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Leakage detection for a transmon-based surface code
B. M. Varbanov, F. Battistel, B. M. Tarasinski, V. P. Ostroukh, T. E. O'Brien, L. DiCarlo, B. M. Terhal
TL;DR
Leakage complicates quantum error correction, motivating detection methods for surface-code qubits. The paper develops HMM-based leakage inference from defect signatures, reporting a sharp local increase in neighboring-stabilizer defect probability associated with leakage.
Problem
Leakage detection is a concern for transmon-based surface-code implementations, where leakage is associated with a sharp local increase in neighboring-stabilizer defect probability.
Source: D. Hidden Markov Models · p. 5Method
The paper develops HMMs for leakage inference using the local defect-probability increase associated with leakage.
Source: D. Hidden Markov Models · p. 5, §II · p. 9Results
Leakage is associated with a sharp local increase in the defect probability of neighboring stabilizers.
Source: D. Hidden Markov Models · p. 5Takeaways & Limitations
The authors describe their HMM approach as low-cost and scalable for leakage detection.
Source: §II · p. 9The simulations omit readout-declaration errors and crosstalk effects, and assume perfectly projective measurements.
Source: §II · p. 9
Abstract
from arXiv · showhide
Leakage outside of the qubit computational subspace, present in many leading experimental platforms, constitutes a threatening error for quantum error correction (QEC) for qubits. We develop a leakage-detection scheme via Hidden Markov models (HMMs) for transmon-based implementations of the surface code. By performing realistic density-matrix simulations of the distance-3 surface code (Surface-17), we observe that leakage is sharply projected and leads to an increase in the surface-code defect probability of neighboring stabilizers. Together with the analog readout of the ancilla qubits, this increase enables the accurate detection of the time and location of leakage. We restore the logical error rate below the memory break-even point by post-selecting out leakage, discarding about 47% of the data. Leakage detection via HMMs opens the prospect for near-term QEC demonstrations, targeted leakage reduction and leakage-aware decoding and is applicable to other experimental platforms.
A. Leakage error model
The leakage error model centers on leakage during transmon CZ gates and includes leakage-induced phases and exchanges. It simplifies the dynamics by omitting effects found negligible in gate simulations and by making additional modeling assumptions.
A. Leakage error model
- The simulations target the Net-Zero CZ gate, using a 30 ns bipolar flux pulse and two 10 ns phase-correction pulses.The pulse tunes the fluxed qubit twice into resonance with the |11⟩↔|02⟩ avoided crossing.Source: A. Leakage error model · p. 2
- Leakage is modeled as an exchange between |11⟩ and |02⟩ with probability L1, while leakage conditional phases describe extra rotations when leaked and non-leaked qubits interact.The model also considers leakage mobility and further leakage to |3⟩ in Appendix H.Source: A. Leakage error model · pp. 2–3
- The model sets the leakage phase and off-diagonal coherence terms to zero for efficiency, assumes no single-qubit or measurement-induced leakage, and randomizes φL stat across runs.The phase randomization captures average behavior because φL stat had not been characterized experimentally.Source: A. Leakage error model · pp. 2–3
- The parametrized CZ error model agrees with full-trajectory simulations to within ±0.1% in average gate fidelity.The model omits several potential errors found to be small in the full-trajectory simulations.Source: A. Leakage error model · p. 3
B. Effect of leakage on the code performance
Density-matrix simulations of Surface-17 show that leakage pushes the logical error rate above the memory break-even point. Both the MWPM decoder and a decoding upper bound show increased logical error, which also depends on leakage conditional phases.
B. Effect of leakage on the code performance
- The simulations model leakage-prone high- and mid-frequency qubits as three-level systems, with ancilla measurements projective in the |0⟩, |1⟩, and |2⟩ basis.Low-frequency qubits remain two-level, and ancillas are not reset between QEC cycles.Source: B. Effect of leakage on the code… · p. 3
- Leakage sharply pushes the logical error rate above the memory break-even point for both MWPM and the decoding upper bound.The logical error rate also depends on the leakage conditional phases for both decoders.Source: Figure 2 · p. 3, B. Effect of leakage on the code… · p. 4
- MWPM uses weights trained on simulated data without leakage, while the decoding upper bound uses complete density-matrix information to infer logical errors.The logical error rate is extracted from the logical fidelity over QEC cycles.Source: B. Effect of leakage on the code… · pp. 3–4
C. Projection and signatures of leakage
Leakage is sharply projected onto the leakage subspace and raises defect probabilities on neighboring stabilizers. Data-qubit leakage acts through stabilizer anti-commutation, whereas ancilla leakage spreads errors and disables its stabilizer check.
