Leakage out of the Computational Subspace

A physical qubit is almost never a true two-level system. Superconducting transmons, trapped-ion and neutral-atom qubits, and semiconductor spins all carry a ladder of energy levels, of which only two are anointed as the computational states $|0\rangle$ and $|1\rangle$ . The rest — the leakage subspace — are nominally idle, but control fields, residual couplings, and thermal processes keep pushing population into them. The moment population leaves the computational subspace $\mathcal{H}_{\mathrm C}=\mathrm{span}\{|0\rangle,|1\rangle\}$ , the tidy abstraction of a "qubit channel" quietly fails. Leakage is qualitatively unlike the noise channels built for genuine two-level systems. An amplitude-damping or depolarizing channel maps qubit density operators to qubit density operators; leakage instead ships the qubit block of the density matrix into a larger space, so its restriction to $\mathcal{H}_{\mathrm C}$ is trace-decreasing. This is exactly what breaks quantum error correction. Stabilizer measurement decomposes errors into Pauli operators acting within the code's registers — but a leaked qubit carries no well-defined logical value, corrupts its neighbours through the very couplings meant to entangle them, and is invisible to a naive decoder. It is a silent error that accumulates round after round and can defeat the threshold theorem unless handled explicitly. Two regimes deserve separation. Coherent leakage comes from unitary control imperfections — chiefly off-resonant driving of the $|1\rangle\!\to\!|2\rangle$ transition during single-qubit gates, set by the anharmonicity, and the deliberate use of higher levels in two-qubit gates (the controlled-phase gate rides the $|11\rangle\!-\!|02\rangle$ avoided crossing). It is reversible in principle and can be engineered away with DRAG pulse shaping. Incoherent leakage comes from the environment — thermal excitation, heating, photon-assisted transitions — and lives in the Lindblad framework. This article quantifies both, derives the off-resonant leakage of a driven anharmonic qubit, and proves the DRAG cancellation condition.

Key takeaways

Real qubits are the lowest two rungs of a multi-level ladder (a weakly anharmonic transmon, or hyperfine manifolds in ions and atoms); population that escapes the chosen span $\{|0\rangle,|1\rangle\}$ into auxiliary levels such as $|2\rangle$ is leakage.
Leakage is fundamentally not a Pauli error: the lost population is not described by any channel acting inside the qubit Hilbert space, so it violates the assumptions of stabilizer error correction and is invisible as an ordinary $X$ , $Y$ , or $Z$ flip.
The dominant coherent mechanism in weakly anharmonic qubits is off-resonant drive of the $|1\rangle\!\to\!|2\rangle$ transition, governed by the anharmonicity $\alpha=\omega_{12}-\omega_{01}$ ; leakage scales as $(\Omega/2\alpha)^2$ , so faster gates leak more — a speed-versus-leakage tradeoff.
DRAG (Derivative Removal by Adiabatic Gate) suppresses coherent leakage to leading order by adding a quadrature drive $\Omega_y=-\dot\Omega_x/\alpha$ plus a dynamic detuning $\delta=\Omega_x^2/2\alpha$ .
Formally, leakage is a CPTP map on the full multi-level space whose restriction to the qubit block is trace-decreasing; the leakage rate $L$ (out per operation) and seepage rate $S$ (back in) are the figures of merit, and incoherent residual leakage is managed by leakage-reduction units (LRUs) and reset that pump population from $|2\rangle$ back into the qubit subspace.
Key idea: Leakage is not an error inside the qubit channel — it is the failure of the channel to stay inside the qubit Hilbert space. Quantify it by a leakage rate $L$ and a seepage rate $S$ , and engineer control so $L$ falls faster than the gate's speed demands.

The core insight

Leakage is not an error inside the qubit channel — it is the failure of the channel to stay inside the qubit Hilbert space. Quantify it by a leakage rate $L$ (population out per gate) and a seepage rate $S$ (population back in), and engineer control so $L$ is suppressed faster than the gate's speed demands.

