DRAG: Derivative Removal by Adiabatic Gate

• DRAG is a quantum-control method that uses a derivative-shaped control pulse to suppress leakage and unwanted phase errors in weakly anharmonic multilevel qubits.
• It leverages counterdiabatic driving principles by augmenting resonant in-phase control with a quadrature proportional to the pulse derivative, effectively nulling off-resonant transitions.
• Experimental and theoretical extensions, such as FAST DRAG and half-derivative protocols, have demonstrated significant improvements in gate fidelity and speed across various qubit platforms.

Derivative Removal by Adiabatic Gate (DRAG) is a quantum-control framework for suppressing unwanted transitions during fast gates in multilevel systems whose qubit subspace is only weakly isolated from nearby states. In its standard form for weakly anharmonic superconducting qubits, a resonant in-phase control envelope is augmented by an orthogonal quadrature proportional to the time derivative of that envelope, sometimes together with a detuning correction, so that finite-bandwidth excitation of noncomputational levels and drive-induced phase errors are reduced. Within the broader control literature, DRAG has also been described as a multi-transition variant of counterdiabatic driving and as a control-constrained realization of superadiabatic cancellation (Theis et al., 2018, Patra et al., 23 Apr 2026).

1. Control problem and error channels

DRAG addresses the fact that many physical qubits are not exact two-level systems. In a transmon or other weakly anharmonic ladder, a pulse intended to drive the $|0\rangle \leftrightarrow |1\rangle$ transition also has spectral weight near $|1\rangle \leftrightarrow |2\rangle$, and fast gates broaden that spectrum further. The result is not a single error mechanism but at least two distinct ones: virtual excursions into higher levels, which appear primarily as unwanted $Z$-axis phase shifts on the computational states, and real population transfer into noncomputational states, which appears as leakage or amplitude error. The early superconducting-qubit implementation makes this distinction explicit and models the phase-imperfect $X_{\pi/2}$ gate as $$X'_{\pi/2}=e^{-i\epsilon' Z_\epsilon X_{\pi/2} Z_\epsilon}$$, with the phase error scaling roughly as $\epsilon \sim \theta^2/t_g$, so that longer gates reduce virtual-transition-induced phase error (Lucero et al., 2010).

A standard three-level transmon description makes the leakage structure explicit. With in-phase and quadrature controls $I(t)$ and $Q(t)$, the rotating-wave Hamiltonian separates the desired computational drive from the off-resonant coupling to $|2\rangle$: $$H_{RWA}=\frac{\hbar}{2}\big(I(t)\sigma_{01}^{x}-Q(t)\sigma_{01}^{y}\big) +\frac{\lambda\hbar}{2}\Big[\big(I(t)+iQ(t)\big)e^{i\Delta t}\sigma_{12}^{+} +\big(I(t)-iQ(t)\big)e^{-i\Delta t}\sigma_{12}^{-}\Big].$$ Here $|1\rangle \leftrightarrow |2\rangle$0 is the anharmonicity-related detuning of the leakage transition. In this formulation, finite bandwidth, weak anharmonicity, and multilevel structure jointly determine the dominant error channels; DRAG is introduced precisely to cancel the leading off-resonant coupling while preserving the intended qubit rotation (Patra et al., 23 Apr 2026).

The same spectral logic reappears across later work. Fast gates are advantageous because they increase the number of operations within a coherence window, but shorter pulses are spectrally broader and therefore more likely to overlap with harmful transitions. This makes DRAG not merely an amplitude-calibration method but a pulse-shaping method directed at leakage, phase accumulation, and other off-resonant effects arising from finite-time control (Hyyppä et al., 2024, De, 2015).

2. Standard DRAG construction and analytical interpretations

In the conventional single-qubit transmon setting, the basic prescription is to choose the quadrature correction as

$|1\rangle \leftrightarrow |2\rangle$1

assuming the pulse begins and ends smoothly, $|1\rangle \leftrightarrow |2\rangle$2. In the Magnus-expansion analysis, this choice cancels the first-order leakage channel into $|1\rangle \leftrightarrow |2\rangle$3 and removes the unwanted first-order $|1\rangle \leftrightarrow |2\rangle$4 term, leaving the desired first-order $|1\rangle \leftrightarrow |2\rangle$5-rotation,

$|1\rangle \leftrightarrow |2\rangle$6

so that setting the pulse area to $|1\rangle \leftrightarrow |2\rangle$7 implements a $|1\rangle \leftrightarrow |2\rangle$8-pulse on the computational subspace (Patra et al., 23 Apr 2026).

