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DRAG calibration

This experiment calibrates the optimal DRAG amplitude, \(\lambda_{\text{DRAG}}\), for a \(\frac{\pi}{2}\)-pulse.

Description

Generally, qubits are confined to a computational subspace consisting of two basis states: \(|0\rangle\) and \(|1\rangle\). However, higher states (e.g. \(|2\rangle\), ...) do exist, and it's possible for a pulse to inadvertently excite the qubit to these higher states, resulting in leakage outside the computational subspace. Phase errors can also occur, where the qubit couples to neighbouring energy levels12. Both of these effects reduce qubit fidelity.

To prevent this from occurring, a specific technique known as Derivative Removal via Adiabatic Gate (DRAG) can be implemented12. This applies a DRAG correction to the drive pulse that's optimised to minimise these effects.

The qubit drive can be expressed as

\[ \varepsilon(t) = \varepsilon_{I}(t) \cos(\omega_d t) + \varepsilon_{Q}(t) \sin(\omega_d t), \]

where \(\omega_d\) is the drive frequency and \(\varepsilon_{I}(t)\) and \(\varepsilon_{Q}(t)\) are the in-phase and quadrature components of the pulse, respectively. These are given by

\[ \varepsilon_{I}(t) = \varepsilon_{\text{Gauss}}(t) \]

and

\[ \varepsilon_{Q}(t) = -\lambda_{\text{DRAG}} \dot{\varepsilon}_{\text{Gauss}}(t). \]

Here, \(\varepsilon_{\text{Gauss}}(t)\) is a Gaussian-shaped pulse, \(\dot{\varepsilon}_{\text{Gauss}}(t)\) is its time derivative, and \(\lambda_{\text{DRAG}}\) is the DRAG amplitude.

To determine the optimal value of \(\lambda_{\text{DRAG}}\), an error amplification sequence is used. Here, a \(\frac{\pi}{2}\)-pulse is applied to the qubit, followed by a \(-\frac{\pi}{2}\)-pulse while DRAG amplitude is varied around \(\lambda_{\text{DRAG}}=0\).

The \(\frac{\pi}{2}\) pulse rotates the qubit onto the \(y\)-axis, creating a superposition state, and the \(-\frac{\pi}{2}\)-pulse rotates it back to the measurement basis. If \(\lambda_{\text{DRAG}}\) is correctly set, the qubit will return to the \(|0\rangle\) state; if not, the qubit will end up in a superposition state and its final state when measured will be projected onto the \(z\)-axis. The probability of finding the qubit in the \(|0\rangle\) or \(|1\rangle\) state varies as a cosine function of \(\lambda_{\text{DRAG}}\), with the minimum representing the optimal value.

Experiment steps

  1. A \(\frac{\pi}{2}\)-pulse (\(R_x(\frac{\pi}{2})\)) is applied, which prepares the qubit in the superposition state \(\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\).

  2. A \(-\frac{\pi}{2}\)-pulse (\(R_x(\frac{-\pi}{2})\)) is applied to return the qubit to the measurement basis.

  3. The readout resonator transmission is measured for each value of the DRAG amplitude, \(\lambda_{\text{DRAG}}\).

Analysis steps

  1. The qubit population is predicted from the IQ data by applying the discriminator trained in the Readout Discriminator Training experiment.

  2. A cosine function is fit to the experimental data, and \(\lambda_{\text{DRAG}}\) is determined as the value that corresponds to a minimum of the cosine.

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  1. F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wilhelm. Simple pulses for elimination of leakage in weakly nonlinear qubits. Phys. Rev. Lett., 103:110501, 2009. doi:10.1103/PhysRevLett.103.110501. ↩↩

  2. Sarah Sheldon, Lev S. Bishop, Easwar Magesan, Stefan Filipp, Jerry M. Chow, and Jay M. Gambetta. Characterizing errors on qubit operations via iterative randomized benchmarking. Phys. Rev. A, 93:012301, Jan 2016. doi:10.1103/PhysRevA.93.012301. ↩↩