Benchmarking trigonometric continuous-variable gate primitives with trapped ions

Technical Summary: Benchmarking Trigonometric Continuous-Variable Gate Primitives with Trapped Ions

Problem Statement
Hybrid continuous-variable (CV) and discrete-variable (DV) quantum architectures offer a flexible route for processing quantum information, particularly for problems involving infinite-dimensional Hilbert spaces such as lattice gauge theories, rotor models, and molecular dynamics. While standard CV universality is typically formulated using finite polynomial functions of canonical quadratures ($\hat{x}, \hat{p}$), many physical systems of interest possess periodic structures (e.g., compact gauge variables, sine-Gordon interactions) that are not naturally represented by low-order polynomials. Approximating these periodic operators with polynomials often requires high-order terms and deep circuits. Recent theoretical work proposed trigonometric continuous-variable gate sets as natural primitives for these tasks. However, prior to this work, there was a lack of experimental gate-level characterization of these non-polynomial primitives, specifically regarding their performance, error profiles, and non-Gaussian generation capabilities on physical hardware.

Methodology
The authors experimentally demonstrated and benchmarked elementary trigonometric CV gates on the QSCOUT trapped-ion quantum platform using $^{171}\text{Yb}^+$ ion chains.

• Hardware: The experiments utilized collective motional modes of three-ion (for single-qumode) and four-ion (for two-qumode) chains as qumodes, with hyperfine states serving as ancillary qubits.
• Gate Implementation: The target gates, $e^{-i\theta \cos(c\hat{x})}$ (single-mode) and $e^{-i\theta \cos(c_1\hat{x}_1 + c_2\hat{x}_2)}$ (two-mode), were realized via finite-step Trotterization. These circuits were decomposed into elementary hybrid operations: conditional displacements (CD) and qubit rotations ($R_z$), implemented using red- and blue-sideband couplings.
• Readout: Since direct motional state readout is unavailable in trapped ions, the authors employed an ancillary "probe" qubit coupled to the motional mode. They reconstructed Fock-state populations by measuring the probe qubit after applying a blue-sideband (BSB) interaction, analyzing the resulting Rabi oscillations to extract transition probabilities.
• Error Modeling: To compare experimental data with theory, the authors developed an open-system simulation model incorporating:
  1. Initialization errors: Modeled as thermal states with mean phonon occupation $\bar{n}$.
  2. Dynamical errors: Dominated by motional dephasing (modeled via Lindblad dynamics with collapse operator $\hat{n}$).
  3. Calibration errors: Modeled as phase rotations in the conditional displacement due to detuning.
• Postselection: The circuits were postselected on the gate qubit remaining in the $|\uparrow\rangle$ state to filter out parity-violating components introduced by finite-step approximations.

Key Contributions

1. Experimental Realization: This work provides the first experimental implementation of elementary trigonometric CV gates (one- and two-qumode cosine gates) on a trapped-ion platform.
2. Gate-Level Characterization: Unlike previous works that used trigonometric gates as components of higher-level algorithms, this study isolates the primitives themselves. It characterizes their action via Fock-space transition probabilities, analyzing dependencies on gate parameters ($\theta, c$), Trotter step counts, and postselection.
3. Analytical Framework: The authors derived analytical expressions for the ideal gate matrix elements, characteristic functions, Wigner functions, and Wigner logarithmic negativity (mana). This connects the measured observables to the non-Gaussian phase-space structures generated by the gates.
4. Error Analysis: The study quantifies the impact of experimental imperfections, identifying motional dephasing as the predominant error source, followed by residual thermal initialization and displacement miscalibration.

Results

• Transition Probabilities: Experimental measurements of Fock-state transition probabilities for single-qumode circuits (with $c \in \{1, 2, 3\}$ and Trotter steps $i \in \{1, 2, 4, 8\}$) showed good agreement with open-system simulations. The data confirmed that increasing the number of Trotter steps drives the system toward the exact ideal gate behavior.
• Two-Mode Dynamics: For the two-qumode gate, mode-resolved marginal distributions demonstrated nontrivial population transfer. Crucially, the data showed an enhancement of odd-Fock-level populations in the two-mode case (compared to their suppression in the single-mode postselected data), serving as an indirect signature of the total-parity conservation inherent to the two-mode cosine gate.
• Phase-Space Diagnostics: The reconstructed characteristic function for a two-step Trotter approximation matched theoretical predictions when including dephasing ($\gamma \approx 0.19$), thermal occupation ($\bar{n} \approx 0.14$), and a displacement miscalibration angle ($\phi \approx -6^\circ$).
• Non-Gaussianity: Theoretical analysis confirmed that these gates generate Wigner negativity (mana) from Gaussian inputs, with the growth of negativity scaling with the gate parameters and approaching asymptotic bounds for large evolution times.

Significance
The paper establishes trigonometric CV gates as reusable, hardware-efficient building blocks for bosonic Hamiltonian simulations and hybrid quantum algorithms requiring intrinsically non-polynomial operations. By demonstrating that these gates can be implemented and characterized on current trapped-ion hardware, the work provides a quantitative foundation for future simulations of periodic bosonic systems, compact gauge fields, and rotor models. The authors note that while current limitations (such as motional dephasing and lack of direct intermode correlation readout) constrain the depth and complexity of simulations, the demonstrated primitives offer a natural alternative to polynomial decompositions for problems with periodic structures.