Where this lives today

Leakage control is now load-bearing infrastructure in every serious quantum-computing roadmap. On single-qubit gates, DRAG-shaped pulses are universal across transmon processors: the in-phase envelope drives the rotation while $\Omega_y=-\dot\Omega_x/\alpha$ and $\delta=\Omega_x^2/2\alpha$ hold coherent leakage below the decoherence-limited error, enabling ~20–30 ns gates at single-qubit fidelities beyond 99.9%. On two-qubit gates the physics inverts: the controlled-phase gate pushes $|11\rangle$ near resonance with $|02\rangle$ to harvest a conditional phase, deliberately visiting the leakage level, and net-zero or adiabatic flux pulses are shaped to return the population coherently since residual $|02\rangle$ occupation is the dominant two-qubit leakage channel. At the system level, surface-code experiments from Google and others interleave leakage-reduction units — reset or swap operations that flush leaked population each cycle — because otherwise leakage accumulates and caps the logical error rate; the 2023 "below threshold" results depend on it. For simulation, the Pauli-twirling approximation replaces residual post-reduction leakage with an effective depolarizing channel, tractable inside stabilizer simulators. Leakage-aware decoders using analog readout to flag leaked qubits, and the leakage budget of fluxonium architectures, remain active fronts.

The physical picture

Picture the qubit as the bottom two rungs of a slightly uneven ladder. To flip $|0\rangle\leftrightarrow|1\rangle$ you push at the rung-spacing frequency $\omega_{01}$ . But the next rung sits at $\omega_{12}=\omega_{01}+\alpha$ , only slightly detuned because the anharmonicity $\alpha$ is small. A short, hard push has a broad frequency spectrum that inevitably rattles the $|1\rangle\!\to\!|2\rangle$ rung as well — that spilled population is leakage, and the faster the gate the worse the spill. The first figure fixes the geometry: the ladder, the two nearby transition frequencies, and the detuning $\alpha$ that is the only thing keeping population out of $|2\rangle$ .

Figure

The transmon as an anharmonic ladder: a drive resonant with ω01/2π = 5.00 GHz also rattles the 1→2 rung at ω12/2π = 4.80 GHz, only α/2π = −200 MHz away, so any drive bandwidth reaching that detuning spills population into level 2 as leakage

DRAG is the trick of pushing partly sideways. By adding a control quadrature shaped like the time-derivative of the main pulse, the spurious amplitude that would have climbed to $|2\rangle$ is made to interfere destructively with itself and return to the qubit subspace by the end of the gate — a coherent cancellation, exact only to leading order in $1/\alpha$ . Distinguish leakage from seepage the way you distinguish evaporation from condensation: leakage $L$ is the rate probability evaporates out of the qubit subspace per operation, seepage $S$ the rate it condenses back in. A passive process with detailed balance has both; an active LRU drives $S$ toward a known state so error correction can proceed. The second figure is the number an experimentalist lives with — how much an unshaped pulse leaks, and how slow it must be to hit a fault-tolerant target.

Figure

Leakage of an unshaped square π-pulse falls as Pleak = Ω²/2α² ∝ 1/tg²: on a α/2π = −200 MHz transmon a 20 ns gate leaks 0.8 percent, and reaching the 10⁻⁴ fault-tolerance target needs 177 ns unshaped — DRAG hits that target already at 20 ns

Why error correction hates leakage: a stabilizer code asks "which Pauli happened?" A leaked qubit answers with a shrug — its state is not in the question's domain. Worse, the residual couplings the code uses to compute syndromes let a leaked qubit corrupt its neighbours every cycle, so a single leakage event spreads. The remedy is to convert the leakage back into a legitimate (if erroneous) qubit state the decoder can handle as an ordinary error.

The core equations

The starting point decomposes the full Hilbert space: only the first two levels are computational, everything above is leakage, and every quantity below is bookkeeping about how population crosses between the two blocks.

\mathcal{H} = \mathcal{H}_{\mathrm C} \oplus \mathcal{H}_{\mathrm L}, \qquad \mathcal{H}_{\mathrm C}=\mathrm{span}\{|0\rangle,|1\rangle\} \qquad (1)

where: $\mathcal{H}_{\mathrm C}$ — computational (qubit) subspace, dimension $d_{\mathrm C}=2$ ; $\mathcal{H}_{\mathrm L}$ — leakage subspace, spanned by $|2\rangle,|3\rangle,\dots$ , dimension $d_{\mathrm L}$ . The device lives in $\mathcal{H}$ ; the algorithm only ever intends to use $\mathcal{H}_{\mathrm C}$ . The two blocks are picked out by orthogonal projectors that measure "how much population is where" independent of any particular state.

P_{\mathrm C} = |0\rangle\!\langle 0| + |1\rangle\!\langle 1|, \qquad P_{\mathrm L} = \mathbb{1} - P_{\mathrm C} \qquad (2)

where: $P_{\mathrm C}$ — projector onto the qubit subspace; $P_{\mathrm L}$ — projector onto the leakage subspace. The resolution of identity $P_{\mathrm C}+P_{\mathrm L}=\mathbb{1}$ , inserted inside a channel, exposes the leakage cross-terms. The average leakage rate is the fraction of population an operation ejects from the qubit block, computed from a maximally mixed qubit input so it does not depend on the starting logical state. This is the Wood–Gambetta definition.