A more general formulation, emphasized in the DRAG review literature, treats the method as constrained counterdiabatic control. The pulse is expanded as a smooth base envelope plus derivative corrections,

$|1\rangle \leftrightarrow |2\rangle$9

so that the unwanted response vanishes at one or several harmful detunings $Z$0. In this perspective, DRAG is a practical version of counterdiabatic driving in which the correction must be synthesized from physically available quadratures, detunings, sidebands, or higher derivatives rather than from an abstract exact adiabatic gauge potential (Theis et al., 2018).

The spectral interpretation is equally central. Differentiation in time multiplies the spectrum by frequency, allowing destructive interference at selected leakage transitions. One later analytic transmon treatment writes the complex envelope as $Z$1 with

$Z$2

Within that framework, $Z$3 defines a DRAG-P regime aimed at canceling AC-Stark phase error, while $Z$4 defines a DRAG-L regime aimed at placing a spectral zero near the $Z$5 transition and suppressing leakage, with the remaining phase then corrected by virtual $Z$6 rotations (Hyyppä et al., 2024).

A recurrent misconception is that DRAG is identical to any form of counterdiabatic driving. A more precise statement is that DRAG is closely related in spirit and formal structure, but not every counterdiabatic construction is standard DRAG. This distinction is explicit in the gate-based adiabatic-gauge-potential literature, where state-selective transition suppression is described as strongly connected to DRAG in principle but not to the specific microwave-control protocol used for superconducting qubits (Vreumingen, 2024).

3. Experimental realization, half-derivative DRAG, and phase-error metrology

An influential superconducting-qubit experiment established DRAG as an experimentally workable method for simultaneously reducing phase error and leakage. That work introduced amplified phase error (APE) pulses as a metrology for small phase shifts and implemented a simplified DRAG variant called half-derivative (HD). The APE construction inserts a pseudo-identity between Ramsey $Z$7 pulses,

$Z$8

so that, after $Z$9 repetitions, the phase error accumulates coherently as $X_{\pi/2}$0. In the reported experiment, $X_{\pi/2}$1 produced about a $X_{\pi/2}$2 amplification, corresponding to an inferred uncorrected phase error of about $X_{\pi/2}$3 per gate in the single-control case (Lucero et al., 2010).

The same experiment compared full DRAG to a reduced-control implementation. Standard DRAG was described there in terms of three controls: a Gaussian $X_{\pi/2}$4, a quadrature correction $X_{\pi/2}$5, and a dynamic detuning $X_{\pi/2}$6. Because $X_{\pi/2}$7 and $X_{\pi/2}$8 were not independent in practice, the experiment simplified the protocol by setting

$X_{\pi/2}$9

which defined the half-derivative protocol. The authors reported that HD performed as well as full DRAG in their device (Lucero et al., 2010).

The reported performance established the practical significance of the method. Phase errors were lowered by about a factor of five to $$X'_{\pi/2}=e^{-i\epsilon' Z_\epsilon X_{\pi/2} Z_\epsilon}$$0 per gate and could be tuned to zero by adjusting the derivative amplitude. Leakage outside the qubit manifold, measured as population in $$X'_{\pi/2}=e^{-i\epsilon' Z_\epsilon X_{\pi/2} Z_\epsilon}$$1, was reduced to $$X'_{\pi/2}=e^{-i\epsilon' Z_\epsilon X_{\pi/2} Z_\epsilon}$$2 for 6 ns HD $$X'_{\pi/2}=e^{-i\epsilon' Z_\epsilon X_{\pi/2} Z_\epsilon}$$3 pulses, about 20% faster than comparable non-HD gates, and the HD pulse yielded about five times lower leakage error than a Gaussian pulse of the same width (Lucero et al., 2010).

This early experiment also clarified the physical content of DRAG. The corrective quadrature does not merely reduce net over-rotation; it suppresses transient excitation pathways, compensates AC-Stark-like phase shifts from virtual excursions, and makes the multilevel system follow the intended two-level trajectory more closely in a transformed frame. That diagnostic-and-correction pairing—APE for metrology and HD DRAG for control—became a template for later superconducting-qubit pulse engineering (Lucero et al., 2010).

