PALEA: Phase-Averaged Leakage Error Amplification

• PALEA is a phase-averaging framework that transforms coherent leakage dynamics into an effectively incoherent, additive error signal, facilitating accurate leakage metrology.
• The protocol employs a sequence design with CZ gates followed by a dynamical-decoupling layer to isolate and amplify leakage effects, enabling robust extraction of small leakage parameters.
• PALEA’s methodology improves calibration in superconducting qubits and is also applicable to correcting rotating leakage biases in wideband radio polarimetry and stabilizer measurements.

Phased-Averaged Leakage Error Amplification (PALEA) is a technical term used in more than one leakage-sensitive research context, with a shared emphasis on how phase or geometric averaging changes the observability of leakage processes. In superconducting quantum control, PALEA is a leakage-focused calibration and metrology protocol for high-fidelity CZ gates between transmons coupled via a tunable coupler; it isolates the coherent exchange in the two-excitation subspace spanned by $|11\rangle$ and $|02\rangle$, then converts that dynamics into a phase-averaged signal that can be fit to extract per-gate leakage (Marxer et al., 22 Aug 2025). In wideband radio polarimetry, the same acronym denotes the amplification of hybrid-induced leakage bias that arises when averaging and parallactic de-rotation act on imperfect pseudo-circular feeds, producing a rotating effective leakage term in the sky frame (Mitra, 31 May 2026). A related superconducting literature on repeated stabilizer measurements, although centered on leakage in error-detection circuits rather than on the later acronym, provides an explicit model for how phase drift transforms coherent leakage dynamics into effectively incoherent error accumulation (Ghosh et al., 2013).

1. Superconducting-qubit meaning of PALEA

In the superconducting implementation, PALEA was developed to tune and quantitatively characterize high-fidelity CZ gates in a device containing two transmon qubits coupled via a tunable floating transmon coupler. The relevant architecture labels the qubits as $Q1$ and $Q2$, with $Q1$ the lower-frequency qubit and $Q2$ the higher-frequency qubit. The diabatic CZ is realized by flux-pulsing $Q1$ and the coupler so that the two-excitation subspace spanned by $\{|11\rangle,|02\rangle\}$ undergoes a controlled exchange that produces the desired conditional $Z$ phase. In this setting, leakage refers specifically to population transferred out of the computational subspace into qubit $|2\rangle$ levels per gate, predominantly into $|02\rangle$0, although ancillary exchanges such as $|02\rangle$1 and $|02\rangle$2 can also occur (Marxer et al., 22 Aug 2025).

The motivation for PALEA is the failure mode of standard leakage amplification near an optimized CZ. Repeating CZ gates on $|02\rangle$3 can amplify leakage, but the oscillation amplitude depends not only on the leakage angle itself but also on uncontrolled phases accumulated by $|02\rangle$4 and $|02\rangle$5. As fidelities approach $|02\rangle$6, the per-gate leakage becomes sufficiently small that the signal-to-noise ratio collapses precisely near the optimal operating point. The same operating regime is also susceptible to bias from coherent interference due to non-target dynamics, including coupler leakage. Existing remedies, such as inserting precisely tuned delays or interleaving calibrated $|02\rangle$7 rotations in Floquet calibration, can restore contrast, but they introduce additional parameters that must be re-optimized whenever the gate changes, which makes them slower and less robust for automated fine calibration (Marxer et al., 22 Aug 2025).

A central misconception addressed by this formulation is that leakage metrology for CZ gates can be treated as a generic two-qubit benchmarking problem. In the cited experiment, PALEA is narrower and more structured: it targets the intended $|02\rangle$8-$|02\rangle$9 mechanism of the diabatic CZ rather than total out-of-subspace population across arbitrary channels.