L \;=\; 1 - \frac{1}{d_{\mathrm C}}\,\mathrm{Tr}\!\left[P_{\mathrm C}\,\mathcal{E}(P_{\mathrm C})\right] \qquad (3)

where: $\mathcal{E}$ — CPTP map of the gate on the full Hilbert space; $d_{\mathrm C}=2$ ; $L$ — average fraction of qubit population lost to $\mathcal{H}_{\mathrm L}$ per operation. Feed in $P_{\mathrm C}/d_{\mathrm C}$ , project the output back onto $\mathcal{H}_{\mathrm C}$ , and $L$ is whatever trace went missing. Seepage is the mirror image: the fraction of leakage-subspace population an operation returns to the qubit block, from a maximally mixed input on $\mathcal{H}_{\mathrm L}$ . A leakage-reduction unit is engineered to make $S$ large.

S \;=\; \frac{1}{d_{\mathrm L}}\,\mathrm{Tr}\!\left[P_{\mathrm C}\,\mathcal{E}(P_{\mathrm L})\right] \qquad (4)

where: $S$ — average fraction of leakage-subspace population returned to $\mathcal{H}_{\mathrm C}$ per operation; $d_{\mathrm L}$ — dimension of the leakage subspace; $P_{\mathrm L}/d_{\mathrm L}$ — maximally mixed state on $\mathcal{H}_{\mathrm L}$ . Together $L$ and $S$ are the transition rates of a two-compartment population model, related by detailed balance in a passive device. For the coherent mechanism we need the driven Hamiltonian. In the frame rotating at the qubit frequency, a resonant drive couples $|0\rangle\!-\!|1\rangle$ and $|1\rangle\!-\!|2\rangle$ , and the auxiliary level sits detuned by exactly the anharmonicity.

\hat H_{\mathrm{rot}} = \begin{pmatrix} 0 & \tfrac{\Omega(t)}{2} & 0 \\[2pt] \tfrac{\Omega^*(t)}{2} & 0 & \tfrac{\sqrt{2}\,\Omega(t)}{2} \\[2pt] 0 & \tfrac{\sqrt{2}\,\Omega^*(t)}{2} & \alpha \end{pmatrix} \qquad (5)

where: $\Omega(t)$ — complex drive envelope (Rabi frequency, $\hbar=1$ ); $\sqrt{2}$ — bosonic enhancement of the $|1\rangle\!\to\!|2\rangle$ matrix element; $\alpha=\omega_{12}-\omega_{01}$ — anharmonicity, the static detuning of $|2\rangle$ (negative for transmons). The single small parameter $\Omega/\alpha$ controls everything — leakage exists because $\alpha$ is not infinite. DRAG cancels the leading leakage amplitude with a quadrature drive proportional to minus the derivative of the in-phase envelope, plus a dynamic detuning that repairs the phase.

\Omega_y(t) = -\,\frac{\dot{\Omega}_x(t)}{\alpha}, \qquad \delta(t) = \frac{\Omega_x^2(t)}{2\alpha} \qquad (6)

where: $\Omega_x(t)$ — in-phase (target) envelope; $\Omega_y(t)$ — quadrature (DRAG) envelope; $\dot\Omega_x$ — its time derivative; $\delta(t)$ — dynamic frequency shift compensating the AC-Stark pull from the virtual $|2\rangle$ excursion. Shape the derivative into the orthogonal channel and the population climbing toward $|2\rangle$ comes back down. Incoherent leakage — the part control cannot fix — obeys a GKSL master equation with jump operators moving population up and down the $|1\rangle\!-\!|2\rangle$ rung at thermally set rates.

\frac{\mathrm{d}\rho}{\mathrm{d}t} = -i[\hat H,\rho] + \Gamma_{\downarrow}\,\mathcal{D}[|1\rangle\!\langle 2|]\rho + \Gamma_{\uparrow}\,\mathcal{D}[|2\rangle\!\langle 1|]\rho \qquad (7)

where: $\mathcal{D}[L]\rho = L\rho L^\dagger-\tfrac12\{L^\dagger L,\rho\}$ — GKSL dissipator; $\Gamma_{\downarrow}$ — seepage (relaxation $|2\rangle\!\to\!|1\rangle$ ); $\Gamma_{\uparrow}$ — thermal leakage (excitation $|1\rangle\!\to\!|2\rangle$ ). Even with perfect control, finite temperature keeps $\Gamma_{\uparrow}\neq0$ — a residual leakage floor.

