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Achieving Optimal-Distance Atom-Loss Correction via Pauli Envelope

Pengyu Liu, Shi Jie Samuel Tan, Eric Huang, Umut A. Acar, Hengyun Zhou, Chen Zhao

arXiv:2603.04156v2quant-ph

TL;DR

Atom loss is a major, nonlinear error source whose delayed and correlated effects complicate fault-tolerant neutral-atom computation. The paper bounds these effects with Pauli Envelopes, designs Mid-SWAP extraction and two envelope-based decoders, and reports improved thresholds, effective distances, and correction of correlated loss.

  • Problem

    Atom loss accounts for over 40% of recent neutral-atom physical errors, while its nonlinear, correlated effects make existing syndrome extraction and decoding methods inefficient, suboptimal, or without provable guarantees.

  • Method

    The paper uses Pauli Envelopes to bound atom-loss effects with low-weight Pauli approximations, then develops Mid-SWAP extraction, Envelope-MLE, and Envelope-Matching decoders.

  • Results

    Circuit-level evaluation shows up to 40% higher thresholds and 30% higher effective distances than prior approaches in the loss-dominated regime.

  • Takeaways & Limitations

    Correlated atom loss is easier to correct than independent loss, with thresholds increasing from 5.15% to 7.82% as the correlated fraction grows in the studied two-qubit-gate setting.

Abstract

from arXiv · show

Atom loss is a major error source in neutral-atom quantum computers, accounting for over 40% of the total physical errors in recent experiments. Its nonlinear and correlated nature poses significant challenges: current syndrome extraction circuits require additional overhead or sacrifice loss tolerance, and existing decoders are computationally inefficient, suboptimal, or lack provable guarantees. To address these challenges, we propose the Pauli Envelope framework, which bounds the effect of atom loss with low-weight, efficiently computable Pauli approximations, generalizing existing loss-to-Pauli methods and enabling rigorous analysis. Guided by this framework, we design improved atom-replenishing syndrome extraction circuits, the Mid-SWAP syndrome extraction, which achieves optimal loss distance and minimal space-time overhead for rotated surface codes. We also propose two decoders: an Envelope-MLE decoder achieving the optimal loss distance d_loss ~ d, and an Envelope-Matching decoder achieving d_loss ~ 2d/3 via Minimum-Weight Perfect Matching (MWPM), surpassing the previous best (d_loss ~ d/2) and readily integrating with fast correlated decoding techniques for transversal logical circuits. Circuit-level simulations demonstrate up to 40% higher thresholds and 30% higher effective distances compared with existing methods in the loss-dominated regime. Moreover, we explore correlated atom loss and show that it is easier to correct than independent loss, with thresholds rising from 5.15% to 7.82%. Remarkably, our Envelope-MLE decoder improves the error suppression factor of a hybrid MLE--machine-learning decoder from Λ= 2.14 to Λ= 2.24 on recent experimental data.

I. INTRODUCTION

Atom loss is a prevalent, nonlinear error mechanism in neutral-atom systems that complicates syndrome extraction and decoding. The paper introduces Pauli Envelopes, Mid-SWAP extraction, and two decoders to obtain rigorous loss-distance guarantees with efficient correction.

  • Motivation: More than 40% of total physical errors in recent neutral-atom experiments arise from atom loss, which removes subsequent gates and creates correlated errors.Loss persists until replenishment and can accumulate across gates and syndrome-extraction rounds.
  • Pauli Envelope framework: The Pauli Envelope framework bounds nonlinear atom-loss effects with low-weight Pauli errors, guaranteeing correctness when the envelope is decoded and strengthening distance guarantees.It addresses loss patterns that cannot be obtained by composing individual loss patterns.
  • Syndrome extraction: Mid-SWAP syndrome extraction achieves optimal loss distance d_loss ∼ d with no additional space-time overhead relative to conventional rotated surface-code extraction.Here d_loss measures the number of simultaneous atom losses tolerated before a logical error occurs.
  • Decoding: Envelope-MLE achieves d_loss ∼ d by enforcing that each atom loss triggers exactly one detector pattern, outperforming the prior Average-MLE scaling d_loss ∼ d/2.The decoder jointly reasons about Pauli-envelope selections and Pauli errors through a mixed-integer linear program.
  • Decoding: Envelope-Matching achieves d_loss ∼ 2d/3 through MWPM, exceeding the prior Marginal-Matching scaling d_loss ∼ d/2 while integrating with fast correlated decoding techniques.It rescales affected edge weights to approximately enforce envelope exclusivity.
  • Evaluation: Circuit-level evaluation reports up to 40% higher thresholds and 30% higher effective distances than prior approaches in the loss-dominated regime.The paper also reports that correlated atom loss is easier to correct than independent loss.

