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arXiv:2412.19561

Single-qubit quantum gate at an arbitrary speed

Seongjin Ahn, Park, Kichan, Daehee Cho, Mikyoung Lim, Taeyoung Choi, A. S. Moskalenko

Published Dec 27, 2024• 0 citations
arXivHTMLPDF

Abstract

Quantum information processing comprises physical processes, which obey the quantum speed limit (QSL): high speed requires strong driving. Single-qubit gates using Rabi oscillation, which is based on the rotating wave approximation (RWA), satisfy this bound in the form that the gate time $T$ is inversely proportional to the Rabi frequency $Ω$, characterizing the driving strength. However, if the gate time is comparable or shorter than the qubit period $T_{0} \equiv 2π/ ω_{0}$, the RWA actually breaks down since the Rabi frequency has to be large compared to the qubit frequency $ω_{0}$ due to the QSL, which is given as $T \gtrsim π/Ω$. We show that it is possible to construct a universal set of single-qubit gates at this strong-coupling and ultrafast regime, by adjusting the central frequency $ω$ and the Rabi frequency $Ω$ of the driving pulse. We observe a transition in the scaling behavior of the central frequency from the long-gate time regime ($T \gg T_{0}$) to the short-gate time ($T \ll T_{0}$) regime. In the former, the central frequency is nearly resonant to the qubit, i.e., $ω\simeq ω_{0}$, whereas in the latter, the central frequency is inversely proportional to the gate time, i.e., $ω\sim π/T$. We identify the transition gate time at which the scaling exponent $n$ of the optimal central frequency $ω\sim T^{n}$ changes from $n=0$ to $n=-1$.

DOI
10.48550/arxiv.2412.19561

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Quantum information processing comprises physical processes, which obey the quantum speed limit (QSL): high speed requires strong driving. Single-qubit gates using Rabi oscillation, which is based on the rotating wave approximation (RWA), satisfy this bound in the form that the gate time TTT is inversely proportional to the Rabi frequency ΩΩΩ, characterizing the driving strength. However, if the gate time is comparable or shorter than the qubit period T0≡2π/ω0T_{0} \equiv 2π/ ω_{0}T0​≡2π/ω0​, the RWA actually breaks down since the Rabi frequency has to be large compared to the qubit frequency ω0ω_{0}ω0​ due to the QSL, which is given as T≳π/ΩT \gtrsim π/ΩT≳π/Ω. We show that it is possible to construct a universal set of single-qubit gates at this strong-coupling and ultrafast regime, by adjusting the central frequency ωωω and the Rabi frequency ΩΩΩ of the driving pulse. We observe a transition in the scaling behavior of the central frequency from the long-gate time regime (T≫T0T \gg T_{0}T≫T0​) to the short-gate time (T≪T0T \ll T_{0}T≪T0​) regime. In the former, the central frequency is nearly resonant to the qubit, i.e., ω≃ω0ω\simeq ω_{0}ω≃ω0​, whereas in the latter, the central frequency is inversely proportional to the gate time, i.e., ω∼π/Tω\sim π/Tω∼π/T. We identify the transition gate time at which the scaling exponent nnn of the optimal central frequency ω∼Tnω\sim T^{n}ω∼Tn changes from n=0n=0n=0 to n=−1n=-1n=−1.
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.
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
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.
Logic gates in superconducting quantum processors are implemented through precise quantum control techniques in the microwave regime. The choice of drive frequency and other control parameters directly determines the duration of quantum gate operations. Because current devices remain too noisy to reach fault tolerance, reducing gate durations, and thereby the overall circuit depth, is of critical importance. In this work, we present a model of single qubit gate execution in both the adiabatic regime, where the rotating wave approximation (RWA) is valid, and the diabatic regime, where the RWA no longer applies. Using parameters representative of superconducting qubits, we investigate how gates can be driven at durations well below conventional timescales, and we examine the associated limitations, performance trade offs. The results demonstrate that ultrashort control pulses in the diabatic regime can achieve fidelities comparable to those obtained under standard RWA conditions, offering a possible route to faster quantum logic without sacrificing accuracy under idealized conditions.
Logic gates in superconducting quantum processors are implemented through precisely shaped microwave pulses, whose durations are constrained by both device coherence and the multilevel structure of artificial atoms. While single-qubit fidelities above 0.999 are routinely achieved, the physical execution time of these gates remains a major bottleneck for deep quantum circuits. Here we examine single-qubit control across adiabatic [where the rotating-wave approximation (RWA) is valid], intermediate, and diabatic driving regimes using both an ideal two-level model and a realistic weakly anharmonic transmon Hamiltonian. By sweeping pulse durations and drive strengths over experimentally relevant ranges, we compare performance trade-offs and identify the conditions under which ultrashort gates remain viable. Our simulations uncover a quantitative speed-limit mechanism intrinsic to weakly anharmonic superconducting qubits: leakage to higher levels rises sharply once the peak Rabi rate exceeds a fixed fraction of the anharmonicity. Across all pulse shapes and driving regimes studied, we find a consistent numerical threshold at Ω max /|Δ|≈0.15−0.20, where Ω max denotes the peak drive (Rabi) amplitude and Δ is the transmon anharmonicity. Beyond this ratio, gate fidelity deteriorates regardless of waveform engineering. This boundary, obtained from full multilevel time-domain simulations spanning adiabatic, intermediate, and diabatic regimes, constitutes a previously unreported characterization of the fundamental leakage barrier in transmon devices. Our results, therefore, provide the first unified quantitative map linking pulse duration, driving strength, and anharmonicity, establishing why realistic superconducting qubits cannot sustain picosecond-scale gates despite ideal two-level predictions, and offering a practical benchmark for next-generation architectures seeking shorter, high-fidelity control.
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.