C. Projection and signatures of leakage
- Data-qubit leakage rises over roughly 3 QEC cycles to a maximum probability of approximately 0.8, with similar behavior on D3, D4, and D5.Its probability stays near unity for a finite period, averaging 9 QEC cycles for the considered parameters.Source: C. Projection and signatures of leakage · p. 4
- When data-qubit leakage peaks, the neighboring-stabilizer defect probability reaches approximately 0.5 because leakage makes the checks effective weight-3 anti-commuting checks.This anti-commutation contributes to the increased logical error rate.Source: C. Projection and signatures of leakage · p. 4
- Ancilla leakage raises neighboring defect probability and effectively disables the corresponding stabilizer measurement.Interactions with a leaked ancilla impart phase rotations to neighboring data qubits; the defect rate abruptly rises after leakage begins.Source: C. Projection and signatures of leakage · p. 5
- Leakage in either data or ancilla qubits correlates with a local increase in defect rate, offering a signature that can be used for leakage detection.The mechanisms differ: data leakage induces anti-commutation, while ancilla leakage spreads errors and disables a check.Source: C. Projection and signatures of leakage · pp. 4–5
D. Hidden Markov Models
The paper uses one two-state HMM per leakage-prone qubit to infer leakage from local defect patterns and, for ancillas, analog readout. Sharp leakage projection supports the two-state model, while local defect increases provide the detection signature.
D. Hidden Markov Models
- Each HMM models transitions between computational and leakage states and updates their probabilities from state-dependent observables.Transition probabilities depend on leakage and seepage probabilities per QEC cycle.Source: D. Hidden Markov Models · p. 5, Figure 4 · p. 5
- The model treats sharp projection of leakage into high-probability states as support for representing each qubit with only computational and leakage hidden states.The figure compares the HMM response over time with density-matrix leakage probabilities and presents data-qubit precision-recall curves.Source: D. Hidden Markov Models · p. 5, Figure 5 · p. 6
- The HMMs use neighboring-stabilizer defects as observables, adding analog in-phase readout for ancilla qubits.The sharp local increase in defect probability is the leakage signature used by the model.Source: D. Hidden Markov Models · p. 5, Figure 4 · p. 5
E. Data-qubit leakage detection
Data-qubit HMMs track leakage probabilities with an average optimality of about 74%, but their response lags by about one QEC cycle and neighboring leakage can cause false detections.
E. Data-qubit leakage detection
- The HMM response closely follows the density-matrix leakage probability, reaching the same maximum with a smaller logistic growth rate.This slower response is expected to produce an average detection delay of about 1 QEC cycle.Source: E. Data-qubit leakage detection · p. 6
- Precision is the fraction of HMM leakage detections that correspond to leakage events, while recall is the fraction of leakage events detected.Source: E. Data-qubit leakage detection · p. 6
- The precision-recall curve summarizes detection across both time and space, with stronger detection shifting it toward higher precision and recall.Source: E. Data-qubit leakage detection · p. 6
- About 74.0% average optimality was extracted for data-qubit HMMs against the density-matrix optimal model.The density-matrix model achieves the best possible precision and recall.Source: E. Data-qubit leakage detection · p. 6
- False detections when a neighboring qubit is leaked are the main limitation on data-qubit HMM optimality.The response delay is only a few QEC cycles.Source: E. Data-qubit leakage detection · p. 6
F. Ancilla-qubit leakage detection
Ancilla HMMs combine neighboring-stabilizer defects with analog readout to detect leakage in time and location. Analog information markedly improves detection over defects alone.