4. Later superconducting-qubit variants and performance regimes

Subsequent work extended DRAG in several directions. One line emphasized that two-quadrature DRAG can be strengthened by simultaneous optimization of detuning and pulse norm. In explicit lab-frame simulations for truncated multilevel transmon models derived from a tight-binding model, optimized DRAG substantially improved fidelity for sharper Gaussian pulses, with infidelity pushed down to $$X'_{\pi/2}=e^{-i\epsilon' Z_\epsilon X_{\pi/2} Z_\epsilon}$$4 for $$X'_{\pi/2}=e^{-i\epsilon' Z_\epsilon X_{\pi/2} Z_\epsilon}$$5 ns in favorable cases. The same study also identified a low-spectral-weight envelope regime, exemplified by $$X'_{\pi/2}=e^{-i\epsilon' Z_\epsilon X_{\pi/2} Z_\epsilon}$$6, in which DRAG is almost not needed and the two-state error fidelities are stable against pulse jitter; this directly counters the assumption that derivative correction is uniformly indispensable (De, 2015).

A later analytic extension introduced Fourier ansatz spectrum tuning derivative removal by adiabatic gate (FAST DRAG) and higher-derivative (HD) DRAG. FAST DRAG shapes the in-phase spectrum over selected frequency windows and then applies standard DRAG, while HD DRAG adds higher even derivatives to create stronger spectral zeros at leakage frequencies. Experimentally, these methods enabled $$X'_{\pi/2}=e^{-i\epsilon' Z_\epsilon X_{\pi/2} Z_\epsilon}$$7 gates with leakage error below $$X'_{\pi/2}=e^{-i\epsilon' Z_\epsilon X_{\pi/2} Z_\epsilon}$$8 down to a gate duration of 6.25 ns without iterative closed-loop optimization, corresponding to a 20-fold leakage reduction compared to a conventional Cosine DRAG pulse. FAST DRAG further achieved an error per gate of $$X'_{\pi/2}=e^{-i\epsilon' Z_\epsilon X_{\pi/2} Z_\epsilon}$$9 at a 7.9-ns gate duration, outperforming conventional pulse shapes in both error and speed (Hyyppä et al., 2024).

A still more aggressive generalization targeted the strong-driving regime in which the dominant errors are no longer limited to nearest-neighbor $\epsilon \sim \theta^2/t_g$0 leakage. In the double recursive DRAG method, or R2D, the squared envelope $\epsilon \sim \theta^2/t_g$1 is itself shaped by recursive derivative-based spectral notching to suppress effective two-photon transitions such as $\epsilon \sim \theta^2/t_g$2 and $\epsilon \sim \theta^2/t_g$3, after which standard DRAG is applied to the linear envelope. For a transmon with $\epsilon \sim \theta^2/t_g$4 GHz and $\epsilon \sim \theta^2/t_g$5 MHz, this yielded total leakage consistently below $\epsilon \sim \theta^2/t_g$6 and fidelities above $\epsilon \sim \theta^2/t_g$7 for both $\epsilon \sim \theta^2/t_g$8 and $\epsilon \sim \theta^2/t_g$9 gates down to 6.8 ns, with more than a factor-of-20 leakage reduction relative to standard single-photon DRAG (Gao et al., 27 Nov 2025).

These extensions suggest that “DRAG” denotes a family of derivative-based spectral-engineering methods rather than a single fixed waveform formula. In one limit the correction is first-order and nearest-neighbor; in another it is multi-derivative, recursive, or combined with explicit spectral-window design. Across these variants, the common objective remains suppression of harmful transition amplitudes without sacrificing fast gate times (Theis et al., 2018, Hyyppä et al., 2024, Gao et al., 27 Nov 2025).

5. Hardware realization, calibration, and practical limitations

The efficacy of DRAG depends not only on the analytic pulse form but also on the control stack that synthesizes it. A standard implementation uses an AWG + LO + IQ mixer chain: the AWG generates the baseband envelope $I(t)$0 and the derivative-shaped quadrature $I(t)$1, the local oscillator upconverts them to the qubit frequency, and the IQ mixer combines them into the physical microwave drive. In ideal form, the drive is $I(t)$2. In practice, finite sampling, DAC sinc roll-off, LO phase noise, mixer imbalance, gain mismatch, and imperfect orthogonality between I and Q distort the effective pulse shape and degrade the intended DRAG cancellation (Patra et al., 23 Apr 2026).

This hardware sensitivity makes calibration intrinsic to DRAG rather than ancillary to it. Experimental implementations tune the derivative amplitude using spectroscopy or Rabi-style calibration sweeps, and later ultrafast work combines amplified amplitude error (AAE), amplified phase error (APE), and amplified leakage error (ALE) for in-situ optimization. ALE was introduced because conventional leakage randomized benchmarking accumulates leakage incoherently and becomes insensitive below about $I(t)$3, whereas the target regime was $I(t)$4; by coherently phase-matching the leaked amplitude, ALE amplifies control leakage while leaving thermal leakage largely unamplified (Gao et al., 27 Nov 2025).