Derivation ALeakage and seepage as the trace deficit of the qubit block

The goal is to prove the structural fact that makes leakage special: a channel perfectly trace-preserving on the full multi-level Hilbert space is nonetheless trace-decreasing when restricted to the qubit block, and the deficit is exactly the leakage rate $L$ . We start from the Kraus form of a CPTP map, insert the resolution of identity to split each Kraus operator into a part that stays in $\mathcal{H}_{\mathrm C}$ and a part that escapes to $\mathcal{H}_{\mathrm L}$ , and read off why no Pauli channel can reproduce the result — the formal statement behind "leakage is invisible to a stabilizer decoder."

Step 1. Write the gate as a CPTP map $\mathcal{E}(\rho)=\sum_k K_k\rho K_k^\dagger$ with Kraus operators obeying the completeness relation $\sum_k K_k^\dagger K_k = \mathbb{1} = P_{\mathrm C}+P_{\mathrm L}$ on the full space. Why: completeness on $\mathcal{H}$ is trace preservation on $\mathcal{H}$ — the channel loses nothing overall; the question is only where the population ends up.

Step 2. Feed in the maximally mixed qubit state $\rho_{\mathrm C}=P_{\mathrm C}/d_{\mathrm C}$ and compute the surviving qubit-block trace $\mathrm{Tr}[P_{\mathrm C}\,\mathcal{E}(\rho_{\mathrm C})]=\sum_k\mathrm{Tr}[P_{\mathrm C}K_k\rho_{\mathrm C}K_k^\dagger]$ . Since $\rho_{\mathrm C}=P_{\mathrm C}\rho_{\mathrm C}P_{\mathrm C}$ the input stays in the block, but the output can land in either. Why: the cross-terms $P_{\mathrm C}K_k\rho_{\mathrm C}K_k^\dagger P_{\mathrm L}$ are exactly the amplitude leaving the qubit subspace.

Step 3. The population that fails to return is $\sum_k\mathrm{Tr}[P_{\mathrm L}K_k\rho_{\mathrm C}K_k^\dagger]$ , so $\mathrm{Tr}[P_{\mathrm C}\,\mathcal{E}(\rho_{\mathrm C})]=1-\sum_k\mathrm{Tr}[P_{\mathrm L}K_k\rho_{\mathrm C}K_k^\dagger]$ , and by definition (3) that second term is exactly $L$ . Why: the qubit-block trace drops by $L$ — the restricted channel is trace-decreasing whenever any Kraus operator has a $P_{\mathrm L}K_kP_{\mathrm C}$ component.

Step 4. A trace-decreasing qubit map has no Pauli-mixture representation, since every Pauli channel is trace-preserving on $\mathcal{H}_{\mathrm C}$ . No syndrome can account for the missing population, so restoring trace preservation requires seepage — an LRU or reset pumping $|2\rangle$ back down as in definition (4). Why: this is the precise reason leakage must be converted, not merely decoded.

Go deeper · Derivation B · 1 minOff-resonant leakage of a driven anharmonic qubit

Derivation BOff-resonant leakage of a driven anharmonic qubit

The goal is to compute how much population a resonant single-qubit pulse pumps into $|2\rangle$ as a function of drive strength $\Omega$ and anharmonicity $\alpha$ , recovering the central $(\Omega/2\alpha)^2$ scaling that makes leakage the dominant coherent error in fast transmon gates. We reduce the three-level dynamics to the $\{|1\rangle,|2\rangle\}$ block, treat it as an off-resonant two-level Rabi problem with detuning $\alpha$ , and expand for weak driving. The payoff is the quantitative origin of the speed-versus-leakage tradeoff — the target DRAG must beat.

Step 1. Isolate the leaky transition. While the drive rotates $|0\rangle\!\leftrightarrow\!|1\rangle$ , the same envelope couples $|1\rangle\!\leftrightarrow\!|2\rangle$ with Rabi frequency $\sqrt{2}\,\Omega/2$ and fixed detuning $\alpha$ , read straight from the bottom-right corner of $\hat H_{\mathrm{rot}}$ . Why: near $|1\rangle$ the problem is a textbook driven two-level system off-resonant by $\alpha$ , so standard Rabi physics applies.

Step 2. Apply the off-resonant Rabi formula. The population reaching $|2\rangle$ oscillates as

|c_2|^2 = \frac{(\sqrt{2}\,\Omega/2)^2}{(\sqrt{2}\,\Omega/2)^2 + \alpha^2}\,\sin^2\!\left(\tfrac{1}{2}\sqrt{(\sqrt{2}\,\Omega/2)^2+\alpha^2}\;t\right),

where the prefactor caps the maximum transferable population and the generalized Rabi frequency sets the oscillation. Why: a detuned drive can never fully invert; the detuning $\alpha$ throttles both the amplitude and the speed of the excursion to $|2\rangle$ .

