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arXiv:2607.16124

Reverse engineering of single-qubit quantum gates

Gustavo Fernandes da Costa, Leonardo K. Castelano, Emanuel Fernandes de Lima

quant-phSubmitted Jul 17, 2026• Announced Jul 20, 2026• 0 citations
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Abstract

In this work, we address the problem of designing single-qubit quantum gates by means of a linearly-polarized field. We show that any desired one-qubit gate corresponding to a special unitary matrix can be generated by a modulated sinusoidal field. The only approximation involved is the rotating-wave-approximation. The formula for the control field is obtained by inverting the equation of motion for the evolution operator and imposing the conditions for the desired gate. We give a simple procedure to obtain closed analytical formulas for the fields in terms of a priori chosen dynamical functions. Additionally, these dynamical functions can depend on tunable free parameters intentionally introduced to meet a desired performance criteria.

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In this work, we address the problem of designing single-qubit quantum gates by means of a linearly-polarized field. We show that any desired one-qubit gate corresponding to a special unitary matrix can be generated by a modulated sinusoidal field. The only approximation involved is the rotating-wave-approximation. The formula for the control field is obtained by inverting the equation of motion for the evolution operator and imposing the conditions for the desired gate. We give a simple procedure to obtain closed analytical formulas for the fields in terms of a priori chosen dynamical functions. Additionally, these dynamical functions can depend on tunable free parameters intentionally introduced to meet a desired performance criteria.
Abstract We studied the dynamics of a pair of single-electron double quantum dots (DQDs) under longitudinal and transverse static magnetic fields and time-dependent harmonic modulation of their interaction couplings. We propose to modulate the tunnel coupling between the QDs to produce one-qubit gates and the exchange coupling between DQDs to generate entangling gates, the set of operations required for quantum computing. We developed analytical approximations to set the conditions to control the qubits and applied them to numerical calculations to test the accuracy and robustness of the analytical model. The results shows that the unitary evolution of the two-electron state performs the designed operations even under conditions shifted from the ideal ones.
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.
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.
The applicability and performance of qubits dressed by classical fields are limited because their control protocols give rise to an undesired counter-rotating term (CRT). This in turn forces operation in a regime where a (dressed) rotating-wave approximation (RWA) is valid, thereby restricting key aspects of their operation. Here, using only a single coupling axis in the laboratory frame, we introduce a dressed-qubit control protocol that optimally removes the CRT, eliminating the need for the RWA and delivering substantial improvements in multiple performance metrics, including single-qubit gate speed, two-qubit gate fidelity, spectroscopic range, clock stability, and coherence preservation. In addition, we provide a general parameterization together with a Floquet-based coherence-time expression, which elucidates the protocol's working principles and lowers the barrier to adoption. Collectively, these advances position our scheme as the state-of-the-art strategy for qubit control, paving the way for a wider class of quantum technologies to be realized using dressed-qubit architectures.
A quantum unitary gate is realized in this paper by perturbing a free charged particle in a one-dimensional box with a time- and position-varying electric field. The perturbed Hamiltonian is composed of a free particle Hamiltonian plus a perturbing electric potential such that the Schro¨\ddot{o}o¨dinger evolution in time TTT, the unitary evolution operator of the unperturbed system after truncation to a finite number of energy levels, approximates a given unitary gate such as the quantum Fourier transform gate. The idea is to truncate the half-wave Fourier sine series to MMM terms in the spatial variable x\mathbf xx before extending the potential as a Dyson series in the interaction picture to compute the evolution operator matrix elements up to the linear and quadratic integral functionals of $ \mathbf V_n(t)'$s. As a result, we used the Dyson series with the Frobenius norm to reduce the difference between the derived gate energy and the given gate energy, and we determined the temporal performance criterion by plotting the noise-to-signal energy ratio (NSER). A mathematical explanation for a quantum gate's magnetic control has also been provided. In addition, we provide a mathematical explanation for a quantum gate that uses magnetic control.