2. Sequence design and phase-averaged signal formation

PALEA models an imperfect CZ gate, restricted to the span of $Q1$0 and ignoring common phase, as

$Q1$1

where $Q1$2 is the over-rotation angle quantifying population transfer between $Q1$3 and $Q1$4, and $Q1$5 are phases acquired during a cycle. After each CZ, PALEA inserts a dynamical-decoupling layer $Q1$6 that flips $Q1$7 and $Q1$8 with an effective $Q1$9 rotation while carrying an adjustable phase difference. Operationally, this is implemented with $Q2$0 in the $Q2$1-subspace of $Q2$2 and $Q2$3 in the $Q2$4-subspace of $Q2$5, with phases $Q2$6 and $Q2$7, so that

$Q2$8

One PALEA cycle is a CZ followed by $Q2$9. Starting from $Q1$0, the $Q1$1 population after $Q1$2 cycles is

$Q1$3

with

$Q1$4

The defining step is phase averaging over $Q1$5, either passively because laboratory-frame phases drift and are not reset between repetitions in CZ-based circuits, or actively by cycling $Q1$6. The averaged signal is

$Q1$7

This averaging suppresses interference terms tied to $Q1$8 and $Q1$9, leaving a high-contrast function of $Q2$0 alone. The paper summarizes the intuition as the conversion of coherent sums into incoherent sums under phase averaging,

$Q2$1

so that leakage becomes effectively additive under repetition (Marxer et al., 22 Aug 2025).

The experimental sequence begins by preparing $Q2$2 with $Q2$3 in $Q2$4 on each qubit. Each cycle then applies the CZ candidate using a flat-top $Q2$5 flux pulse with cosine edges of approximately $Q2$6, total duration approximately $Q2$7, and a coupler flux pulse with a $Q2$8 Slepian-shaped envelope; the pulses are pre-compensated with exponential filters, and $Q2$9 buffers are used on each side, for a total CZ duration of $Q1$0. The DD layer follows immediately. Measurement uses three-state single-shot discrimination over $Q1$1, $Q1$2, and $Q1$3, with assignment correction. For even $Q1$4, $Q1$5 is the wanted outcome; for odd $Q1$6, $Q1$7 is wanted. The associated unwanted-population metric is

$Q1$8

3. Analytical model, sensitivity regime, and extracted leakage parameter

The phase-averaged PALEA signal admits a closed-form expression in terms of Legendre polynomials,

$Q1$9

which the experiment evaluates efficiently with Clenshaw recursion. In the small-angle, moderate-$\{|11\rangle,|02\rangle\}$0 regime, the transformed quantity

$\{|11\rangle,|02\rangle\}$1

obeys the approximation

$\{|11\rangle,|02\rangle\}$2

with derivative

$\{|11\rangle,|02\rangle\}$3

The reported sensitivity scales near-Heisenberg as $\{|11\rangle,|02\rangle\}$4 for $\{|11\rangle,|02\rangle\}$5 and as $\{|11\rangle,|02\rangle\}$6 at large $\{|11\rangle,|02\rangle\}$7. This identifies the practical regime $\{|11\rangle,|02\rangle\}$8 as the useful operating window for leakage calibration (Marxer et al., 22 Aug 2025).

The extracted over-rotation angle $\{|11\rangle,|02\rangle\}$9 is related directly to per-gate leakage into $Z$0,

$Z$1

Under phase-averaged repetition, the leaked population after $Z$2 cycles obeys

$Z$3

and, in an independent-error picture,

$Z$4

PALEA’s $Z$5-averaging enforces the additivity consistent with these relations while the exact fit to $Z$6 yields $Z$7 and hence $Z$8 robustly (Marxer et al., 22 Aug 2025).

The protocol is designed to remove the phase fragility of standard amplification. In the standard method, oscillation amplitude depends on the uncontrolled cycle $Z$9-angle $|2\rangle$0, which can drive the signal to low contrast near the optimum. PALEA instead makes the cycle axis nearly $|2\rangle$1-like and removes $|2\rangle$2 dependence through averaging. Formally,

$|2\rangle$3

so $|2\rangle$4 remains tightly near $|2\rangle$5 for small $|2\rangle$6, yielding large contrast that is insensitive to $|2\rangle$7. This feature is what enables PALEA to be embedded in black-box pulse-shape optimization procedures, including Nelder–Mead and Bayesian GP, without continual retuning of compensating $|2\rangle$8 rotations or delays. The reported comparison states that, relative to standard leakage amplification, PALEA reduces the residual calibrated leakage by at least a factor of two using the same number of repetitions, and simulations indicate $|2\rangle$9 at fixed $|02\rangle$00, whereas non-adaptive Floquet calibration can require up to $|02\rangle$01 shots to reach comparable precision (Marxer et al., 22 Aug 2025).