Step 3. Expand for weak driving. When $\Omega\ll|\alpha|$ the detuning dominates the generalized Rabi frequency, the prefactor becomes $(\sqrt2\,\Omega/2)^2/\alpha^2$ , and the peak leakage is

P_{\mathrm{leak}} \;\lesssim\; \frac{(\sqrt{2}\,\Omega/2)^2}{\alpha^2} = \left(\frac{\sqrt{2}\,\Omega}{2\alpha}\right)^2 = \frac{\Omega^2}{2\alpha^2}.

Why: in the regime every real qubit operates in, the peak leakage is set purely by the ratio of drive amplitude to anharmonicity, with no dependence on pulse details.

Step 4. Read off the tradeoff. Leakage grows quadratically with $\Omega$ ; a square π-pulse has $\Omega=\pi/t_{\mathrm g}$ , so $P_{\mathrm{leak}}=\pi^2/(2\alpha^2 t_{\mathrm g}^2)\propto 1/t_{\mathrm g}^2$ — halving the gate time quadruples the leakage. Why: this is the exact curve plotted in the second figure and the quantitative pressure that DRAG is engineered to relieve. *As a quick check: with $|\alpha|/2\pi=200$ MHz and $t_{\mathrm g}=20$ ns, one has $\Omega/2\pi=25$ MHz and $P_{\mathrm{leak}}=\tfrac12(0.125)^2=7.8\times10^{-3}$ , and the formula correctly gives $P_{\mathrm{leak}}\to0$ as $\Omega\to0$ .*

Go deeper · Derivation C · 1 minDRAG cancellation to leading order

Derivation CDRAG cancellation to leading order

The goal is to show that adding a quadrature envelope $\Omega_y=-\dot\Omega_x/\alpha$ cancels the leading $1/\alpha$ leakage amplitude, and that a companion detuning $\delta=\Omega_x^2/2\alpha$ repairs the phase the correction disturbs. The strategy is an adiabatic frame transformation that dresses $|1\rangle$ with a small admixture of $|2\rangle$ set by the instantaneous drive; the frame change generates a geometric term against which the physical quadrature control is tuned to interfere destructively. This turns the previous derivation's speed-versus-leakage wall into a soft one.

Step 1. Adiabatically eliminate $|2\rangle$ . Define a time-dependent unitary $V(t)$ that diagonalizes the instantaneous coupling to $|2\rangle$ , dressing $|1\rangle$ with an admixture $\propto \Omega(t)/\alpha$ so the fast virtual excursion is absorbed into the effective qubit states. Why: leakage is a real transition riding on a virtual excursion; the frame change separates the reversible virtual part from the irreversible leaked part.

Step 2. Track the residual transition amplitude. Because $V(t)$ is time dependent, transforming the generator produces a geometric contribution $\partial_t V$ , and the net $|1\rangle\!\to\!|2\rangle$ drive becomes

\langle 2|\,V^\dagger\!\left(-i[\hat H,\cdot]+\partial_t\right)V\,|1\rangle \;\propto\; \Omega_y(t) + \frac{\dot\Omega_x(t)}{\alpha}.

The physical control supplies $\Omega_y$ ; the frame supplies $\dot\Omega_x/\alpha$ ; their sum is the leftover leakage drive. Why: the derivative term is unavoidable geometry, so the only lever is to feed in a quadrature that exactly opposes it.

Step 3. Impose cancellation. Setting the bracket to zero gives the leading-order DRAG condition $\Omega_y(t)=-\dot\Omega_x(t)/\alpha$ , which zeroes the leakage amplitude over the whole pulse rather than only on average. Why: shaping the quadrature as minus the in-phase derivative makes the induced and geometric amplitudes cancel instant by instant, not just at the endpoints.

Step 4. Repair the phase. The virtual $|2\rangle$ excursion also pushes $|1\rangle$ down by a second-order AC-Stark shift $\Omega_x^2/2\alpha$ , mistuning the intended rotation. Compensate it with the dynamic detuning $\delta(t)=\Omega_x^2(t)/2\alpha$ . With both corrections the residual leakage falls to $\mathcal{O}(1/\alpha^2)$ or higher, buying roughly a factor of a few in gate speed at fixed leakage. Why: cancelling leakage without fixing the induced phase would trade a leakage error for a coherent over/under-rotation — DRAG must do both to be useful.

