Single-Qubit Gates Beyond the Rotating-Wave Approximation for Strongly Anharmonic Low-Frequency Qubits

Single-qubit gates are in many quantum platforms applied using a linear drive resonant with the qubit transition frequency which is often theoretically described within the rotating-wave approximation (RWA). However, for fast gates on low-frequency qubits, the RWA may not hold and we need to consider the contribution from counter-rotating terms to the qubit dynamics. The inclusion of counter-rotating terms into the theoretical description gives rise to two challenges. Firstly, it becomes challenging to analytically calculate the time evolution as the Hamiltonian is no longer self-commuting. Moreover, the time evolution now depends on the carrier phase such that, in general, every operation in a sequence of gates is different. In this work, we derive and verify a correction to the drive pulses that minimizes the effect of these counter-rotating terms in a two-level system. We then derive a second correction term that arises from non-computational levels for a strongly anharmonic system. We experimentally implement these correction terms on a fluxonium superconducting qubit, which is an example of a strongly anharmonic, low-frequency qubit for which the RWA may not hold, and demonstrate how fast, high-fidelity single-qubit gates can be achieved without the need for additional hardware complexities.

💡 Research Summary

The paper addresses a fundamental challenge in quantum control: how to implement fast, high‑fidelity single‑qubit gates on low‑frequency, strongly anharmonic superconducting qubits when the rotating‑wave approximation (RWA) breaks down. In conventional platforms—NV centers, quantum dots, trapped ions, and most superconducting circuits—gate operations are designed using a linear drive resonant with the qubit transition, and the counter‑rotating (CR) terms are discarded under the RWA. This approximation holds only when the drive amplitude Ω is much smaller than the drive frequency ω_d. For low‑frequency qubits such as fluxonium (transition frequencies in the 100 MHz–1 GHz range) and for fast gates (10–100 ns), Ω becomes comparable to ω_d, making the CR terms non‑negligible. The authors identify two major complications arising from the inclusion of CR terms: (i) the Hamiltonian no longer commutes with itself at different times, preventing a simple analytical solution for the time evolution; (ii) the evolution now depends explicitly on the carrier phase ϕ, so each gate in a sequence would require a different pulse shape if the phase changes.

To overcome these issues, the authors employ a Magnus‑Taylor expansion of the time‑ordered evolution operator, dividing the total gate time into “Magnus intervals” of length t_c = π/ω_d (the period of the fast oscillating terms). By matching the evolution generated by the full non‑RWA Hamiltonian to that of the ideal RWA Hamiltonian over an integer number of Magnus intervals, they derive corrected pulse parameters that cancel the leading CR contributions. The analysis is carried out explicitly for the zeroth‑order and first‑order Magnus terms.

For a concrete pulse shape, they choose a cosine envelope for the in‑phase component, E_I(t) = Ω_I