III. PAULI ENVELOPE

The Pauli Envelope framework bounds nonlinear atom-loss effects with Pauli error sets, allowing atom-loss decoding guarantees to be transferred to envelope decoding. It defines corresponding logical-error and envelope-failure probabilities and proves the former is upper-bounded by the latter.

  • The Pauli Envelope bounds every detector-observable outcome from a loss configuration by outcomes generated from a set of Pauli errors without loss.This converts atom-loss decoding into Pauli-error decoding while preserving the relevant detector and observable behavior.
  • A decoder that correctly handles every Pauli-envelope outcome also correctly decodes the corresponding atom-loss outcomes.
  • The framework defines atom-loss logical error and envelope-failure probabilities under separate Pauli and loss distributions, with the envelope replacing the loss configuration in the latter.
  • The logical error rate with atom loss is upper-bounded by the failure probability of decoding the corresponding Pauli envelopes.

B. Construction of Pauli Envelope for Atom Loss

The paper constructs low-weight Pauli envelopes for single and multiple atom losses in Clifford circuits. Linearity lets multi-loss envelopes be assembled from single-loss envelopes rather than analyzed through exponentially many loss scenarios.

  • A single atom-loss envelope is constructed from Pauli sets at the loss location, after subsequent Hadamards, and immediately before measurement.Each relevant location contributes the set {I, X, Y, Z}, whose compositions form the envelope.
  • The envelope construction models a lost atom using reset-like effects after subsequent gates, omitting redundant resets across gates that preserve |0⟩.
  • Pauli-envelope linearity composes envelopes for multiple losses and unions envelopes for multiple loss configurations producing the same readout.
  • This linearity avoids direct analysis of exponentially many multi-loss scenarios and avoids reasoning about atom-loss nonlinearity in each combination.
  • For a data-qubit loss, the paper explicitly constructs the envelope as compositions of Pauli sets at circuit locations.

IV. LOSS-REPLENISHING SYNDROME EXTRACTION FOR SURFACE CODES

The Mid-SWAP circuit moves atom shuttling into each syndrome-extraction round, preventing a single loss from producing both hook and data errors. This doubles the effective loss distance of SWAP extraction without additional hardware overhead.

  • B. Mid-SWAP Syndrome Extraction: Mid-SWAP doubles the effective distance against atom loss compared with SWAP syndrome extraction without additional hardware overhead.
  • A. SWAP Syndrome Extraction: SWAP syndrome extraction is limited to d_loss ∼d/2 because an early ancilla-phase loss can create both a hook error and a data error.
  • B. Mid-SWAP Syndrome Extraction: Mid-SWAP inserts atom shuttling after the first CNOT in each round, allowing a replenished atom to act first as data and then as ancilla.
  • B. Mid-SWAP Syndrome Extraction: In Mid-SWAP, a loss during the data phase causes a data error without a hook error, while a loss during the ancilla phase causes a hook error without a data error.
  • B. Mid-SWAP Syndrome Extraction: A Mid-SWAP loss triggers at most one detector edge or detector pair, producing the exclusive structure used by the decoder analysis.

V. ENVELOPE-MLE DECODER

The Envelope-MLE decoder jointly selects Pauli-envelope loss patterns and Pauli errors through a MILP subject to detector and loss-exclusivity constraints. For Mid-SWAP, it achieves optimal loss distance d_loss ∼d, while Average-MLE is limited to d_loss ∼d/2.

  • Envelope-MLE jointly optimizes Pauli-envelope selections and Pauli errors to find a minimum-weight configuration consistent with observed detector outcomes.
  • The decoder enforces detector satisfaction and selects exactly one loss-induced detector pattern for each loss-resolving readout.
  • Envelope-MLE is optimal because the actual error configuration is feasible for its MILP and the decoder returns a solution with no greater Pauli weight.
  • The Envelope-MLE decoder achieves loss distance d_loss ∼d for Mid-SWAP syndrome extraction, which is optimal for code distance d.
  • Average-MLE lacks loss exclusivity and linearizes nonlinear multi-loss patterns, limiting it to d_loss ∼d/2.