4. Calibration workflow, reported device performance, and stated limitations

The full CZ calibration procedure reported alongside PALEA has three parts. First, single-qubit virtual $|02\rangle$02 rotations on both qubits and the conditional $|02\rangle$03 over-rotation are calibrated with MEADD-type phase estimation. Second, leakage to $|02\rangle$04 is minimized with PALEA by sweeping coupler flux pulse amplitude and fitting $|02\rangle$05 across $|02\rangle$06 to extract $|02\rangle$07; in practice, averaging $|02\rangle$08 over $|02\rangle$09 produces a sharp Lorentzian-like dip versus coupler amplitude, allowing rapid center finding. Third, the $|02\rangle$10 and coupler pulse widths are swept, PALEA is rerun at each point, and iterative interleaved RB is used for benchmarking. The optimum was found at a total CZ duration of $|02\rangle$11, using a $|02\rangle$12 Slepian coupler pulse overlapping a $|02\rangle$13 $|02\rangle$14 pulse with $|02\rangle$15 buffers (Marxer et al., 22 Aug 2025).

A deliberately detuned operating point illustrates the fitting power of the protocol. At coupler amplitude $|02\rangle$16 in arbitrary device units, fitting $|02\rangle$17 yields $|02\rangle$18, corresponding to $|02\rangle$19 per gate. Near the calibrated optimum, at amplitude approximately $|02\rangle$20, $|02\rangle$21 and leakage is strongly suppressed. After PALEA-based leakage tuning and MEADD-based phase tuning, iterative interleaved RB gives an average CZ error over $|02\rangle$22 hours of $|02\rangle$23, corresponding to $|02\rangle$24 fidelity, with the best observed value $|02\rangle$25. The quadratic iterative-IRB fit is reported to separate coherent and incoherent contributions and to remove non-gate offsets; the calibrated gate is described as lying near the incoherent limit set by hybridized $|02\rangle$26 during the gate. Leakage randomized benchmarking with three-state discrimination yields total per-gate leakage into $|02\rangle$27 states of both qubits $|02\rangle$28, dominated by $|02\rangle$29 leakage at approximately $|02\rangle$30. In the same device, simultaneous single-qubit gate fidelities of $|02\rangle$31 and readout fidelities over $|02\rangle$32 are reported (Marxer et al., 22 Aug 2025).

The paper also calibrates single-excitation SWAP error in the span $|02\rangle$33 using a PALEA/MEADD-style sequence with two phase-stable $|02\rangle$34 gates as DD. The measured angle is $|02\rangle$35, implying

$|02\rangle$36

which is described as negligible relative to the incoherent floor (Marxer et al., 22 Aug 2025).

The stated assumptions and limitations are narrow and explicit. PALEA assumes that the dominant coherent exchange during a cycle is effectively two-level, namely $|02\rangle$37; substantial simultaneous three-level dynamics such as concurrent $|02\rangle$38 and $|02\rangle$39 would require extensions such as qutrit DD layers. Spectator leakage into the coupler is not directly measured because the coupler has no XY drive or readout in the device; it is instead probed by delay-based Floquet spectroscopy. Phase tracking within a sequence is necessary so that $|02\rangle$40 remains constant gate to gate. Device-specific constraints, including transmon anharmonicity, coupler nonlinearity, and mixer compression, limit the feasible gate durations and repetition counts.

5. PALEA in wideband radio polarimetry

A different literature uses the same acronym for a leakage-amplification mechanism in pseudo-circular radio feeds. In that setting, orthogonal linear receptors $|02\rangle$41 are combined by analog quadrature hybrids to synthesize circular polarization $|02\rangle$42. The standard assumption in many calibration pipelines is that instrumental polarization leakage can be represented as a static complex offset independent of parallactic angle. The cited work shows that this assumption fails when the hybrid has amplitude imbalance and phase error, because the hybrid operator $|02\rangle$43 and parallactic rotation $|02\rangle$44 do not commute. The ideal quadrature hybrid is

$|02\rangle$45

whereas the imperfect hybrid is modeled as

$|02\rangle$46

with small fractional amplitude imbalance $|02\rangle$47 and phase error $|02\rangle$48. The non-commutativity condition is

$|02\rangle$49

and becomes exact equality only in the ideal case $|02\rangle$50 (Mitra, 31 May 2026).

