Benchmarking trigonometric continuous-variable gate primitives with trapped ions

Hybrid quantum processors that combine discrete-variable (DV) qubits with continuous-variable (CV) bosonic oscillators (qumodes) offer a powerful framework for simulating complex physical systems. While standard CV operations are typically limited to polynomial functions of quadrature operators, many physical phenomena—such as lattice gauge theories, rotor models, and anharmonic dynamics—require periodic, non-polynomial interactions. This paper investigates the experimental implementation of trigonometric CV gates, specifically cosine gates, as a solution to this limitation.

The authors utilize the QSCOUT trapped-ion platform at Sandia National Laboratories, employing collective motional modes of three- and four-ion 171Yb+ chains as qumodes. By using hyperfine qubit states as ancillae, they realize trigonometric gates through hybrid qubit-qumode operations and conditional phase-space displacements. The study focuses on characterizing these primitives at the gate level, measuring Fock-space transition probabilities and comparing them against analytical models that account for thermal initialization and motional dephasing.

The researchers successfully implemented and benchmarked both one-qumode and two-qumode cosine gates. They derived analytical expressions for gate matrix elements, characteristic functions, and Wigner functions, providing a clear map between the experimental observables and the non-Gaussian structure generated by these gates. The results demonstrate that trigonometric gates can be effectively synthesized using finite-step Trotter circuits, establishing them as reusable building blocks for hybrid quantum algorithms that require intrinsically non-polynomial operations.

This work bridges the gap between theoretical proposals for trigonometric CV gates and their physical realization. By providing a quantitative foundation for these gates, the study enables more efficient simulations of bosonic systems that are otherwise difficult to approximate with low-order polynomials. This approach is particularly relevant for future quantum simulations of gauge fields and molecular dynamics, where periodic potentials are central to the physics.