VI. ENVELOPE-MATCHING DECODER

The Envelope-Matching decoder uses Pauli-envelope information to reweight matching edges, approximately enforcing loss-event exclusivity while retaining MWPM efficiency. For Mid-SWAP extraction, it achieves loss distance d_loss ∼2d/3, exceeding prior matching-based decoders.

  • Algorithm: Envelope-Matching initializes a standard detector matching graph, processes each triggered loss readout, and runs MWPM with updated weights.Edge weights begin as log-likelihood weights derived from their trigger probabilities.
  • Method: The decoder rescales loss-affected edge weights to discourage selecting multiple edges from the same Pauli envelope.Space-like edges receive 0.5w and time-like edges receive 0.25w before MWPM.
  • Performance: d_loss ∼2d/3 is achieved for Mid-SWAP extraction, improving on the d_loss ∼d/2 of previous matching-based decoders.The result is established theoretically for independent atom loss and Pauli errors.
  • Comparison: The reweighting avoids the asymptotic failure of constant-weight loss edges, which can cause marginal matching to over-select loss-induced edges as Pauli noise vanishes.Marginal-Matching is consequently limited to d_loss ∼d/2.

VII. EVALUATION

Circuit-level evaluation finds that the proposed circuits and decoders improve thresholds and effective distances over prior approaches, including on experimental data and transversal logical circuits. Correlated loss is easier to correct than independent loss, while the framework also improves hybrid decoding performance.

  • Overall results: 40% higher thresholds and 30% higher effective distances are achieved in the loss-dominated regime than with prior approaches.The evaluation uses a circuit-level noise model combining depolarizing Pauli noise and atom loss.
  • Envelope-MLE: The Envelope-MLE decoder with Mid-SWAP raises the threshold from 3.46% to 4.85%, while achieving deff ≈7 for distance-7 extraction at η = 1.At p = 1.5% and η = 90%, it provides 6× error suppression over previous work.
  • Correlated loss: Correlated-loss thresholds increase from 5.15% to 7.82% as the correlated fraction grows, indicating easier correction than independent loss.The reported correlated-loss analysis considers atom loss on two-qubit gates.
  • Experimental data: On recent experimental data, Envelope-MLE raises Λ from 1.69 ± 0.08 to 2.01 ± 0.11 alone and from 2.14 ± 0.13 to 2.24 ± 0.14 with machine learning.The latter comparison shows complementary gains when Envelope-MLE is combined with a machine-learning decoder.
  • Envelope-Matching: Envelope-Matching with Mid-SWAP raises the threshold from 3.42% to 4.38% over Marginal-Matching and performs better on transversal logical circuits.Marginal-Matching uses erasure errors with full depolarization as its closest supported approximation to atom loss.
  • Outlook: The paper identifies extending the methods beyond surface codes and developing intermediate decoders between Envelope-MLE and Envelope-Matching as open directions.It also suggests applications to dual-rail superconducting qubits and machine-learning decoder features.

Appendix A: Proofs from Pauli Envelope Section

The appendix proves that atom-loss effects can be enclosed by Pauli-error sets and that these envelopes support the paper’s decoder guarantees. It also formalizes envelope composition and the optimality argument for Envelope-MLE.

  • Pauli Envelope Section: The logical error rate is upper-bounded by the decoder’s failure probability under the Pauli-envelope formulation.The proof maps every loss-induced detector-observable outcome to one achievable with the corresponding envelope and no loss.
  • Pauli Envelope Section: A single loss event is enveloped by Pauli sets at the loss location, after subsequent Hadamards, and before measurement.The construction yields E = L1 ⊕ L2 ⊕ ··· ⊕ Lk for the relevant locations.
  • Pauli Envelope Section: The envelope construction is justified by inserting resets, removing redundant resets after CZ and S gates, and replacing the resulting resets with Pauli mixtures.These circuit transformations preserve the set of possible outcomes and establish S(p,l) ⊆ S(p ⊕ E,∅).
  • Pauli Envelope Section: Pauli envelopes compose across multiple loss configurations, ambiguous loss readouts, and multiple loss events through union and composition operations.The appendix derives these rules using the distributive property of set composition.
  • Envelope-MLE Decoder: For Envelope-MLE, the actual error configuration is feasible for the MILP, so the minimizing solution has no greater Pauli weight than the actual configuration.This establishes the decoder’s optimality property from the envelope inclusion guarantee.

