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

TL;DR

The paper tackles the breakdown of the rotating-wave approximation for fast single-qubit gates on strongly anharmonic, low-frequency qubits by developing a Magnus-Taylor expansion framework to design corrected drive pulses. It derives zeroth- and first-order Magnus corrections and a non-computational-level leakage correction to yield high-fidelity gates using standard hardware, validated through simulations and fluxonium experiments. Key findings include a zeroth-order algebraic solution that cancels non-RWA terms across the full gate (under symmetry), and a first-order Magnus treatment that matches full dynamics for gates longer than several Magnus periods; leakage to higher levels can be suppressed with a time-dependent detuning and a small set of calibration parameters. The work demonstrates experimentally that deterministic calibration protocols can realize sub-coherence-limited gate errors for moderate gate durations, paving the way for fast, hardware-efficient quantum control in strongly anharmonic qubits and offering a framework applicable to other low-frequency platforms.

Abstract

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.

Figures (11)

• Figure 1: (a) Circuit diagram of the fluxonium qubit. (b) Potential and the wavefunctions offset by the eigenenergies of the first four levels for $E_C/h=E_L/h=$ 1 GHz and $E_J/h=$ 5 GHz at the sweet spot $\varphi_\text{ext}=0.5$.
• Figure 2: Error between the zeroth-order Magnus expanded non-RWA time evolution and the desired RWA time evolution for $\omega_{01}/2\pi = \omega_d/2\pi = 80$ MHz. Errors are truncated at $10^{-12}$ for visibility purposes. (a) Gate-error as a function of the gate duration $t_g$. The uncorrected pulse parameters correspond to setting $\mathcal{P}=\mathcal{P}_\text{RWA}$. The solid corrected line corresponds to the algebraic result, i.e. $\Omega_I=\Omega_{I,\text{RWA}}$ and $\lambda=1/2\omega_d$ and the dashed line correspond to the truncated series solution for $k\leq 14$. (b) Corrected and uncorrected error as a function of time for a 20 ns gate. In (a) and (b) the carrier phase $\phi=0$. (c)-(f) Heatmaps of the same simulation as in (b) as a function of the carrier phase. (c) and (d) show the error for the uncorrected pulse parameters and algebraic result respectively. (e) and (f) show the results for the truncated series solution with $k\leq 14$ corresponding to the parameter choices $\beta=\pi$ and $t_0=0$ respectively. The dashed lines in (d) and (e) indicate the times for which $t=t_0+nt_c$.
• Figure 3: (a)-(d) Optimized error, drive strength $\Omega_I$, PPP $\lambda$ and detuning $\Delta$ respectively for $\omega_{01}/2\pi = 80$ MHz. The cost function for the optimizer averages over 12 different carrier phases, i.e. $N=11$ in Eq. \ref{['eq:cost_function']}. In (a), the solid lines indicate the mean error and the shaded area indicates the minimum and maximum error for the different carrier phases. The dashed lines indicate the errors for the uncorrected pulse parameters, i.e. $\mathcal{P}=\mathcal{P}_\text{RWA}$. The solid lines in (b)-(d) indicate the pulse parameters calculated using the approach detailed in App. \ref{['app:first-order-magnus-expansion']}. The PPP in (c) is calculated in units of $1/2\omega_d$, where $\omega_d$ is computed using the detunings shown in (d).
• Figure 4: Minimum error versus $\lambda$ and $\Delta$ for a two-level system with $\omega_{01}/2\pi = 80$ MHz. For each point, the error is minimized as a function of the drive strength $\Omega_I$ according to the cost function in Eq. \ref{['eq:cost_function']} with $N=12$. The first (second) row shows the results for a 40 (80) ns gate, and the first (second) column shows the results for an $X_{\pi/2}$ ($X_\pi$) gate. The crossings at $\Delta=0$ and $\lambda=1/4\omega_{01}$ are indicated with black solid lines. The red dashed lines indicate the analytically calculated pulse parameters corresponding to $t_0=0$ and varying $\beta$.
• Figure 5: Error between the time evolutions corresponding to a four-level and two-level system for a fluxonium qubit with $E_J/h =$ 5 GHz, $E_L/h =$ 1 GHz and as a function of $E_C$ and the gate duration $t_g$. The carrier phase is fixed to 0. (a) Uncorrected error, i.e. $\Omega_\Delta=0$ and $\epsilon=1$. (b) Optimized error as a function of $\Omega_\Delta$ and $\epsilon$. (c) Leakage to non-computational states for the optimized parameters calculated using Eq. \ref{['eq:leakage_definition']}. (d) Optimized $\Delta(t)$ (solid lines) and $\Delta^\prime(t)$ (dashed lines) calculated using Eqs. \ref{['eq:time_dependent_detuning']} and \ref{['eq:time_dependent_detuning2']} respectively for four points in the parameter space as indicated in (b).