At sub-10-ns timescales, microwave distortions themselves become a principal limitation. Analytical DRAG extensions have shown that non-Markovian coherent errors caused by temporal microwave control pulse distortions may be a significant source of error for sub-10-ns single-qubit gates unless corrected using predistortion. In that context, I-distortion and C-distortion were explicitly identified, and an exponential inverse-filter predistortion improved FAST DRAG-L gate errors by $I(t)$5 across 6–11 ns durations (Hyyppä et al., 2024).

DRAG also serves as a useful baseline for assessing what it does not optimize. In a study of transmon gates robust to parameter fluctuations, a calibrated 128 ns Gaussian pulse with DRAG corrections achieved nominal error $I(t)$6, comparable to two GRAPE-derived robust pulses. Under amplitude drift, however, the robust AROG pulse suppressed coherent added errors over 15 times more than DRAG, and during increases in dephasing robust pulses showed up to 1.7 times less added error than DRAG. The point was not that DRAG is ineffective, but that standard DRAG is calibrated for a nominal operating point rather than explicitly optimized over an ensemble of fluctuations (Wright et al., 27 Nov 2025).

6. Generalizations beyond conventional microwave single-qubit control

The logic of DRAG has proved portable across platforms, though the implementation details can differ sharply from the canonical microwave transmon setting. In single-flux-quantum control, the method has been “digitalized”: the derivative correction is realized not by a continuous analog quadrature but by additional timed SFQ kicks placed in specific clock slots. For a transmon with $I(t)$7, $I(t)$8, and fixed kick angle $I(t)$9, compact SFQ schedules stored in 22 bits or fewer were reported to achieve gate fidelities exceeding 99.99% (Shillito et al., 2023).

In tunable-coupler two-qubit gates, DRAG has been reformulated as $Q(t)$0-DRAG. Here the derivative shaping is applied not to a complex microwave quadrature pair but to the real-valued effective coupling $Q(t)$1 generated by flux control. For symmetric anharmonicities the second-order rule

$Q(t)$2

creates spectral nulls at the leakage detuning, while asymmetric anharmonicities motivate a fourth-order extension. Reported numerical results showed leakage suppression below $Q(t)$3 for fast entangling gates, with effectiveness retained within the fifteen nanosecond range (Georgiadis et al., 11 Jun 2026).

Neutral-atom and Rydberg systems provide two further adaptations. For a Rydberg-blockade entangling gate, analytic DRAG pulses were constructed from a smooth in-phase base pulse plus even derivatives so that the pulse spectrum vanished at selected leakage detunings. With additional constant detuning to correct phase errors and with optimized blockade, Bell states with fidelity $Q(t)$4 were reported in a 300 K environment for a gate time of only $Q(t)$5 (Theis et al., 2016). In a cavity-coupled Rydberg superatom, DRAG was adapted to suppress doubly excited leakage states during preparation of a collective single excitation. There, optimized DRAG improved the single excitation probability from the previous theoretical benchmark of 77% to 91.9%, and benchmarking against GRAPE showed that DRAG operated close to the optimal-control limit while retaining smooth, experimentally feasible pulse shapes (Aggarwal et al., 20 Feb 2026).

The portability of the underlying idea also clarifies its boundaries. In semiconductor spin-qubit CZ control, DRAG is treated as the canonical leakage-suppression strategy for short spectrally broad pulses, written as $Q(t)$6, but it is also described as infeasible for positive-valued baseband exchange pulses because implementing a DRAG pulse requires access to the complex domain through IQ bandpass modulation. That limitation motivated delayed leakage reduction (DLR), a baseband-friendly notch-forming method based on delayed pulse replicas rather than derivative quadrature (Polat et al., 4 Aug 2025). Similarly, in gate-based counterdiabatic algorithms, regularized adiabatic gauge potentials are said to be DRAG-like in spirit—because they selectively suppress transitions out of a target state—but not instances of the specific named microwave-control protocol (Vreumingen, 2024).

Taken together, these developments position DRAG as both a specific transmon pulse recipe and a broader design principle. In the narrow sense, it is the derivative quadrature $Q(t)$7 used to reduce $Q(t)$8 leakage and AC-Stark-induced phase error in weakly anharmonic superconducting qubits. In the broader sense, it is a family of derivative-based, leakage-aware, counterdiabatic pulse constructions that engineer spectral zeros or approximate transitionless dynamics under realistic control constraints (Theis et al., 2018, Patra et al., 23 Apr 2026).