Limiting cases

Leakage severity is fixed by the interplay of anharmonicity, gate speed, and temperature; the four corners below bracket every real device.

Case	Result	Meaning
Large anharmonicity (ideal qubit)	$\vert\alpha\vert \gg \Omega_{\max} \Rightarrow L\to0$	When the anharmonicity dwarfs the drive amplitude, the $\vert1\rangle\!\to\!\vert2\rangle$ transition is far off-resonant and leakage is negligible — the two-level approximation is excellent (ions, atoms).
Weakly anharmonic transmon	$L\sim(\Omega/2\alpha)^2$	Off-resonant leakage scales as the squared drive-to-anharmonicity ratio; with $\alpha/2\pi\approx-200$ to $-300$ MHz this forces either slow gates or DRAG.
Fast (broadband) gate limit	$t_{\mathrm g}\vert\alpha\vert \lesssim 1 \Rightarrow L=\mathcal{O}(1)$	If the gate time approaches the inverse anharmonicity, the pulse bandwidth blankets $\omega_{12}$ and leakage becomes order unity — the hard speed limit for unshaped pulses.
Thermal (incoherent) limit	$\Gamma_{\uparrow}=\Gamma_{\downarrow}\,e^{-\hbar\omega_{12}/k_{\mathrm B}T}$	Detailed balance fixes thermal excitation into $\vert2\rangle$ relative to relaxation out; leakage persists even with perfect control whenever $T>0$ .

Common trapLeakage is just another Pauli or depolarizing error

It is not — at least not until you make it one. Leaked population lives outside the qubit space, so the qubit-block map is trace-decreasing on $\mathcal{H}_{\mathrm C}$ and no Pauli operator takes $|1\rangle$ to $|2\rangle$ . A stabilizer decoder assuming a trace-preserving Pauli mixture is blind to the missing population, while the leaked qubit silently corrupts its neighbours every cycle. Only after a leakage-reduction or reset step converts the population back into a definite (possibly erroneous) qubit state is the depolarizing approximation justified — and even then it must respect the seepage rate $S$ . Modelling residual leakage as a depolarizing channel from the outset overstates the code's protection and hides its most dangerous failure mode.

Worked examples

1 · Leakage of an unshaped fast gate (fully worked)

A transmon has anharmonicity $\alpha/2\pi=-200\ \mathrm{MHz}$ . You drive a single-qubit $\pi$ rotation with an unshaped square pulse of duration $t_{\mathrm g}=20\ \mathrm{ns}$ . Estimate the peak leakage into $|2\rangle$ . Then find how slow the gate must be to reach the fault-tolerant target $L=10^{-4}$ , and how much DRAG buys you.

Step 1. Fix the drive amplitude. A square $\pi$ -pulse has $\Omega t_{\mathrm g}=\pi$ , so $\Omega/2\pi=1/(2t_{\mathrm g})=1/(40\ \mathrm{ns})=\mathbf{25\ MHz}$ and the small parameter is $\Omega/\alpha=25/200=\mathbf{0.125}$ .

Step 2. Apply the scaling from Derivation 2: $P_{\mathrm{leak}}=\tfrac12(\Omega/\alpha)^2=\tfrac12(0.125)^2=\mathbf{7.8\times10^{-3}}$ . An unshaped 20 ns gate leaks about 0.8% per operation — two orders of magnitude above the fault-tolerant target.

Step 3. Invert for the target. Setting $\tfrac12(\Omega/\alpha)^2=10^{-4}$ gives $\Omega/\alpha=0.01414$ , so $\Omega/2\pi=2.83\ \mathrm{MHz}$ and $t_{\mathrm g}=1/(2\times2.83\ \mathrm{MHz})=\mathbf{177\ ns}$ — roughly 9× slower than the shaped gate.

Step 4. Compare with DRAG. By cancelling the leading $1/\alpha$ amplitude, DRAG makes residual leakage scale as $\mathcal{O}((\Omega/\alpha)^4)$ instead of $(\Omega/\alpha)^2$ ; at $\Omega/\alpha=0.125$ that is $\sim(0.125)^4\sim2\times10^{-4}$ , in practice below $10^{-4}$ at the same 20 ns. The shaped pulse hits at 20 ns the target the unshaped pulse cannot reach before 177 ns — the value proposition of DRAG in one comparison.

2 · The thermal leakage floor (detailed balance)

The same transmon has $\omega_{01}/2\pi=5.0\ \mathrm{GHz}$ , so the $|1\rangle\!\to\!|2\rangle$ transition sits at $\omega_{12}/2\pi=4.8\ \mathrm{GHz}$ . In a dilution refrigerator at $T=20\ \mathrm{mK}$ , and with perfect coherent control, what is the residual thermal population of $|2\rangle$ relative to $|1\rangle$ ?