For a single defective antenna $|02\rangle$51 against an ideal reference antenna $|02\rangle$52, the measured RL cross-hand is

$|02\rangle$53

After standard parallactic de-rotation into the sky frame,

$|02\rangle$54

with effective leakage

$|02\rangle$55

Thus a static antenna-frame defect becomes a rotating sky-frame leakage vector. Written explicitly in the Stokes $|02\rangle$56 plane,

$|02\rangle$57

The measured EVPA is therefore biased according to

$|02\rangle$58

and, to first order for $|02\rangle$59,

$|02\rangle$60

The work emphasizes that this is not merely a calibration artifact but a systematic error that can corrupt Faraday rotation measurements. A representative numerical example uses $|02\rangle$61, $|02\rangle$62, $|02\rangle$63, and $|02\rangle$64, yielding $|02\rangle$65. For $|02\rangle$66 and $|02\rangle$67, $|02\rangle$68, so the cross-hand leakage term is several percent of Stokes $|02\rangle$69 (Mitra, 31 May 2026).

In this polarimetric meaning, PALEA is the amplification of hybrid-induced leakage bias caused by averaging or phased beamforming after de-rotation. Because $|02\rangle$70 for typical tracks, the rotating leakage does not cancel under time averaging. In phased arrays, the array-level leakage becomes

$|02\rangle$71

and post-beamforming calibration at a single reference angle $|02\rangle$72 leaves residuals

$|02\rangle$73

The mitigation proposed in that paper is Static Offset Pre-correction (SOP), applied in the antenna frame before parallactic de-rotation:

$|02\rangle$74

To first order, SOP restores

$|02\rangle$75

so the non-commutative rotating leakage term is removed rather than re-fitted as a static sky-frame offset (Mitra, 31 May 2026).

6. Relation to leakage amplification in repeated stabilizer measurements

A closely related superconducting analysis studies leakage in repeated ancilla-assisted measurement of the single-qubit stabilizer $|02\rangle$76. The register consists of ancilla $|02\rangle$77 and data $|02\rangle$78, each treated as a qutrit with levels $|02\rangle$79, $|02\rangle$80, and $|02\rangle$81. Each measurement cycle resets the ancilla to $|02\rangle$82, applies a Hadamard $|02\rangle$83, applies a CZ gate, applies a second Hadamard, and then measures the ancilla. The reported timing is $|02\rangle$84, $|02\rangle$85, and $|02\rangle$86. The extended qutrit CZ carries dynamical phases $|02\rangle$87 on non-computational channels, and the phase

$|02\rangle$88

controls the ancilla’s behavior during a leakage episode (Ghosh et al., 2013).

If the data qubit leaks to $|02\rangle$89, the dynamics reduce to the two-state sector $|02\rangle$90, and the ancilla undergoes an effective rotation

$|02\rangle$91

Starting from $|02\rangle$92, the ancilla state before measurement is

$|02\rangle$93

so

$|02\rangle$94

The detectability metric

$|02\rangle$95

gives the mean spacing between measured ancilla “1” outcomes during a leakage episode. When $|02\rangle$96, the ancilla becomes paralyzed and repeatedly outputs “0”; when $|02\rangle$97, it is approximately randomized (Ghosh et al., 2013).

The paper introduces a discrete-time Markov description in which leakage is initiated with probability $|02\rangle$98 per cycle and ends with recovery probability $|02\rangle$99, giving steady-state leaked fraction

$Q1$00

mean dwell time $Q1$01, and steady-state error rate

$Q1$02

In the PALEA interpretation attached to this work, phase drift across cycles averages the coherent ancilla rotation into an effectively incoherent error process. The resulting amplification factor is

$Q1$03

For uniform averaging over $Q1$04, $Q1$05, so $Q1$06. The cited numerical example with $Q1$07 and $Q1$08 gives $Q1$09 cycles, $Q1$10, and $Q1$11 per cycle. This makes explicit how a rare leakage event can generate many consecutive faulty rounds unless active reset increases $Q1$12 (Ghosh et al., 2013).

Taken together, these literatures indicate that PALEA is best understood as a phase-averaging framework rather than a single universally fixed protocol. In the superconducting CZ-calibration setting, phase averaging is deliberately engineered to suppress nuisance coherent terms and reveal the targeted leakage angle. In radio polarimetry and in repeated stabilizer measurements, by contrast, averaging makes a rotating or drifting leakage contribution survive at the level of observables and appear additive across time. The common structure is the conversion of phase-sensitive coherent leakage into a robust, accumulative signature; the practical meaning of that conversion, however, is field-dependent.