F. Ancilla-qubit leakage detection
- Ancilla HMMs use neighboring-stabilizer defects and the ancilla’s analog measurement outcome as observables.The analog input is the in-phase response Im.Source: F. Ancilla-qubit leakage detection · pp. 6, 8
- Defects alone yield about 35% optimality for bulk ancillas and no leakage-detection ability for boundary ancillas.Boundary ancillas have one neighboring stabilizer, compared with three for bulk ancillas in Surface-17.Source: F. Ancilla-qubit leakage detection · pp. 6–7
- Defect-only performance varies among same-type bulk ancillas because gate ordering and noncommuting CZ operations break apparent Surface-17 symmetry.Source: F. Ancilla-qubit leakage detection · p. 7
- Defect-based signatures depend on randomized φL flux, and tuning phases to improve detection could increase logical error rate for a leakage-unaware decoder.Source: F. Ancilla-qubit leakage detection · p. 7
- With analog readout, ancilla HMMs reach about 96% average optimality and detect leakage with an expected delay of at most 2 QEC cycles.The simulations use projective measurements, so the density-matrix optimality is 1 for ancilla qubits.Source: F. Ancilla-qubit leakage detection · p. 8
- At moderate discrimination fidelity of about 80%, ancilla leakage can be accurately identified for both bulk and boundary qubits.At fidelity of 60% or less, detection remains possible but is very limited, especially for boundary ancillas.Source: F. Ancilla-qubit leakage detection · p. 8
G. Improving code performance via post-selection
Post-selection using detected leakage restores the MWPM decoder’s performance below the memory break-even point, with about 47% of data discarded. HMM-based selection agrees with selection based on density-matrix leakage probabilities.
G. Improving code performance via post-selection
- A run is discarded if any qubit’s leakage probability exceeds the threshold in any QEC cycle.This criterion is sensitive to detection accuracy and response timing, especially for short-lived leakage events.Source: G. Improving code performance via… · p. 8
- The optimization compares logical error rate against the fraction of discarded data, using an evolutionary algorithm to identify the Pareto front for both decoders.The HMM predictions include analog measurement information at experimentally extracted discrimination fidelity.Source: G. Improving code performance via… · p. 8
- About 47% of data must be discarded to restore MWPM performance below the quantum memory break-even point.The HMM and density-matrix post-selection methods agree; the paper attributes the gain to discarding runs with leakage rather than runs with only regular errors.Source: G. Improving code performance via… · pp. 8–9
- For the upper-bound decoder, post-selection reaches the logical error rates from simulations without leakage after discarding about 43% of data.The corresponding MWPM fraction is about 57%.Source: G. Improving code performance via… · p. 9
II. DISCUSSION
Density-matrix simulations of Surface-17 show that leakage detection with HMMs can support post-selection below the memory break-even point, while the discussion identifies conditions and boundaries for broader use. The simulations use a leakage-unaware MWPM decoder and a restricted set of logical-memory settings.
II. DISCUSSION
- Post-selecting detected leakage restores Surface-17 performance below the memory break-even point while discarding about 47% of the data.The HMMs use defect signatures and analog ancilla measurements to identify leakage time and location.Source: §II · p. 9
- Readout-declaration errors and crosstalk are omitted; the authors expect declaration errors to affect MWPM performance and slightly affect HMM optimality.The simulations also assume perfectly projective measurements, though small deviations are not expected to substantially change leakage projection or its signatures.Source: §II · p. 9
- Data-qubit leakage is detectable when H acts as the identity in the leakage subspace or commutes with CZ there; ancilla leakage needs readable leakage states or detectable error-spreading signatures.These are the stated conditions for applying HMM detection beyond the simulated platform.Source: §II · p. 9
- Identifying leakage could enable targeted leakage-reduction units, avoiding their hardware, time, or inadvertent-leakage costs when applied without confirmed leakage.The discussion names swap, resonator-reset, leakage-reversal, and |1⟩↔|2⟩ pulse options.Source: §II · p. 9
- Leakage-aware decoder gains are expected to be limited for low-distance codes such as Surface-17 because leakage can reduce effective code distance.The discussion suggests accounting for leakage-induced errors and remaining stabilizer information, while noting that leakage can remove data qubits or disable ancillas that spread errors.Source: §II · p. 10
Appendix A: Simulation protocol
- The memory simulations prepare |0⟩L, maintain it for up to 20 or 50 cycles, and use MWPM with weights estimated from simulations without leakage.The study focuses on Z-basis memory states; it does not simulate X-basis logical states or use a leakage-aware decoder.Source: Appendix A · p. 12
Appendix B: Error model and parameters
The error model combines qubit damping with gate- and flux-dependent decoherence, then treats leakage as incoherent in the simulations. Measurement modeling excludes several non-ideal effects.