Step 1. Thermal energy scale: $k_{\mathrm B}T/h=20.84\ \mathrm{GHz/K}$ , so at $T=20\ \mathrm{mK}$ , $k_{\mathrm B}T/h=\mathbf{0.417\ GHz}$ .

Step 2. By detailed balance (equation 7), $P_2/P_1=\Gamma_{\uparrow}/\Gamma_{\downarrow}=e^{-\hbar\omega_{12}/k_{\mathrm B}T}=e^{-4.8/0.417}=e^{-11.5}$ .

Step 3. Evaluate: $e^{-11.5}=\mathbf{1.0\times10^{-5}}$ . Thermal physics alone parks about one part in $10^{5}$ of any $|1\rangle$ population in $|2\rangle$ — a hard floor no pulse shaping can remove. Leakage has an irreducible incoherent component set by temperature, which is why reset and LRUs, not just DRAG, are needed, and why a higher qubit frequency or colder fridge lowers the floor.

Physics of the portrait

Leakage control is where the abstract theory of quantum channels meets the concrete engineering of a microwave pulse. Its lineage runs from the weakly anharmonic transmon, through the derivative-quadrature trick, to the modern definitions of leakage and seepage.

Origins · 1 minDRAG pulse shaping and how this was discovered

DRAG pulse shaping

Derivative Removal by Adiabatic Gate, Ωy = −(dΩx/dt)/α, introduced by Motzoi–Gambetta–Rebentrost–Wilhelm, 2009 · coherent leakage control in weakly anharmonic qubits

DRAG — Derivative Removal by Adiabatic Gate — is the pulse-shaping idea that made fast, low-leakage single-qubit gates possible, and its story is inseparable from the transmon. In 2007 Jens Koch and collaborators at Yale introduced the transmon, which traded charge-noise sensitivity for a shrunken anharmonicity of a few hundred megahertz, so the $|1\rangle\!\to\!|2\rangle$ transition sat dangerously close to the qubit frequency and every fast gate flirted with leakage. In 2009 Felix Motzoi, Jay Gambetta, Patrick Rebentrost, and Frank Wilhelm published the resolution: driving the orthogonal quadrature with $\Omega_y=-\dot\Omega_x/\alpha$ makes the virtual excursion to $|2\rangle$ close on itself, cancelling the leading-order leakage amplitude, while $\delta=\Omega_x^2/2\alpha$ repairs the induced AC-Stark phase. Gambetta and coworkers extended the framework in 2011 to higher order. A decade later Christopher Wood and Jay Gambetta defined the average leakage and seepage rates as gauge-invariant channel quantities — the $L$ and $S$ of this article. Today DRAG is the default single-qubit waveform on essentially every transmon platform.

History

2007

— Koch and collaborators introduce the transmon, a deliberately weakly anharmonic qubit

2009

— Motzoi, Gambetta, Rebentrost, and Wilhelm introduce DRAG pulse shaping

2018

— Wood and Gambetta define the average leakage and seepage rates as channel quantities

2021

— Google demonstrates reset-based leakage-reduction units in a repetition and surface code

2023

— Surface-code experiments below threshold rely on explicit leakage removal each cycle

Going further

Texts and papers. Nielsen & Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Ch. 8 (quantum operations, CPTP maps) and Ch. 7 (driven two-level dynamics) · Preskill, Lecture Notes for Physics 219, Ch. 3 (quantum channels) · Wilde, Quantum Information Theory, 2nd ed. (channels and Kraus representations) · Watrous, The Theory of Quantum Information (completely positive maps) · Kitaev, Shen & Vyalyi, Classical and Quantum Computation (fault tolerance) · Koch et al., PRA 76, 042319 (2007) — the transmon and its anharmonicity · Motzoi, Gambetta, Rebentrost & Wilhelm, PRL 103, 110501 (2009) — DRAG · Gambetta et al., PRA 83, 012308 (2011) — analytic leakage control · Wood & Gambetta, PRA 97, 032306 (2018) — leakage and seepage rate definitions. The key worked numbers throughout this article are: $t_{\mathrm g}=20$ ns with $|\alpha|/2\pi=200$ MHz gives $\Omega/2\pi=25$ MHz and $L=7.8\times10^{-3}$ ; reaching the $10^{-4}$ fault-tolerance target unshaped requires $t_{\mathrm g}=177$ ns; and the 20 mK thermal leakage floor for $\omega_{12}/2\pi=4.8$ GHz is $e^{-11.5}=1.0\times10^{-5}$ .