2. Proof of Loss Pattern Weight Requirement

The proof decomposes decoder failure into Pauli errors and loss-pattern mismatches, then shows each loss mismatch reduces the required Pauli weight by at most one. This yields the tight condition 2n_p+n_l<d for successful decoding and establishes optimal loss distance for Envelope-MLE.

  • Error decomposition: The error difference between the actual configuration and decoder output is decomposed into a Pauli component and syndrome differences from mismatched loss patterns.Both configurations satisfy the same detector constraints, so their symmetric difference has empty syndrome when decoding fails.
  • Loss-pattern weight lemma: Each loss-pattern mismatch reduces the Pauli weight required for an undetectable logical error by at most one, giving |C_P|≥d−n_l.The argument proceeds inductively over the number of loss events after analyzing adjacent, non-adjacent, and empty-pattern cases.
  • Base cases: For a single Z-type ancilla loss, possible detector patterns form a Z-shaped S1–S2–S3 path, while an X-type ancilla loss produces one possible edge.These patterns provide the base cases for the inductive weight argument.
  • Envelope-MLE decoder: 2n_p+n_l<d is sufficient for successful decoding, and d losses can erase logical information, proving the bound is tight.Here n_l is the number of atom losses, n_p the number of Pauli errors, and d the code distance.
  • Loss-distance consequence: The Envelope-MLE decoder achieves optimal loss distance d_loss∼d for Mid-SWAP extraction under independent atom loss and Pauli errors.The result follows from the failure condition and the tight d-loss configuration.

5. Extension to Loss-Resolving Errors

The paper extends the envelope-based decoding analysis to loss-resolving errors and establishes weight properties underlying Envelope-Matching. The extension is theoretically supported but omitted from numerical evaluation for Envelope-MLE and lowers the Pauli-error threshold for Envelope-Matching.

  • Envelope-MLE extension: The loss-resolving Envelope-MLE extension preserves the original failure-probability scaling by treating false negatives as errors with effective probability p_loss^2.False positives are already covered by the original decoder, while false negatives contribute like Pauli errors.
  • Evaluation scope: The Envelope-MLE extension was not used in numerical evaluation because it significantly increases MILP problem sizes.This is an implementation scope limitation rather than a theoretical failure of the extension.
  • Envelope-Matching construction: Envelope-Matching assigns affected space-like edges weight 0.5w and time-like edges weight 0.25w, reducing each loss-induced path by at most 0.5w.These reweightings support the decoder’s failure-condition analysis.
  • Failure analysis: The Envelope-Matching failure condition follows from bounding both the matching cost of loss patterns and the reduction in logical-cycle weight.The proof compares the decoder correction with the actual error under MWPM’s minimum-weight property.

2. Proof of Loss-Distance Scaling for Envelope-Matching decoder

The Envelope-Matching analysis assigns effective weight 2n_p+1.5n_l to mixed Pauli-loss configurations and uses connected-error counting to derive loss-distance scaling. It proves d_loss∼2d/3, improving over prior decoder bounds while retaining an MWPM-based implementation.

  • Loss-distance result: d_loss∼2d/3 for Envelope-Matching under independent atom loss and Pauli errors.The effective-weight argument sets each loss contribution to 1.5 and each Pauli error contribution to 2.
  • Scaling proof: The failure-probability bound sums connected configurations with effective weight ω≥d, and the resulting series converges below a threshold p_th=1/μ̃.The counting uses polynomial lattice factors and a constant overlap parameter.
  • Loss-resolving extension: The loss-resolving extension uses w_T=0.5w and preserves the stated scaling, with undetected losses contributing like Pauli errors.A false negative requires both atom loss and failed loss detection, yielding an effective probability proportional to p_loss^2.
  • Prior decoder comparison: The conventional matching decoder reaches only d_loss∼d/3 because it interprets certain loss-induced patterns as higher-weight Pauli errors.The failure construction uses d/3 such patterns.

3. SWAP syndrome extraction

The Mid-SWAP construction inserts a SWAP after the first ancilla-data CNOT, enabling atom replenishment while controlling loss-induced data errors. Its analysis extends from surface codes to CSS codes through hook-error and segment-hook conditions.

