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AIP Advances

Calibrating single-qubit gates by a two-dimensional Rabi oscillation

You Huang, Mohammad T. Amawi, Francesco Poggiali, Fazhan Shi, Jiangfeng Du, Friedemann Reinhard

Published Mar 1, 2023• 2 citations
Publisher

Abstract

We present and analyze a simple scheme to calibrate single–qubit gates. It determines the amplitude and phase difference between a quadrature pair of drives, as well as their common detuning from the qubit resonance. The method is based on a two–dimensional Rabi oscillation, a sequence of two pulses of varying length sourced from the drive pair. We demonstrate error diagnosis using this scheme on an ensemble of nitrogen-vacancy centers in diamond and point out subtle pitfalls in its implementation.

DOI
10.1063/5.0139454

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We present and analyze a simple scheme to calibrate single–qubit gates. It determines the amplitude and phase difference between a quadrature pair of drives, as well as their common detuning from the qubit resonance. The method is based on a two–dimensional Rabi oscillation, a sequence of two pulses of varying length sourced from the drive pair. We demonstrate error diagnosis using this scheme on an ensemble of nitrogen-vacancy centers in diamond and point out subtle pitfalls in its implementation.
Rabi oscillations describe the coherent evolution of a two-level quantum system under the influence of a resonant driving field. They are central to quantum computing and determine the utility of qubits. Reliable control of these oscillations requires precise calibration of the driving field; however, experimental implementations are subject to imperfections that stray from what is ideal. One source of error is amplitude miscalibration, where the strength of the applied control pulse fluctuates around its intended value.In this study, the effects of amplitude error on Rabi oscillations were investigated using a formulaic simulation of a single-qubit system. The system was modeled using the Cirq quantum circuit framework, where Rabi oscillations were represented as rotations on the Bloch sphere. Amplitude divergence was incorporated by modifying the Rabi drive strength with Gaussian-distributed noise. The resulting state evolution was compared with the ideal theoretical oscillation in order to visualize the impact of stochastic amplitude variations on qubit dynamics.This model provides a simplified framework for examining how imperfect calibration influences coherent quantum state evolution in two-level systems.
We propose and demonstrate a quantum control scheme for hybrid quantum registers that can reduce the operation time, and therefore the effects of relaxation, compared to existing implementations. It combines resonant excitation pulses with periods of free precession under the internal Hamiltonian of the qubit system. We use this scheme to implement quantum gates like controlled-NOT operations on electronic and nuclear spins of the nitrogen-vacancy center in diamond. As a specific application, we transfer population between electronic and nuclear spin qubits and use it to measure the Rabi oscillations of a nuclear spin in a system with multiple coupled spins.
Precise quantum control relies on a deep understanding of the dynamics of quantum systems under external drives. This study investigates the impact of anharmonicity on qubit dynamics under conditions typical for two-qubit entangling gates activated by weak near-resonant microwave drives. We measure the Rabi oscillation frequency as a function of drive amplitude and detuning. Our results reveal a linear dependence of the squared Rabi frequency on the squared drive amplitude, which relates to the ratio of detuning to anharmonicity, demonstrating strong agreement between experimental data and analytical predictions. Additionally, we analyze the leakage and phase errors arising from inaccurate Rabi frequency adjustments in the CZ gate implementation on fluxonium qubits driven by a microwave signal applied to the coupler [Simakov et al., PRX Quantum 4, 040321 (2023)].
The implementation of a high-fidelity two-qubit quantum logic gate remains an outstanding challenge for isolated solid-state qubits such as nitrogen-vacancy (NV) centers in diamond. In this work, we show that by driving pairs of NV centers to undergo photon scattering processes that flip their qubit states simultaneously, we can achieve a unitary two-qubit gate conditioned upon a single photon-detection event. Further, by exploiting quantum interference between the optical transitions of the NV centers' electronic states, we realize the existence of two special drive frequencies: a ``magic'' point where the spin-preserving elastic scattering rates are suppressed and a ``balanced'' point where the state-flipping scattering rates are equal. We analyzed four different gate operation schemes that utilize these two special drive frequencies, and various combinations of polarizations in the drive and collection paths. Our theoretical and numerical calculations show that the gate fidelity can be as high as 98%. The proposed unitary gate, combined with available single-qubit unitary operations, forms a universal gate set for quantum computing.
Ensemble nitrogen-vacancy (NV) centers are widely used for quantum sensing and magnetometry, where performance depends on the accurate implementation of microwave pulse sequences. In these systems, inhomogeneous broadening and short coherence times limit reliable multi-pulse control, while hardware imperfections and off-resonant excitation distort implemented rotations, making direct calibration of arbitrary gates tedious. To address this, we implement a control framework with a ZXZXZ gate decomposition which uses fixed-angle, phase-parameterized pulses, enabling universal single-qubit control from a single calibrated $ π/2$ primitive (Xπ/2X_{π/2}Xπ/2​) along with virtual Z operations. We compare a standard pulse with a composite pulse designed for high-power operation in a regime where the Rabi frequency is comparable to the hyperfine splitting, leading to non-uniform evolution across hyperfine levels. Using pre-determined combinations of π/2π/2π/2 pulse sequences, we observe that standard pulses rapidly deviate from ideal behavior under repeated application, while the composite pulse suppresses this error accumulation by more than 70%\%% for longer sequences despite its longer duration. In contrast, randomized benchmarking yields similar decay factors for both implementations, highlighting its limited sensitivity to coherent control errors. These results establish a practical route to robust control in short-coherence ensemble systems and emphasize the importance of pulse-level characterization.
We present a scheme for the implementation of fast arbitrary qubit gates in the ground state of the negatively charged nitrogen-vacancy (NV) defect in diamond [1]. The protocol is especially useful for sensing in two regimes: on the one hand, in the low-field limit where the Zeeman splitting of the NV-center is smaller than the MW Rabi frequency; on the other hand, for the detection of high-frequecy signals, comparable to the Zeeman splitting of the NV-center. It constitutes an extension to the NV-ERC technique [2, 3], which has demonstrated efficient initialization and readout of the double quantum transition with no leakage to any third level thanks to an effective Raman coupling. Here we derive a full theoretical framework of the scheme, identifying the complete unitary associated to the approach, and more specifically the relevant basis transformation for each of two characteristic pulse durations. Based on this insight, we propose a scheme to perform fast single qubit gates in the double quantum transition. We study its robustness with respect to pulse-timing errors resulting from faulty identification of system parameters or phase-control limitations. We finally demonstrate that the technique can also be implemented in the presence of unknown electric or strain fields and numerically test its effectiveness in a Hahn echo sequence in the high-frequency or low-field regime.