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arXiv:2610.40595

Optimal detuning and gate time for strongly driven single-qubit gates beyond the rotating-wave approximation

Alejandro D. Bendersky, Sergio S. Gomez, Rodolfo H. Romero

quant-phSubmitted Oct 29, 2026• Announced Oct 30, 2026• 0 citations
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Abstract

Single-qubit quantum gates are implemented as the time-controlled unitary evolution of a two-level system under harmonic driving. The rotating-wave approximation (RWA) provides closed-form gate parameters for weak coupling. Larger amplitudes, desirable to speed up the dynamics, make the RWA-designed gates inaccurate because of the neglected counter-rotating term, which also shifts the resonance (Bloch-Siegert shift). Here we study theoretically the infidelity of gates beyond the RWA, using the effective Magnus-Taylor Hamiltonian through second order in the inverse of the driving frequency. By detuning the driving frequency by an amplitude-dependent shift of Bloch-Siegert type and rescaling the operation time, the RWA gate conditions are recovered with corrected parameters. Numerical solutions of the Schrodinger equation show that these corrections lower the gate infidelity by several orders of magnitude, extending the range of usable amplitudes

AI Summary

The paper asks how to correct single-qubit gates implemented by harmonic driving when counter-rotating terms invalidate rotating-wave-approximation (RWA) gate parameters. It studies the ideal two-level Hamiltonian $H(t)=\omega_0\sigma_z/2+A\sin(\omega t)\sigma_x$, using an existing second-order Magnus–Taylor effective Hamiltonian to derive corrected parameters for $Z$, $X$ and Hadamard gates. For a fixed drive phase and stroboscopic gate endpoints, the prescription is an amplitude-dependent detuning $\Delta=\omega_0-\omega\simeq-3A^2/(4\omega)$ and a gate-time rescaling $t_g\simeq\tau_g[1-A^2/(16\omega^2)]$, together with corresponding discrete amplitude conditions. These restore the RWA gate conditions through second order in $\lambda=A/\omega$. The gate-optimal detuning is distinguished from the conventional spectral Bloch–Siegert shift, whose leading coefficient is $1/4$ rather than $3/4$. Analytical average-gate-infidelity calculations explain why uncorrected RWA gates have leading errors proportional to $\lambda^4$ for $Z$ but $\lambda^2$ for $X$ and Hadamard: the latter are sensitive to a tilt of the effective rotation axis. Direct Schrödinger-equation simulations for $Z$ and Hadamard show orders-of-magnitude reductions in infidelity, including at least three orders for Hadamard near $\lambda=1/14$. Numerical optimal detunings follow approximately $\Delta/\omega=-0.7\lambda^{2.0}$. The reported numerical exploration extends to $\lambda\lesssim1$, beyond a strictly weak-drive regime. However, the derivation remains perturbative, with residual effective-Hamiltonian terms of order $A\lambda^3$. The protocol depends on drive phase and stroboscopic timing; robustness to calibration errors and noise remains unanalysed.

Little sign of AI-generated textIncrementalMissing comparisonStrong assumptions

Error found: The discussion after the corrected gate conditions misleadingly says the corrected amplitude decreases: A/omega decreases, but A/omega_0—and therefore the physical drive amplitude at fixed qubit gap—increases to the stated perturbative order.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

