Quantum Gate Dynamics Beyond the Rotating-Wave Approximation using Multi-Timescale Quantum Averaging Theory

TL;DR

The paper develops Quantum Averaging Theory (QAT), a unitarity-preserving, multi-timescale framework to model quantum gate dynamics beyond the rotating-wave approximation, capturing both near-resonant and off-resonant contributions. By factorizing the interaction propagator into a fast part and an effective slow part via an exponential Lie transform, QAT provides an order-by-order effective Hamiltonian $\hat{H}_{I,\mathrm{eff}}(\tau;\lambda)$ governed by a slow time $\tau=\lambda s$, with partial averaging regularization (PETS) that isolates fast dynamics while preserving unitary evolution. The framework is demonstrated on the Mølmer-Sørensen gate in trapped ions, delivering explicit higher-order corrections (up to fourth order) and showing how pulse shaping and additional tones can suppress parasitic off-resonant processes, achieving near-perfect gate fidelities even in strong coupling. This analytic, scalable approach yields physical insight into dominant gate interactions and leakage channels, reduces computational cost for simulating bosonic modes, and integrates naturally with numerical optimal control for robust, high-fidelity quantum computation beyond the RWA.

Abstract

We present a quantum averaging theory (QAT) for analytically modeling unitary gate dynamics in driven quantum systems beyond the rotating-wave approximation. QAT addresses the simultaneous presence of distinct timescales by generating a rotating frame with a dynamical phase operator that toggles with the high-frequency dynamics and yields an effective Hamiltonian for the slow degree of freedom. By accounting for the fast-varying effects, we demonstrate that high-fidelity two-qubit gates in strongly driven systems are achievable by going beyond the validity of first-order approximations. The QAT results rapidly converge with numerical calculations of a fast-entangling Mølmer-Sørensen trapped-ion-qubit gate in the strong coupling regime, illustrating QAT's ability to simultaneously provide both an intuitive, effective-Hamiltonian model and high accuracy.

Figures (3)

• Figure 1: Level diagram: Two ion qubits coupled to a single phonon mode, symmetrically driven near the first motional sidebands by blue- ($+\Delta$) and red- ($-\Delta$) detuned drives.
• Figure 2: Effective Hamiltonian dynamics for population in the Bell state $\ket{\boldsymbol{\varphi}_{+}} = (\ket{ee} + \ket{gg})/\sqrt{2}$ of two ions under a strongly-coupled Mølmer-Sørensen gate in the carrier interaction picture are compared with numerical integration (dark gray), where $\ket{\bm{\varphi}_\mathrm{eff}^{[N]}}$ captures $N$th-order dynamics [eq. \ref{['eq:qat:QAT_interaction_approx']}]. The system is initialized in $\ket{g g}$ with motional coherent state $\ket{\alpha=i\sqrt{5}}$ ($\bar{n}=5$). Strong off-resonant and motionally-sensitive interactions shift the optimal entanglement time, and neglecting the carrier further reduces gate fidelity. Nonetheless, the effective dynamics closely track the envelope of the numerical solution, with deviations highlighted by shading. Parameters: $\eta = 0.1$, $\Omega = \nu$, $\phi_+ = \pi/4$, $\phi_- = 0$, and $\delta \approx 0.383\nu$.
• Figure 3: Full QAT dynamics for population in the Bell state $\ket{\boldsymbol{\varphi}_{+}} = (\ket{ee} + \ket{gg})/\sqrt{2}$ of two ions under a strongly-coupled Mølmer-Sørensen gate in the carrier interaction picture as compared to numerical integration (black). The dynamics including the off-resonant carrier is shown for reference (light gray). The system is initialized in $\ket{g g}$ with a motional coherent state $\ket{\alpha=i\sqrt{5}}$ ($\bar{n}=5$). Left: includes off-resonant interactions neglected in the effective Hamiltonian description of Fig. \ref{['fig:MSgate_Eff']}. Right: By adding pulse shaping, a power-of-sine window suppresses off-resonant effects, while a second-sideband tone decouples from leading-order thermal effects; parameters: $\Omega_2/\Omega_1 \approx0.7885$, $\delta_1/3=\delta_2 \approx 0.107\nu$, $\omega=\delta_2/3$, $s_g = 2\pi/\Lambda_\omega$. Bottom: shows the rapid convergence of the QAT approximation to the numerical result via instantaneous process fidelity, i.e., the average fidelity between numerical and approximate dynamics over a Pauli eigenstate 2-design nielsenQuantumComputationQuantum2010nielsenSimpleFormulaAverage2002a.