Why not use a simple square pulse?

A square pulse has a sinc frequency spectrum with high sidelobes that can excite transitions to non-computational levels (|2> state in a transmon). These leakage transitions reduce gate fidelity. The Blackman envelope's steep sidelobe rolloff (-58 dB first sidelobe) keeps the drive spectrum tightly confined to the 0-1 transition frequency.

What frequencies are involved?

Transmon qubit drive frequencies are typically 4-8 GHz, with anharmonicity of 200-300 MHz. The Blackman pulse envelope modulates a carrier at the qubit frequency. Pulse durations are 10-50 ns. The envelope bandwidth must be much smaller than the anharmonicity to avoid leakage: BW << alpha = 200-300 MHz.

Quantum Computing RF

Blackman Pulse

Q: Blackman vs Gaussian vs DRAG?

Gaussian pulses have good spectral confinement but non-zero tails requiring truncation. Blackman pulses have naturally finite support with lower sidelobes (-58 dB vs -13 dB for rectangular). DRAG (Derivative Removal by Adiabatic Gate) adds a quadrature correction to either envelope to cancel leakage to first order, improving gate fidelity from 99.5% to 99.9%+.

A Blackman Pulse is a smooth, finite-duration waveform envelope used to shape qubit drive signals in superconducting quantum computers. The Blackman window's steep spectral rolloff (−58 dB first sidelobe) minimizes leakage to non-computational energy levels (|2⟩ state) in transmon qubits, improving single-qubit gate fidelity. The pulse modulates a microwave carrier at the qubit transition frequency (typically 4-8 GHz) for durations of 10-50 ns.

Category: Quantum Computing RF

1st Sidelobe: −58 dB

Understanding Blackman Pulses

Transmon qubits are weakly anharmonic oscillators: the |0⟩→|1⟩ and |1⟩→|2⟩ transition frequencies differ by only the anharmonicity α ≈ 200-300 MHz. A drive pulse intended for the 0-1 transition must avoid exciting the 1-2 transition. This requires the pulse spectrum to be narrower than α, which demands a smooth, spectrally compact envelope.

The Blackman window achieves this with three cosine terms that produce extremely low spectral sidelobes, keeping leakage below 10⁻⁴ for typical gate durations.

Blackman Pulse Envelope

w(t) = 0.42 − 0.5·cos(2πt/T) + 0.08·cos(4πt/T)
0 ≤ t ≤ T (gate duration)

Drive signal:
s(t) = A·w(t)·cos(2πf₀₁t + φ)
f₀₁ = qubit transition frequency

Pulse Envelope Comparison

Envelope	1st Sidelobe	Main Lobe Width	Leakage
Rectangular	−13 dB	Narrowest	High
Gaussian	−43 dB	Medium	Low
Blackman	−58 dB	Wider	Very low
Blackman + DRAG	−58 dB	Wider	Minimal

Common Questions

Frequently Asked Questions

Why not a square pulse?

Square pulses have −13 dB sinc sidelobes that excite |2⟩ leakage. Blackman's −58 dB sidelobes keep the spectrum within the qubit's anharmonicity window.

Blackman vs Gaussian vs DRAG?

Gaussian has good confinement but requires truncation. Blackman has naturally finite support with lower sidelobes. DRAG adds quadrature correction for 99.9%+ gate fidelity.

What frequencies?

Transmon drives at 4-8 GHz, anharmonicity 200-300 MHz. Pulse durations 10-50 ns. Envelope bandwidth must be much less than the anharmonicity.

Related Terms

← Black Box Model Blackman Window →

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