Appendix B: Error model and parameters
- The simulations include amplitude and phase damping, with additional dephasing for fluxed qubits during CZ gates; single-qubit gates are assumed not to cause leakage.Single-qubit gates act trivially on leaked states, and the model omits deviations from ideal CZ phases.Source: Appendix B · pp. 12–13
- The listed simulation parameters cover qubit decoherence times and gate, measurement, and QEC-cycle durations.The supplied table caption identifies these categories but gives no parameter values.Source: Table I · p. 13
- Setting the coherence block ρcoh to zero leaves the reported leakage projection, signatures, and logical error rate unchanged, motivating an incoherent leakage model.The authors attribute this to leakage projection leaving mostly an incoherent mixture of computational and leakage subspaces.Source: Appendix B · p. 14
- Measurement is modeled as damping before and after a Born-rule projection, while declaration errors, measurement-induced leakage, and non-ideal projective measurements are excluded.The model also omits measurement-induced dephasing and a Purcell-related decrease in relaxation time.Source: Appendix B · p. 14
Appendix C: Transmon measurements in experiment
The measurement appendix describes calibrating transmon readout from analog I/Q responses and projecting those responses onto quadratures to distinguish states. The reported discrimination fidelity is higher for |0⟩ versus |1⟩ than for |1⟩ versus |2⟩.
Appendix C: Transmon measurements in experiment
- Repeated measurements of prepared |0⟩, |1⟩, and |2⟩ states characterize each state's two-dimensional Gaussian distribution in the I/Q plane.The distributions are described by their mean and standard deviation.Source: Appendix C · p. 14
- With equally likely state priors and Gaussian readout responses, the appendix expresses the state probability given I and Q using Bayes' rule.The analog response can be projected onto an axis joining distribution centers to use a single quadrature for pairwise discrimination.Source: Appendix C · pp. 14–15
- The |0⟩–|1⟩ readout discrimination fidelity is 99.6%, while the |1⟩–|2⟩ fidelity is 88.4%.The first pair uses a rotated quadrature maximizing fidelity; the second is projected along an in-phase axis.Source: Figure 9 · p. 14, Appendix C · p. 15
Appendix D: Leakage-induced anti-commutation
Data-qubit leakage makes neighboring X- and Z-type checks anticommute, while products of the resulting weight-3 gauge checks form weight-6 superchecks. The HMM formalism models each qubit’s computational or leaked state from stabilizer defects, with analog readout added for ancilla leakage.
Appendix D: Leakage-induced anti-commutation
- Figure 10 tracks supercheck defect probability against the leakage conditional phase; simulations find about 50% for randomized or fixed phases, with phase-dependent increases reaching 50%.The phase rotations can undermine extraction of the weight-6 parity, so controlling φL stat would benefit decoding when superchecks are used.Source: Figure 10 · p. 15, Appendix D · pp. 16–17
- The anti-commutation result relies on the Hadamard acting trivially in the leakage subspace; it continues to hold when that operation commutes with CZ there.The derivation assumes single-qubit Hadamard gates do not induce leakage and act trivially on the leakage subspace.Source: Appendix D · p. 16
- Leakage makes neighboring X- and Z-type checks anticommute, randomizing ancilla outcomes; paired same-type weight-3 gauge checks form weight-6 superchecks, while a leaked data qubit reduces Surface-17’s dressed distance by one.For a boundary leak, only one supercheck can be defined; the other must be ignored for decoding.Source: Appendix D · p. 16
Appendix E: HMM formalism
- Each HMM tracks two hidden states, computational and leaked, using neighboring-stabilizer defects as observables and transition and emission probabilities to infer state probabilities.Defect-emission probabilities for the leaked state are extracted from cycles when the density-matrix leakage probability exceeds 0.5.Source: Appendix E · p. 17
- For ancilla leakage, HMMs can add the analog measurement outcome as an observable; data-qubit leakage produces an extracted defect probability near 0.5 across leakage conditional phases.The analog outcome likelihoods are modeled from the readout distributions for computational and leaked states.Source: Appendix E · p. 17
Appendix F: HMM error budget
The HMM error budget identifies correlated regular errors and leakage on neighboring qubits as sources of overestimation, with neighboring data-qubit leakage the dominant limitation. An alternative ancilla pulse scheme improves ancilla detection but increases crosstalk and lowers data-qubit HMM optimality.