  • CSS-code scope: For general CSS codes, the construction assumes a one-to-one correspondence between ancilla qubits and data qubits appearing first in stabilizers, with further structural assumptions for Mid-SWAP.The paper notes these assumptions may be relaxable using different alternating syndrome-extraction orders.
  • Hook-error framework: A hook error is the propagated data error from an ancilla error, and hook-error distance counts tolerable combinations of hook and regular Pauli errors.The hook-error weight is at most half the stabilizer weight after applying an equivalent stabilizer.
  • Circuit construction: The Mid-SWAP circuit inserts a SWAP between the first and subsequent CNOTs in each X- or Z-type stabilizer interaction.For X-type ancillas the sequence is CNOT, SWAP, then remaining CNOTs; the Z-type sequence is analogous with reversed CNOT direction.
  • Readout dependence: Without loss-resolving readouts, Mid-SWAP corrects floor((d_h+1)/2) atom losses by reducing their effects to hook errors or single-data-qubit errors.The bound follows from the corresponding hook-error tolerance.
  • Loss tolerance: With loss-resolving readouts, a code correcting d_s segment-hook errors also corrects d_s atom losses.A segment-hook error acts on a contiguous segment of data qubits associated with one stabilizer.
  • Surface-code consequence: Rotated surface codes satisfy d_s=d_h=d, giving optimal atom-loss tolerance with Mid-SWAP.The same condition is stated for HGP codes, including unrotated surface codes.

Appendix G: Extension to Correlated Atom Loss

The appendix extends the Pauli Envelope framework to correlated atom loss, jointly modeling correlated, independent, and Pauli error patterns while enforcing loss exclusivity. It shows that correlated loss can be tolerated without loss-resolving readouts and can provide additional location information when such readouts are available.

  • ⌊(d_h + 1)/2⌋ correlated atom loss errors are tolerable without loss-resolving readouts.
  • With loss-resolving readouts, the Mid-SWAP circuit can identify locations in a single-pair loss case, reducing the envelope to one hook error.The text states that this additional information supports decoding multiple correlated losses under the stated conditions.
  • The correlated-loss decoder jointly minimizes correlated-loss and independent-loss events while prioritizing the total loss count over Pauli errors.A small positive coefficient ensures loss errors are minimized first in the MILP objective.
  • Correlated loss is modeled by assigning Pauli envelopes and binary variables to both individual and correlated loss patterns.The decoder also assigns binary variables to Pauli error mechanisms before enforcing detector constraints.
  • Loss exclusivity enforces exactly one detector pattern for each loss-resolving readout.

3. Predict logical observable: Let x∗

The appendix evaluates correlated-loss behavior, teleportation-based syndrome extraction, error sensitivity, decoder hyperparameters, and practical decoder scaling. It reports higher correlated-loss thresholds, inferior teleportation performance, theoretically validated reweighting coefficients, and a gap between proven and observed decoder scaling.

  • The correlated-loss threshold increases from 0.05 to 0.08 as the correlated contribution η rises from 0 to 1, while effective distance remains approximately 7.
  • Correlated atom loss is easier to correct than independent loss and provides a higher threshold when loss-resolving readouts reveal more location information.
  • The teleportation circuit has substantially lower threshold and effective distance and considerably higher space-time overhead than the SWAP and Mid-SWAP circuits.Its threshold is 0.5% to 1% lower than Mid-SWAP, with a slightly lower effective distance.
  • Mid-SWAP becomes more sensitive as reset errors increase, whereas SWAP becomes more sensitive as measurement errors increase.The comparison fixes the sum of reset and measurement error rates while varying their relative magnitudes.
  • Theoretical Envelope-Matching coefficients of 0.5 for space-like and 0.25 for time-like edges closely match empirically optimized Mid-SWAP values.The agreement is reported at code distance d = 13 across loss contributions.
  • Removing Pauli-envelope conversion significantly worsens SWAP logical error rates, supporting the need to model loss nonlinearity explicitly.
  • Although Average-MLE and Marginal-Matching are proven to achieve d_loss ~ d/2, experiments show practical scaling closer to d_loss ~ 2d/3.The text attributes this possible improvement to alternating syndrome extraction orders and leaves its analysis for future work.
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