Source-verifiedLevel 4

Leakage

Leakage is the escape of quantum population out of the computational subspace {|0⟩, |1⟩} into higher levels such as |2⟩ — the price of short pulses whose wide spectra overlap the ω12 transition.

What it means

Leakage is the escape of quantum population out of the computational subspace {|0⟩, |1⟩} into higher levels such as |2⟩.Its root cause is Fourier's time–bandwidth relation: a pulse of duration Δt necessarily has spectral width Δf ≳ 1/Δt.As gates get shorter — modern entangling gates run around 40 ns — the pulse spectrum broadens until its tails overlap the ω12 transition, which in a transmon lies only 200–300 MHz below the qubit frequency.This is physics, not engineering: it cannot be avoided, only shaped.Smooth Gaussian envelopes narrow the spectrum but are insufficient on their own; the DRAG technique adds the envelope's derivative on a quadrature component to systematically cancel the unwanted transition, and optimal-control pulses have experimentally cut leakage to one-seventh of the best DRAG result at equal gate time.Leakage is qualitatively worse than a bit error, because the state leaves the game board entirely, breaking the error model assumed by standard quantum error correction.Together with decoherence it defines a U-shaped total-error curve: fast gates are leakage-limited, slow gates are coherence-limited, and pulse design lives at the valley's minimum.

Everyday analogy

Close a door gently and only that door moves. Slam it — BANG! — and every window in the house rattles. A fast pulse is that slam: you meant to call only the floor 1→2 elevator, but the sound is so broad — the spectrum so wide — that the elevator to floor 3 answers too. That is how the state leaks into |2⟩.
'Leak' comes from Old Norse leka, 'to drip through' — a word born in the hulls of Viking ships. A thousand years apart, shipwrights and quantum engineers share the same worst enemy: something precious seeping out of the vessel that was meant to hold it.

Common misconceptions

  • Leakage is NOT just another bit error — the state leaves the computational subspace entirely, which breaks the error-model assumptions of standard quantum error correction and makes leakage qualitatively worse than a bit flip.
  • Faster gates are NOT always better — the time-bandwidth limit Δt·Δf ≳ 1 is physics, not engineering, so leakage from short pulses can only be shaped (Gaussian smoothing, DRAG, optimal control), never avoided outright.

Key takeaways

  • Δt·Δf ≳ 1: a short pulse necessarily has a wide spectrum — below ~40 ns the spectral tails start knocking on the ω12 transition, only 200–300 MHz away.
  • Total gate error forms a U-shaped curve: fast gates are leakage-limited (left wall), slow gates are decoherence-limited (right wall), and the optimum sits in the valley between them.
  • Pulse shaping pushes the leakage wall back: Gaussian envelopes narrow the spectrum, DRAG cancels the unwanted transition, and optimal control has cut leakage 7-fold versus the best DRAG at equal gate time (Werninghaus et al., 2021).

Check your understanding

Why do very short control pulses cause leakage into |2⟩?

  1. A.They deposit too much total energy in the qubit
  2. B.Their spectral width Δf ≳ 1/Δt becomes wide enough for the pulse spectrum to overlap the ω12 transition
  3. C.They heat the dilution refrigerator above 15 mK
  4. D.They accidentally excite the readout resonator
Show the answer

Answer: B. Their spectral width Δf ≳ 1/Δt becomes wide enough for the pulse spectrum to overlap the ω12 transition

Why: By the Fourier time-bandwidth relation, a pulse of duration Δt has spectral width of at least ~1/Δt. Shortening the gate broadens the spectrum until its tails reach ω12 — only 200–300 MHz below the qubit frequency — and drive population into |2⟩.

Why is leakage considered worse than an ordinary bit-flip error?

  1. A.It occurs far more frequently than bit flips
  2. B.It permanently damages the qubit hardware
  3. C.The state leaves the computational subspace entirely, violating the error model assumed by standard quantum error correction
  4. D.It can only be detected after warming up the chip
Show the answer

Answer: C. The state leaves the computational subspace entirely, violating the error model assumed by standard quantum error correction

Why: A bit flip stays inside {|0⟩, |1⟩}, where standard error-correcting codes know how to handle it. A leaked state sits in |2⟩ — off the game board — outside the error model those codes assume, which is why recent work still calls leakage the bottleneck of control precision.

Builds on

TransmonAnharmonicity

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

Time-bandwidth leakage mechanism per Motzoi 2009; 1/7 leakage experimental result per Werninghaus et al., npj QI 7, 14 (2021), doi:10.1038/s41534-020-00346-2; arXiv:2606.29854 verified 2026-07-03.

Learn it hands-on

This concept is part of a 46-level curriculum with an interactive simulator and Lumen, a tutor whose answers are verified before you see them. Levels 1–5 are free.

Start freeSee the curriculum