Comments
7 pages, 2 figures

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Single-qubit quantum gates are implemented as the time-controlled unitary evolution of a two-level system under harmonic driving. The rotating-wave approximation (RWA) provides closed-form gate parameters for weak coupling. Larger amplitudes, desirable to speed up the dynamics, make the RWA-designed gates inaccurate because of the neglected counter-rotating term, which also shifts the resonance (Bloch-Siegert shift). Here we study theoretically the infidelity of gates beyond the RWA, using the effective Magnus-Taylor Hamiltonian through second order in the inverse of the driving frequency. By detuning the driving frequency by an amplitude-dependent shift of Bloch-Siegert type and rescaling the operation time, the RWA gate conditions are recovered with corrected parameters. Numerical solutions of the Schrodinger equation show that these corrections lower the gate infidelity by several orders of magnitude, extending the range of usable amplitudes
The paper asks how to correct single-qubit gates implemented by harmonic driving when counter-rotating terms invalidate rotating-wave-approximation (RWA) gate parameters. It studies the ideal two-level Hamiltonian H(t)=ω0σz/2+Asin⁡(ωt)σxH(t)=\omega_0\sigma_z/2+A\sin(\omega t)\sigma_xH(t)=ω0​σz​/2+Asin(ωt)σx​, using an existing second-order Magnus–Taylor effective Hamiltonian to derive corrected parameters for ZZZ, XXX and Hadamard gates.
For a fixed drive phase and stroboscopic gate endpoints, the prescription is an amplitude-dependent detuning Δ=ω0−ω≃−3A2/(4ω)\Delta=\omega_0-\omega\simeq-3A^2/(4\omega)Δ=ω0​−ω≃−3A2/(4ω) and a gate-time rescaling tg≃τg[1−A2/(16ω2)]t_g\simeq\tau_g[1-A^2/(16\omega^2)]tg​≃τg​[1−A2/(16ω2)], together with corresponding discrete amplitude conditions. These restore the RWA gate conditions through second order in λ=A/ω\lambda=A/\omegaλ=A/ω. The gate-optimal detuning is distinguished from the conventional spectral Bloch–Siegert shift, whose leading coefficient is 1/41/41/4 rather than 3/43/43/4.
Analytical average-gate-infidelity calculations explain why uncorrected RWA gates have leading errors proportional to λ4\lambda^4λ4 for ZZZ but λ2\lambda^2λ2 for XXX and Hadamard: the latter are sensitive to a tilt of the effective rotation axis. Direct Schrödinger-equation simulations for ZZZ and Hadamard show orders-of-magnitude reductions in infidelity, including at least three orders for Hadamard near λ=1/14\lambda=1/14λ=1/14. Numerical optimal detunings follow approximately Δ/ω=−0.7λ2.0\Delta/\omega=-0.7\lambda^{2.0}Δ/ω=−0.7λ2.0.
The reported numerical exploration extends to λ≲1\lambda\lesssim1λ≲1, beyond a strictly weak-drive regime. However, the derivation remains perturbative, with residual effective-Hamiltonian terms of order Aλ3A\lambda^3Aλ3. The protocol depends on drive phase and stroboscopic timing; robustness to calibration errors and noise remains unanalysed.
In many quantum platforms, single-qubit gates are 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 counterrotating terms to the qubit dynamics. The inclusion of counterrotating terms into the theoretical description gives rise to two challenges. First, 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 counterrotating terms in a two-level system. We then derive a second correction term that arises from noncomputational 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 and calibration complexities.
We present a theoretical study of single-qubit operations by oscillatory fields on various semiconductor platforms. We explicitly show how to perform faster gate operations by going beyond the universally used rotating-wave approximation (RWA) regime, while using only two sinusoidal pulses. We first show for specific published experiments how much error is currently incurred by implementing pulses designed using standard RWA. We then show that an even modest increase in gate speed would cause problems in using RWA for gate design in the singlet-triplet (ST) and resonant-exchange (RX) qubits. We discuss the extent to which analytically keeping higher orders in the perturbation theory would address the problem. More strikingly, we give a new prescription for gating with strong coupling far beyond the RWA regime. We perform numerical calculations for the phases and the durations of two consecutive pulses to realize the key Hadamard and $\frac{\ensuremath{\pi}}{8}$ gates with coupling strengths up to several times the qubit splitting. Working in this manifestly non-RWA regime, the gate operation speeds up by two to three orders of magnitude and nears the quantum speed limit without requiring complicated pulse shaping or optimal control sequences.
