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Matching and maximum likelihood decoding of a multi-round subsystem quantum error correction experiment

Neereja Sundaresan, Theodore J. Yoder, Youngseok Kim, Muyuan Li, Edward H. Chen, Grace Harper, Ted Thorbeck, Andrew W. Cross, Antonio D. Córcoles, Maika Takita

arXiv:2203.07205v2quant-ph

TL;DR

Quantum error correction seeks to protect computation from hardware faults, while fault-tolerant syndrome processing requires both capable hardware and classical decoding. This work combines real-time feedback with maximum-likelihood decoding on a superconducting heavy-hexagon code. The decoders show a consistent performance ordering, with maximum likelihood outperforming the matching variants in the reported comparison.

  • Problem

    The work addresses experimental improvement of logical-state survivability by combining real-time feedback with maximum-likelihood decoding, which had not previously been explored experimentally.

    Source: §I · p. 2

  • Method

    The experiment combines real-time conditional reset and maximum-likelihood decoding, with leakage post-selection used to assess decoding on runs without detected leakage.

    Source: §I · p. 2, §IV · p. 5

  • Results

    The decoder comparison shows consistent improvement from matching uniform to matching analytical to maximum likelihood, and leakage-post-selected |0⟩ data have a fitted error rate of 0.041(1).

    Source: §IV · p. 5

  • Takeaways & Limitations

    The comparison provides a starting point for understanding the trade-off between decoder scalability and performance under experimental noise.

    Source: §V · p. 6

    The maximum-likelihood decoder comparison is limited to r = 4 syndrome-measurement rounds.

    Source: §IV · p. 5

Abstract

from arXiv · show

Quantum error correction offers a promising path for performing quantum computations with low errors. Although a fully fault-tolerant execution of a quantum algorithm remains unrealized, recent experimental developments, along with improvements in control electronics, are enabling increasingly advanced demonstrations of the necessary operations for applying quantum error correction. Here, we perform quantum error correction on superconducting qubits connected in a heavy-hexagon lattice. The full processor can encode a logical qubit with distance three and perform several rounds of fault-tolerant syndrome measurements that allow the correction of any single fault in the circuitry. Furthermore, by using dynamic circuits and classical computation as part of our syndrome extraction protocols, we can exploit real-time feedback to reduce the impact of energy relaxation error in the syndrome and flag qubits. We show that the logical error varies depending on the use of a perfect matching decoder compared to a maximum likelihood decoder. We observe a logical error per syndrome measurement round as low as $\sim0.04$ for the matching decoder and as low as $\sim0.03$ for the maximum likelihood decoder. Our results suggest that more significant improvements to decoders are likely on the horizon as quantum hardware has reached a new stage of development towards fully fault-tolerant operations.

I. INTRODUCTION

Quantum error correction aims to protect logical information by correcting hardware faults faster than they accumulate, but fault-tolerant computation requires several operations and classical processing alongside syndrome extraction. This work combines real-time feedback with maximum-likelihood decoding in a superconducting heavy-hexagon subsystem code.

I. INTRODUCTION

  • Quantum error correction encodes information in protected logical degrees of freedom and corrects faults faster than they accumulate to enable fault-tolerant computation.The broader process includes logical-state preparation, logical gates, repeated syndrome measurements, and decoding.Source: §I · p. 1
  • Subsystem codes are attractive for heavy-hexagon hardware because they suit qubits with reduced connectivity.Other code choices may offer high fault-tolerance thresholds or many transversal logical gates, but their space and time overhead can hinder scalability.Source: §I · p. 1
  • Syndrome processing must keep pace with measurement data to avoid a growing backlog, while some protocols require real-time feed-forward.Initializing syndrome and flag qubits through feedback between measurement cycles can help mitigate errors.Source: §I · p. 1
  • Quantum error-correction progress spans demonstrations of logical-state preparation, repeated syndrome extraction, and logical operations across several platforms.The cited superconducting-qubit work includes distance-3 surface-code realizations, while other cited demonstrations include fault-tolerant gates and protected memory.Source: §I · p. 1
  • The work combines real-time feedback and maximum-likelihood decoding experimentally to improve logical-state survivability using a distance-3 heavy-hexagon subsystem code.Conditionally resetting syndrome and flag qubits after each cycle addresses errors from energy-relaxation noise asymmetry.Source: §I · p. 2

II. THE HEAVY-HEXAGON CODE AND MULTI-ROUND CIRCUITS

The section specifies a distance-3 heavy-hexagon code and its multi-round syndrome-measurement procedure. Separate subcircuits measure X- and Z-gauge operators, with the Z circuit also measuring flag qubits.

