DRAG — Derivative Removal by Adiabatic Gate — is an analytic pulse-shaping technique for qubits that are only weakly separated from other levels. Superconducting transmons are the prime example. Adding a component proportional to the time derivative of the pulse envelope, in the out-of-phase (quadrature) channel, removes to leading order both leakage out of the qubit subspace and the associated phase errors. Introduced by Motzoi, Gambetta, Rebentrost and Wilhelm [1], it is part of the standard single-qubit gate on most superconducting processors.

The problem: weak anharmonicity

A transmon is essentially an anharmonic oscillator [2]. Its qubit transition \(0\leftrightarrow1\) and its leakage transition \(1\leftrightarrow2\) differ only by the anharmonicity \(\Delta/2\pi\), typically a few hundred MHz. A fast pulse has a broad spectrum. If it is short compared with \(1/|\Delta|\), part of its spectral weight drives the \(1\leftrightarrow2\) transition, causing leakage to \(|2\rangle\). The virtual coupling to \(|2\rangle\) also shifts the qubit levels (an AC Stark shift), causing phase errors. Without correction, this sets a lower bound on gate duration [3]; see Superconducting circuits.

The DRAG solution

Drive the qubit with an in-phase envelope \(\Omega_x(t)\) and add a quadrature component proportional to its derivative,

$$\Omega_y(t) = -\beta\,\frac{\dot\Omega_x(t)}{\Delta},$$

where \(\beta\) is of order one at leading order and is calibrated in practice. A time-dependent detuning compensates the remaining Stark shift. In the frequency domain, the derivative term reshapes the spectrum so that it cancels the drive at the leakage transition. In the time domain, it is the first correction of an adiabatic expansion in a frame that removes the coupling to \(|2\rangle\) [1]. The general theory gives a whole family of analytic solutions, from which one can choose what is practical, and extends to several leakage transitions [4].

In experiments

  • Phase-error correction. A derivative pulse on the quadrature channel removes the phase error caused by higher levels, enabling few-nanosecond single-qubit gates [5]. A simplified “half-derivative” variant, calibrated with error-amplifying sequences, reduced phase errors several-fold together with leakage [6].
  • Leakage at the \(10^{-5}\) level. Refining DRAG with a pulse detuning gives gate errors consistently below \(10^{-3}\) and leakage rates around \(10^{-5}\). Much of the remaining leakage then comes from incoherent heating [7].
  • Beyond DRAG. Closed-loop optimization of every sample of the waveform has produced a 4.16 ns single-qubit pulse with 99.76 % fidelity and 0.044 % leakage, faster than DRAG allows [8]; see Closed-loop & in-situ optimization.

Generalizations

The same derivative-based idea suppresses other unwanted, off-resonant transitions: drive crosstalk onto neighbouring qubits, additional ladder transitions, and sidebands of the drive [9]. In frequency-crowded multi-qubit (qutrit) systems, analytic few-parameter pulses with DRAG-like and composite features achieve simultaneous gates with errors below \(10^{-4}\) [10]. See Quantum gate synthesis.

DRAG and numerical optimal control

DRAG is a compact, interpretable pulse family. It is often the starting point for numerical methods: GOAT can optimize its few parameters exactly, and GRAPE can refine it with filtering for rapidly varying Hamiltonians [11]. Numerically optimized pulses that penalize leakage typically contain similar quadrature components; see Problem formulation & figures of merit.