Appendix F: HMM error budget
- Independent per-stabilizer defect models omit correlations from regular errors, while leakage on neighboring qubits can make an HMM overestimate leakage on its tracked qubit.The models could add hidden states for regular errors or next-nearest-neighbor stabilizer observables, at increased complexity and overhead.Source: Appendix F · pp. 17–18
- Ancilla HMMs are insensitive to leakage on other qubits, but data HMMs can overestimate leakage; crosstalk scales with shared observables and depends on expected defect probabilities.The analog readout helps distinguish leaked ancillas from computational states, whereas data-HMM crosstalk remains a concern.Source: Appendix F · p. 18
- Post-selecting ancilla-leakage runs raises average data-HMM optimality from 74.0% to 80.9%, and also removing other-data-qubit leakage raises it to 95.9%.Neighboring data-qubit leakage is the dominant limitation; regular errors and transition/output probability parametrization account for remaining suboptimality.Source: Appendix F · p. 18
Appendix G: An alternative scheme for enhancing ancilla-qubit leakage detection
- Ancilla-HMM optimality reaches 58.5% and 51.5% for bulk X and Z qubits, and 74.0% and 43.0% for boundary X and Z qubits; data-HMM optimality falls from 74.0% to 39.8%.The scheme applies a π pulse to each ancilla every other QEC cycle; it was assessed in post-processing, which does not capture increased amplitude damping.Source: Appendix G · pp. 18–19
Appendix H: Second-order leakage effects
Appendix H analyzes second-order leakage processes in the CZ gate; Appendix I assesses their effects on leakage detection and code performance. Appendix J estimates leakage buildup across QEC cycles and compares its predictions with density-matrix simulations.
Appendix H: Second-order leakage effects
- Superleakage can be large for some pulse and system parameters, even when φCZ = π, through transfer at the |12⟩↔|03⟩ crossing and subsequent exchange.The two halves of a Net-Zero pulse can interfere, changing exchange population and leakage conditional phases.Source: Appendix H · p. 20
- Including |3⟩ can increase leakage mobility through a two-excitation exchange between resonant |03⟩ and |21⟩, virtually mediated by |12⟩.The exchange strength depends on anharmonicities; without |3⟩, mobility is small but non-negligible and the fluxing qubit does not cross |12⟩↔|21⟩ on its way to the CZ point.Source: Figure 12 · p. 19, Appendix H · p. 20
Appendix I: Effects of leakage mobility and superleakage on leakage detection and code performance
- Leakage mobility has negligible effects on logical performance and HMM optimality in the modeled range, while reducing the average duration of leakage events.The authors attribute its limited impact to the low probability that a qubit is already leaked, given the low L1 per CZ gate.Source: Appendix I · p. 20
- Surface-17 simulations including |3⟩ were not performed because of prohibitive computational cost, although superleakage may increase defects and logical error rates.The authors expect leakage detection to remain largely unaffected: neighboring-stabilizer defects remain at probability 0.5 for leaked data qubits, while propagated bit flips may signal leaked ancillas.Source: Appendix I · p. 20
Appendix J: Leakage steady state in the surface code
- The predicted leakage-population evolution agrees within error bars with density-matrix results for the high-frequency data qubits.The comparison averages extracted populations over 4 × 10^4 runs; error bars are 95% bootstrap confidence intervals.Source: Appendix J · p. 21, Figure 13 · p. 21
- For L3 = 10%, the model predicts steady-state |3⟩ populations of 5.1% for D4 and 3.8% for D3 and D5, comparable to the respective |2⟩ populations.The authors conclude that superleakage should be included when optimizing surface-code performance over many QEC cycles.Source: Appendix J · p. 21