II. THE HEAVY-HEXAGON CODE AND MULTI-ROUND CIRCUITS

  • The example is an n = 9 qubit heavy-hexagon code with minimum distance d = 3 [1].Its Z and X gauge and stabilizer groups are generated by the operators introduced in the section.Source: §II · p. 2
  • Separate subcircuits measure the X- and Z-gauge operators, and the Z-gauge circuit also measures flag qubits.The described fault-tolerant circuit is one particular implementation, though the approach can be used with other codes and circuits.Source: §II · p. 2
  • Each experiment prepares nine qubits, performs r rounds of alternating gauge measurements, then reads out all nine code qubits.The measurement order depends on the prepared logical basis: Z-gauge then X-gauge for |0⟩, and X-gauge then Z-gauge for |+⟩.Source: §II · p. 2

III. DECODING ALGORITHMS

The section introduces decoders as algorithms that use syndrome measurements to determine corrections. It names perfect matching and maximum-likelihood decoding as the two algorithms discussed.

III. DECODING ALGORITHMS

  • A decoder takes syndrome measurements as input and outputs a correction to qubits or measurement data.Source: §III · p. 2
  • The section describes two decoding algorithms: perfect matching and maximum likelihood.Source: §III · p. 2
  • Decoding is presented as part of fault-tolerant quantum computing, where syndrome information guides correction.Source: §III · p. 2

A. The decoding hypergraph

The decoding hypergraph represents circuit information available to the decoder as error-sensitive events and correlations caused by faults. The section describes how those events and hyperedges are constructed and how flag information can be used to simplify the graph.

A. The decoding hypergraph

  • The decoding hypergraph represents error-sensitive events as vertices and fault-induced correlations between them as hyperedges.It concisely describes information gathered by a fault-tolerant circuit for use by a decoding algorithm.Source: A. The decoding hypergraph · p. 2
  • Deterministic measurements define error-sensitive events by comparing each outcome with its error-free predicted value; nonzero events indicate at least one circuit error.In these circuits, events include flag measurements and differences between subsequent measurements of the same stabilizer.Source: A. The decoding hypergraph · p. 2
  • The fault model assigns probabilities to circuit components including CX gates, measurement, initialization, and reset.It distinguishes idle operations during unitary gates from idling during measurement and reset, and models Pauli errors at initialization, reset, measurement, and gates.Source: A. The decoding hypergraph · p. 2
  • Circuit faults create hyperedges from the subsets of error-sensitive events they make non-trivial, with each hyperedge assigned the summed first-order probability of its represented faults.Source: A. The decoding hypergraph · p. 2
  • A propagated fault can anticommute with logical operators, and the hypergraph tracks those effects with logical labels.For codes with distance at least three, each hyperedge has a unique logical label.Source: A. The decoding hypergraph · pp. 2–3
  • Figure 1 maps gauge operators onto the 23 qubits and shows the CX-gate order and direction, alongside a one-round syndrome-measurement circuit.The circuit diagram indicates permitted parallelization using barriers.Source: Figure 1 · p. 3
  • Deflagging uses flag measurements to apply virtual Pauli Z corrections before further gates and adjusts subsequent error-sensitive events accordingly.The resulting hyperedges can be obtained through stabilizer simulation incorporating those corrections.Source: A. The decoding hypergraph · p. 3

B. Perfect Matching Decoding

The perfect-matching decoder reduces the decoding hypergraph to separate graphs for X- and Z-errors, then pairs nontrivial events using minimum-weight edges. Edge probabilities and logical labels support the matching weights and the resulting correction.

B. Perfect Matching Decoding

  • Matching uses separate X- and Z-error graphs built from stabilizer events and size-two hyperedges, with boundary edges representing singleton hyperedges.The graphs are subgraphs of the decoding hypergraph.Source: B. Perfect Matching Decoding · p. 3
  • A perfect matching pairs nontrivial events, adding the boundary when needed, while minimizing total edge weight.Analytical weights are log((1-pe)/pe); the uniform method assigns every edge weight one.Source: B. Perfect Matching Decoding · p. 3
  • The matching's edge logical labels determine a logical correction, or its edges can specify qubit or measurement corrections.Source: B. Perfect Matching Decoding · pp. 3–4
  • The three-round decoding graphs distinguish Z- and X-stabilizer events, with Pauli Y edges connecting the graphs and size-four hyperedges shown for top and bottom Z-stabilizers.Source: Figure 2 · p. 4

C. Maximum Likelihood Decoding

Maximum likelihood decoding uses the decoding hypergraph to accumulate probabilities for error-sensitive events and logical corrections. It selects the most likely logical correction, extending decoding to circuit noise at the cost of scalability.