Prerequisites. Open Quantum Systems and Reduced Dynamics · Quantum Operations and CPTP Maps · Kraus Representation of Quantum Channels · Lindblad / GKSL Master Equation. Leads to. Depolarizing Channel · Amplitude Damping Channel · Markovian vs Non-Markovian Dynamics.

Ask the tutor: Want me to walk through how DRAG's derivative pulse actually closes the virtual excursion to $|2\rangle$ , or how a surface-code decoder turns a leaked qubit into a correctable erasure?

Is leakage the same thing as decoherence?

No, though they often occur together. Decoherence (dephasing and relaxation) acts within the qubit subspace — it scrambles the relative phase or damps $|1\rangle$ toward $|0\rangle$ , but the population stays in $\mathcal{H}_{\mathrm C}$ and the qubit map stays trace-preserving. Leakage moves population out of the qubit subspace, so the restricted map is trace-decreasing and qubit-channel language fails. A relaxation event ( $T_1$ ) is a legitimate error a decoder can handle; a leakage event is silent until seepage or an LRU brings the population back. The cures differ too: decoherence is fought with better materials and dynamical decoupling, leakage with pulse shaping and active reset.

Why does the anharmonicity carry a minus sign for transmons?

The transmon is a slightly anharmonic oscillator whose levels bunch together as you climb, so the $|1\rangle\!\to\!|2\rangle$ spacing is smaller than the $|0\rangle\!\to\!|1\rangle$ spacing. With $\alpha=\omega_{12}-\omega_{01}$ that makes $\alpha$ negative, typically $\alpha/2\pi\approx-200$ to $-300$ MHz. The sign matters for DRAG, since $\Omega_y=-\dot\Omega_x/\alpha$ and $\delta=\Omega_x^2/2\alpha$ both flip with it. But the magnitude of leakage is set by $|\alpha|$ — a larger gap in either direction pushes the leakage transition further off-resonance and suppresses the spill.

Why is there a √2 in the driven three-level Hamiltonian?

Because the transmon is nearly a bosonic mode, and a charge or dipole drive couples adjacent levels through the ladder operator $\hat a$ , with matrix elements $\langle n{+}1|\hat a^\dagger|n\rangle=\sqrt{n+1}$ . For $|0\rangle\!\to\!|1\rangle$ that factor is $1$ , but for $|1\rangle\!\to\!|2\rangle$ it is $\sqrt2$ — so the leakage transition is driven more strongly than the computational one at the same amplitude. This $\sqrt2$ is why leakage is worse than a naive two-level estimate, and it shows up squared (a factor of 2) in $P_{\mathrm{leak}}=\Omega^2/2\alpha^2$ .

Do trapped ions and neutral atoms leak too?

Yes, but usually far less from the coherent channel. Ion and atom qubits are encoded in hyperfine or optical levels whose nearest unwanted transition is detuned enormously compared with the Rabi frequency, so off-resonant leakage is tiny. Their leakage is instead incoherent: off-resonant scattering into other hyperfine states, spontaneous emission, or imperfect optical pumping shelves population outside the qubit subspace. The formalism is identical — a trace-decreasing map with rates $L$ and $S$ — but the dominant mechanism shifts from control bandwidth (transmons) to photon scattering (atoms and ions).

How does a decoder actually cope with residual leakage?

Two complementary tricks. First, conversion: an LRU or reset each cycle pumps $|2\rangle$ population back into the qubit subspace, turning a silent leakage error into an ordinary Pauli-like error the decoder understands and bounding how far it spreads. Second, approximation: whatever survives between reductions is Pauli-twirled into an effective depolarizing channel the decoder treats as extra noise. Advanced schemes use analog readout to flag likely-leaked qubits, feeding the decoder an erasure — easier to correct than an unknown Pauli. The seepage rate $S$ sets how quickly conversion restores the qubit picture.

Can leakage ever be useful rather than just harmful?

Yes — two-qubit gates on transmons rely on it. The controlled-phase gate tunes $|11\rangle$ into near-resonance with $|02\rangle$ , letting the pair swap virtually and pick up a conditional phase; the whole gate is a controlled visit to the leakage subspace. The challenge is to make the visit return — net-zero and adiabatic flux pulses shape the trajectory so population that climbed to $|02\rangle$ comes back to $|11\rangle$ , leaving only the desired phase. Higher levels are simultaneously the resource behind fast entangling gates and the liability that makes two-qubit leakage the dominant error when the pulse is imperfect.