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DRAG Pulse

DRAG (Derivative Removal by Adiabatic Gate) is a pulse-shaping technique that adds the derivative of the main envelope as an orthogonal quadrature component, cancelling unwanted transitions to the leakage state |2⟩ in weakly anharmonic qubits like the transmon.

What it means

A transmon is not a perfect two-level system but a weakly anharmonic ladder: the |1⟩→|2⟩ transition sits only slightly below the |0⟩→|1⟩ transition (the gap is the anharmonicity).Any fast pulse therefore risks driving population out of the computational subspace into |2⟩ — this is leakage.The problem is pure Fourier physics: time-frequency uncertainty (Δt·Δf ≳ 1) means a shorter pulse necessarily has a wider spectrum, and that spectral width overlaps the |1⟩→|2⟩ line.Smoothing the envelope into a Gaussian narrows the spectrum, but at the gate speeds needed to beat decoherence it is not enough.DRAG (Motzoi, Gambetta, Rebentrost & Wilhelm, 2009) is the elegant fix: alongside the in-phase envelope ε_I(t), send its time derivative, scaled by the anharmonicity, on the quadrature channel — ε_Q(t) ∝ −dε_I/dt.The derivative component destructively interferes with exactly the spectral weight that would drive |1⟩→|2⟩, suppressing leakage without lengthening the gate.DRAG is the standard workhorse of superconducting qubit control today; fully numerically optimized pulses can go further — Werninghaus et al.(2021) demonstrated 1/7 the leakage of the best DRAG pulse at equal gate time.

Everyday analogy

Think of closing a door gently versus slamming it. A sudden slam — bang! — shakes every window in the house: the abrupt motion contains many 'frequencies', and some of them rattle things you never meant to touch. DRAG is the clever way of closing the door: as you push, you add a second, precisely timed counter-motion (the derivative of the push) that cancels the shaking before it starts. The door still closes just as fast — but the windows (the |2⟩ state) stay quiet.
DRAG is an acronym: Derivative Removal by Adiabatic Gate. 'Derivative' comes from Latin derivare, 'to draw water off from a stream' (de- + rivus 'stream') — a derivative is a new flow drawn from the original function, and DRAG literally sends that drawn-off flow down the second (quadrature) channel. 'Adiabatic' is Greek a-dia-batos, 'not passable through' — no population is allowed to pass through into the leakage level.

Common misconceptions

  • Shortening pulses is NOT free — time-frequency uncertainty (Δt·Δf ≳ 1) means a shorter pulse has a wider spectrum that overlaps the |1⟩→|2⟩ transition and excites |2⟩. This is Fourier physics, unavoidable in principle, and only manageable by shaping the pulse.
  • A smooth Gaussian envelope alone does NOT solve leakage — Gaussian smoothing narrows the spectrum, but at gate speeds fast enough to beat decoherence it still leaves too much spectral weight at the leakage transition. That is precisely why the DRAG correction was invented.

Key takeaways

  • DRAG adds the time derivative of the main envelope, scaled by the anharmonicity, as an orthogonal quadrature component: ε_Q(t) ∝ −dε_I(t)/dt.
  • The derivative component destructively interferes with the spectral weight at the |1⟩→|2⟩ transition, suppressing leakage without lengthening the gate.
  • Gaussian smoothing narrows the pulse spectrum but is not enough on its own at decoherence-beating gate speeds.
  • Fully optimized control pulses achieved 1/7 the leakage of the best DRAG pulse at equal gate time (Werninghaus et al., npj Quantum Information, 2021).

Check your understanding

Why does DRAG add the derivative of the envelope on the quadrature channel?

  1. A.To make the pulse shorter so more gates fit before decoherence
  2. B.To destructively cancel the spectral weight driving the unwanted |1⟩→|2⟩ transition
  3. C.To increase the rotation angle without increasing amplitude
  4. D.To compensate for cable delays between the AWG and the chip
Show the answer

Answer: B. To destructively cancel the spectral weight driving the unwanted |1⟩→|2⟩ transition

Why: The derivative component, scaled by the anharmonicity, interferes destructively with exactly the frequency content that would drive |1⟩→|2⟩, suppressing leakage while keeping the gate just as fast.

Why do shorter pulses leak more into the |2⟩ state?

  1. A.Because the DAC cannot output short pulses accurately
  2. B.Because shorter pulses deliver more total energy to the qubit
  3. C.Because time-frequency uncertainty (Δt·Δf ≳ 1) gives short pulses a wider spectrum that overlaps the |1⟩→|2⟩ line
  4. D.Because short pulses heat the dilution refrigerator
Show the answer

Answer: C. Because time-frequency uncertainty (Δt·Δf ≳ 1) gives short pulses a wider spectrum that overlaps the |1⟩→|2⟩ line

Why: A pulse compressed in time is necessarily broadened in frequency — Fourier physics. The broadened spectrum overlaps the nearby |1⟩→|2⟩ transition of the weakly anharmonic transmon, driving leakage. It cannot be avoided, only managed by pulse shaping such as DRAG.

Builds on

Microwave Pulse

Primary source: Motzoi, Gambetta, Rebentrost & Wilhelm, Phys. Rev. Lett. 103, 110501 (2009), doi:10.1103/PhysRevLett.103.110501

DRAG origin paper; experimental leakage reduction per Werninghaus et al., npj QI 7, 14 (2021).

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