølmer-Sørensen gates on the Quantum Scientific Computing Open User Testbed (QSCOUT), a low-level trapped-ion processor. To generate the arbitrary entangling angles, $θ$, we simply scale the amplitude of light used to generate the entanglement. However, doing so requires careful consideration of amplifier saturation as well as the variable light shifts that result. As such, we describe a method to calibrate and cancel the dominant fourth-order effects, followed by a dynamic virtual phase advance during the gate to cancel any residual light shifts, and find a linear scaling between $θ$ and the residual light shift. Once, we have considered and calibrated these effects, we demonstrate performance improvement with decreasing $θ$. Finally, we describe nuances of hardware control to transform the XX-type interaction of the arbitrary-angle Mølmer-Sørensen gate into a phase-agnostic and crosstalk-mitigating ZZ interaction.

Figures (16)

• Figure 1: Illustration of sideband transitions and Raman detunings used in an $MS(\theta)$ gate. The hyperfine states $\ket{0}$ and $\ket{1}$ are separated with an energy splitting of $\omega_{\ket{1}-\ket{0}} = 12.643$ GHz. For a chain of N ions, there are also 2N radial motional modes $k$ where $\{k:0 ... 2N-1\}$, and each blue (red) mode is characterized by frequency $\nu_{k} (-\nu_{k})$. The Raman tones applied to the IA beams are symmetrically detuned from the blue (red) motional mode $k$ by $\delta_{k} (-\delta_{k})$. We also denote $\delta_c (-\delta_c)$ indicating the equivalent blue (red) detunings from the carrier transition such that $\delta_k = \delta_c - \nu_k$.
• Figure 2: The RF amplitude ($a$) applied to the global beam AOM is varied and the resulting Rabi oscillation fit to Eq. \ref{['eq:ampscan-fitfn']} in order to determine saturation parameters. Uncertainty interval shown is a $1\sigma$ Wilson score and are roughly the size of the points.
• Figure 3: Repeated applications of the MS gate show less dephasing per gate for small $\theta$. Ideal performance would show oscillation between $\ket{00}$ and $\ket{11}$ in steps of $\theta = \frac{\pi}{2}, \frac{\pi}{8},$ and $\frac{\pi}{32}$ for variable gate number (M) for the a), b) and c), respectively.
• Figure 4: Raman transitions and frequency combs to generate $MS(\theta)$. In a), the level structure for the $MS(\theta)$ Raman transitions. $\ket{n}$ is the initial phonon number in a particular motional mode, and red- and blue-sideband transitions are $\ket{n-1}$ and $\ket{n+1}$ respectively. The Raman transitions operate through virtual states to address the qubit motional modes. The grey arrow is the global beam $\omega_g$ acting as one leg of the transition, while the individual beams complete the transition as symmetric red-(blue-) detuned transitions $\omega_r$ ($\omega_b$). As the qubits never occupy the virtual intermediate states, the population oscillates between $\ket{00}$ and $\ket{11}$. b) A graphical representation of the three frequency combs involved in the $MS(\theta)$ gate. $f_{rep}$ indicates the repetition rate of the laser, $\delta_{AOM}/2\pi$ represents the median frequency shift of the red and blue combs relative to the global comb. These are then shifted further from $\delta_{AOM}/2\pi$ by $\pm\delta_c/2\pi$ (not denoted here). To generate the necessary transitions to drive the gate near $\omega_{\ket{1}-\ket{0}}/2\pi$, tooth $j$ of the global comb and teeth $j+105$ of the red and blue combs combine. Comb tooth separation of 105 is represented by a smaller number of comb teeth in the figure for graphical purposes.
• Figure 5: Light shift as function of blue/red amplitude ratio ($\zeta_{\rm br}$). a-d) Ramsey measurements with $\zeta_{\rm br} = 0.6, 0.9, 1.05, 1.1$, respectively. f) The accumulated phase per gate (teal diamonds) g) and coherence decay constants (blue circles) were extracted from Ramsey measurements for various blue/red ratios, using a cosine with a Gaussian decay profile. Each of the shaded regions correspond to data extracted from plots a-d.