C. Maximum Likelihood Decoding

  • MLD is optimal but non-scalable, and its circuit-noise formulation uses the decoding hypergraph rather than only phenomenological noise.Source: C. Maximum Likelihood Decoding · p. 4
  • MLD computes probabilities for event patterns and logical corrections by updating an initial zero-error distribution for each independent hyperedge.The hyperedge update is applied once for every hyperedge in the decoding hypergraph.Source: C. Maximum Likelihood Decoding · p. 4
  • For an observed event pattern, MLD chooses the most likely logical correction to adjust logical-operator measurements.Source: C. Maximum Likelihood Decoding · p. 4

IV. EXPERIMENTAL DISCUSSION

The experiment compares three decoders on a superconducting processor and reports progressively improved decoding from uniform matching through analytical matching to maximum likelihood decoding. Leakage post-selection lowers the fitted logical error rates for the reported logical states.

IV. EXPERIMENTAL DISCUSSION

  • The distance-three heavy-hexagon code ran on a 27-qubit processor, with simultaneous measurements and conditional resets taking 768 ns per round.An Xπ-Xπ dynamical-decoupling sequence was added during code-qubit idling periods.Source: §IV · p. 5
  • Leakage can make the decoder's assumed Pauli depolarizing error model inaccurate, so runs with detected leakage can be excluded.Source: §IV · p. 5
  • 0.059(2) (0.058(3)) for |0⟩(|1⟩) and 0.113(5) (0.107(4)) for |+⟩(|−⟩) are the fitted error rates without post-selection.Rates were fitted using data through 10 syndrome-measurement rounds.Source: §IV · p. 5
  • 0.041(1) (0.044(4)) for |0⟩(|1⟩) and 0.088(3) (0.085(3)) for |+⟩(|−⟩) are the fitted error rates after leakage post-selection.The same analytical matching decoder was applied to the post-selected data.Source: §IV · p. 5
  • Decoding improves consistently from matching uniform to matching analytical to maximum likelihood on post-selected data, with fits limited to rounds r=0–4.The round limit reflects the current limitation of MLD.Source: §IV · p. 5

V. CONCLUSIONS AND OUTLOOK

The results emphasize joint progress in quantum hardware and classical processing, including mid-circuit measurements and conditional operations. The decoder comparison frames a trade-off between scalability and performance, while larger-distance codes will depend on operational quality and device noise.

V. CONCLUSIONS AND OUTLOOK

  • Mid-circuit measurements and conditional operations are demonstrated as components of the quantum error-correction protocol.The conclusions emphasize the joint progress of hardware and classical information processing.Source: §V · p. 5
  • Figure 3a plots matching-decoded logical error against syndrome-measurement rounds and reports substantial overall error reduction after leakage post-selection.Source: Figure 3 · p. 5
  • Figure 3b compares fitted error per round across the matching uniform, matching analytical, and maximum-likelihood decoders.The comparison covers all four logical states and fits up to four rounds.Source: §V · p. 5
  • The decoder comparison is a starting point for understanding the trade-off between scalability and performance under experimental noise.Source: §V · p. 6
  • Larger-distance code performance will depend on real-time reset, leakage removal, teleportation, decoding quality, and device noise levels.Source: §V · p. 6

SUPPLEMENT

This supplement reports simulated logical error rates under realistic and average depolarizing noise models, and describes the matching and maximum-likelihood decoder implementations used in the experiments.

A. Minimum weight perfect matching edge probabilities

B. Maximum likelihood implementations

  • Offline MLD stores the full probability distribution, whereas online MLD prunes it to entries consistent with each observation string and is preferable to offline MLD for three rounds or more.The choice depends on problem size: online MLD is the feasible option when |V| is large but n and k are small.Source: B. Maximum likelihood implementations · pp. 8–9

C. Simulation Details

  • The realistic simulation model uses device-exported noise parameters, while the average model applies average error rates throughout the device.The modeled noise includes gate, measurement, initialization, reset, and idling errors.Source: C. Simulation Details · pp. 9–10
  • 0.059 (0.038) for logical |0⟩ and 0.152 (0.106) for logical |+⟩ were the simulated per-round logical error rates under realistic (average) depolarizing noise, respectively.The simulations cover circuits with up to 10 syndrome measurement rounds.Source: C. Simulation Details · p. 10

D. ibm peekskill and experimental details

The experiments used 23 qubits of the 27-qubit ibm peekskill Falcon processor, with calibrated operation and reset procedures. The supplement also describes static-coupling mitigation, decoder-parameter optimization, and the fit used to estimate logical error per syndrome round.