We propose a general approach to implement ultrafast nonadiabatic geometric single- and two-qubit gates by employing counter-rotating effects. This protocol is compatible with most optimal control methods used in previous rotating-wave approximation (RWA) protocols; thus, it is as robust as (or even more robust than) the RWA protocols. Using counter-rotating effects allows us to apply strong drives. Therefore, we can improve the gate speed by 5--10 times compared to the RWA counterpart for implementing high-fidelity ($\ensuremath{\ge}\phantom{\rule{0.2em}{0ex}}99.99\mathrm{%}$) gates. Such an ultrafast evolution (nanoseconds, even picoseconds) significantly reduces the influence of decoherence (e.g., the qubit dissipation and dephasing). Moreover, because the counter-rotating effects no longer induce a gate infidelity (in both the weak and strong driving regimes), we can achieve a higher fidelity compared to the RWA protocols. Therefore, in the presence of decoherence, one can implement ultrafast geometric quantum gates with $\ensuremath{\ge}\phantom{\rule{0.2em}{0ex}}99\mathrm{%}$ fidelities.
Semiconductor double quantum dot (DQD) qubits coupled via superconducting microwave resonators provide a powerful means of long-range manipulation of the qubits' spin and charge degrees of freedom. Quantum gates can be implemented by parametrically driving the qubits while their transition frequencies are detuned from the resonator frequency. Long-range two-qubit CZ gates have been proposed for the DQD spin qubit within the rotating-wave approximation (RWA). Rapid gates demand strong coupling, but RWA breaks down when coupling strengths become significant relative to system frequencies. Therefore, understanding the detrimental impact of time-dependent terms ignored by RWA is critical for high-fidelity operation. Here, we go beyond RWA to study CZ gate fidelity for both DQD spin and charge qubits. We propose a novel parametric drive on the charge qubit that produces fewer time-dependent terms and show that it outperforms its spin counterpart. We find that drive amplitude - a parameter dropped in RWA - is critical for optimizing fidelity and map out high-fidelity regimes. Our results demonstrate the necessity of going beyond RWA in understanding how long-range gates can be realized in DQD qubits, with charge qubits offering considerable advantages in high-fidelity operation.
Semiconductor double quantum dot (DQD) qubits coupled via superconducting microwave resonators provide a powerful means of long-range manipulation of the qubits' spin and charge degrees of freedom. Quantum gates can be implemented by parametrically driving the qubits while their transition frequencies are detuned from the resonator frequency. Long-range two-qubit controlled-Z (CZ) gates have been proposed for the DQD spin qubit within the rotating-wave approximation (RWA). Rapid gates demand strong coupling, but RWA breaks down when coupling strengths become significant relative to system frequencies. Therefore, understanding the errors arising from approximations used is critical for high-fidelity operation. Here, we go beyond RWA to study CZ gate fidelity for both DQD spin and charge qubits. We propose a novel parametric drive on the charge qubit that produces smaller errors and show that the fidelity of the CZ gate outperforms its spin counterpart, resulting in a much smaller fidelity loss of 0.05%0.05\%0.05% compared to 0.80%0.80\%0.80% for the spin qubit, and greater robustness against qubit dephasing and photon loss. We find that drive amplitude -- a parameter dropped in RWA -- is critical for optimizing fidelity, with the charge qubit exhibiting better tolerance to drive amplitude variations. Our results demonstrate the necessity of going beyond RWA in understanding how long-range gates can be realized in DQD qubits, with charge qubits offering considerable advantages in high-fidelity operation.
Performing qubit gate operations as quickly as possible can be important to minimize the effects of decoherence. For resonant gates, this requires applying a strong ac drive. However, strong driving can present control challenges by causing leakage to levels that lie outside the qubit subspace. Strong driving can also present theoretical challenges because preferred tools such as the rotating wave approximation can break down, resulting in complex dynamics that are difficult to control. Here we analyze resonant XXX rotations of a silicon-quantum-double-dot hybrid qubit within a dressed-state formalism, obtaining results beyond the rotating wave approximation. We obtain analytic formulas for the optimum driving frequency and the Rabi frequency, which are both affected by strong driving. While the qubit states exhibit fast oscillations due to counter-rotating terms and leakage, we show that they can be suppressed to the point that gate fidelities above 99.9999.99%99.99 are possible, in the absence of decoherence. Hence decoherence mechanisms, rather than strong-driving effects, should represent the limiting factor for resonant-gate fidelities in quantum dot hybrid qubits.