D. ibm peekskill and experimental details

  • The d=3 circuits used 23 of ibm peekskill’s 27 qubits, and all non-virtual single-qubit gates had a 35.55 ns duration.The selected qubits exclude QF 0, QF 6, QF 20, and QF 26.Source: D. ibm peekskill and experimental… · p. 10
  • To mitigate static ZZ coupling, the experiment applied an Xπ-Xπ dynamical-decoupling sequence and characterized simultaneous gate errors under scheduling conditions resembling the stabilizer checks.The additional characterization did not include gates and qubits that idled throughout the experiments.Source: D. ibm peekskill and experimental… · p. 11
  • Each measurement and conditional-reset cycle lasted 768 ns, while initialization used repeated conditional resets to reduce f-state population.The initialization sequence used three conditional-reset rounds, an Xπ e−f pulse, and three further conditional-reset rounds; reset error was measured after a single conditional reset.Source: D. ibm peekskill and experimental… · p. 11
  • The authors optimized analytical matching-decoder input parameters against experimental logical error rates and fit logical errors across rounds using a model with SPAM error A and per-round error ε.The fit defines τ = −1/ln(1−2ε).Source: D. ibm peekskill and experimental… · pp. 11–12

E. Leakage in the system

The section measures leakage into states outside the computational subspace, which the code cannot correct, and tracks how leakage accumulates during syndrome rounds. It reports measurement-induced leakage rates and discusses trade-offs in reducing leakage from measurements and two-qubit gates.

E. Leakage in the system

  • Leakage into |2⟩ or higher states lies outside the code’s correctable computational space and threatens fault-tolerant computing.Frequency collisions during cross-resonant gates can induce leakage.Source: E. Leakage in the system · p. 12
  • Measurement-induced leakage is estimated by repeatedly measuring after an Xπ/2 pulse, which samples |0⟩ and |1⟩ before the measurement tone.The measured outcomes are classified using |0⟩, |1⟩, and |2⟩ calibration data with readout-error mitigation.Source: E. Leakage in the system · p. 13
  • 6.54×10−3 average and 4.86×10−3 median ΓL per measurement quantify the measured leakage rate.ΓL is the probability of leaking during a measurement.Source: E. Leakage in the system · p. 13
  • The estimated leakage total sums gate and measurement leakage probabilities across the syndrome and flag qubits and final code-qubit measurements.The counts depend on the logical basis and number of syndrome rounds.Source: E. Leakage in the system · p. 13
  • Measurement leakage contributes most in early rounds and tends to saturate, while two-qubit-gate leakage becomes significant later.The analysis identifies both sources as targets for leakage reduction.Source: E. Leakage in the system · p. 14
  • Reducing leakage involves trade-offs: slower two-qubit gates and longer readout can increase circuit time or idle time for other qubits.Increasing measurement power can also make qubits more prone to leakage.Source: E. Leakage in the system · p. 14

F. Post-selection method

The post-selection procedure identifies leakage using three-state readout classification and discards runs in which any qubit is classified as |2⟩. The section also describes a two-round comparison of decoder results on full and leakage-post-selected data.

F. Post-selection method

  • Calibration uses a three-cluster Gaussian Mixture Model with independent diagonal covariance matrices, then defines each state’s region within 3σ on each I/Q axis.The method addresses overlap between |1⟩ and |2⟩ distributions across the 23 qubits used.Source: F. Post-selection method · p. 14
  • Each measurement is classified using state regions and centroid proximity, and runs are discarded if any qubit is classified as |2⟩.Shots with a qubit in |1⟩ after initialization are retained as correctable errors.Source: F. Post-selection method · p. 14
  • After initialization, QF 12 populations were 0.9910, 0.0071, and 0.0019 for |0⟩, |1⟩, and |2⟩; after the first X-syndrome, they were 0.4972, 0.4962, and 0.0066.Source: F. Post-selection method · p. 15

G. Error for r = 2 rounds

  • The comparison covers state preparation and two syndrome rounds for |+⟩L, |−⟩L, |0⟩L, and |1⟩L.The matching-decoder analytical-input results for |0⟩L and |+⟩L correspond to Fig. 3a at r = 2.Source: G. Error for r = 2 rounds · p. 15
  • Table S5 compares logical errors from three decoders on full and leakage-post-selected datasets for r = 2 rounds.The full datasets contain 500,000 shots; the table reports retained shot counts for post-selected datasets.Source: Table S5 · p. 15

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