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Demonstration of logical qubits and repeated error correction with better-than-physical error rates

A. Paetznick, M. P. da Silva, C. Ryan-Anderson, J. M. Bello-Rivas, J. P. Campora, A. Chernoguzov, J. M. Dreiling, C. Foltz, F. Frachon, J. P. Gaebler, T. M. Gatterman, L. Grans-Samuelsson, D. Gresh, D. Hayes, N. Hewitt, C. Holliman, C. V. Horst, J. Johansen, D. Lucchetti, Y. Matsuoka, M. Mills, S. A. Moses, B. Neyenhuis, A. Paz, J. Pino, P. Siegfried, A. Sundaram, D. Tom, S. J. Wernli, M. Zanner, R. P. Stutz, K. M. Svore

arXiv:2404.02280v3quant-ph

TL;DR

Large-scale fault-tolerant quantum computing requires suppressing errors to levels inversely proportional to computation size. This paper uses fault-tolerant logical entanglement and repeated error correction on a trapped-ion QCCD processor, reporting logical error rates below physical baselines.

  • Problem

    Large-scale fault-tolerant quantum computing requires suppressing errors to levels inversely proportional to computation size.

  • Method

    The experiments compare fault-tolerant logical circuits with unencoded physical baselines, using Steane and Carbon codes with error correction, detection, and selective rejection.

  • Results

    9.8 times and 500 times lower error rates are reported for Steane-code error correction and error detection, respectively, than for physical experiments.

  • Takeaways & Limitations

    The demonstrations show logical circuits whose observed error rates are below corresponding physical circuit baselines.

Abstract

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The promise of quantum computers hinges on the ability to scale to large system sizes, e.g., to run quantum computations consisting of more than 100 million operations fault-tolerantly. This in turn requires suppressing errors to levels inversely proportional to the size of the computation. As a step towards this ambitious goal, we present experiments on a trapped-ion QCCD processor where, through the use of fault-tolerant encoding and error correction, we are able to suppress logical error rates to levels below the physical error rates. In particular, we entangled logical qubits encoded in the [[7,1,3]] code with error rates 9.8 times to 500 times lower than at the physical level, and entangled logical qubits encoded in a [[12,2,4]] code based on Knill's C4/C6 scheme with error rates 4.7 times to 800 times lower than at the physical level, depending on the judicious use of post-selection. Moreover, we demonstrate repeated error correction with the [[12,2,4]] code, with logical error rates below physical circuit baselines corresponding to repeated CNOTs, and show evidence that the error rate per error correction cycle, which consists of over 100 physical CNOTs, approaches the error rate of two physical CNOTs. These results signify a transition from noisy intermediate scale quantum computing to reliable quantum computing, and demonstrate advanced capabilities toward large-scale fault-tolerant quantum computing.

I. METHODOLOGY

The methodology benchmarks complete physical and fault-tolerantly encoded circuits by comparing their processed experimental outputs. It uses parity-based success metrics and specialized resource-state preparations on Quantinuum’s H2 trapped-ion QCCD processor.

  • Circuit benchmarking: Complete circuits are benchmarked by comparing outputs from unencoded physical circuits with corresponding fault-tolerantly encoded circuits on the same hardware.The approach follows Gottesman’s proposal while allowing a different comparison metric and more general state preparations.
  • Circuit benchmarking: The metric adds classical processing of measurement outputs to classify each run as successful or failed.For Bell-state preparation, parity agreement indicates success; this produces lower-uncertainty estimates than total variation distance.
  • State preparation: Resource-state experiments allow qubits to begin in a finite set of fixed states rather than only |0⟩.In the unencoded setting, preparation reduces to initializing |0⟩ states and applying gates.
  • Hardware platform: The experiments were performed on Quantinuum’s H2 trapped-ion processor, a shuttling-based QCCD device with high-fidelity SPAM, two-qubit gates, long-range connectivity, and mid-circuit measurement and reset.H2 reports 0.15% SPAM error, 0.14(1)% two-qubit-gate error, and crosstalk errors ≤2 × 10^-5 for mid-circuit measurement and reset.
  • Logical entanglement: Logical Bell-state preparation is used as a baseline for evaluating improved entanglement with the [[7, 1, 3]] Steane code and a [[12, 2, 4]] code.The study positions logical Bell-state preparation as a baseline analogous to physical Bell-state demonstrations.

A. Steane code

The Steane-code experiments prepare and measure logical Bell states using fault-tolerant encoding, transversal Clifford operations, flagged syndrome extraction, and destructive logical measurements. Error correction and error detection are analyzed against unencoded physical circuits, with the [[12, 2, 4]] comparison restricted to X and Z parities.

  • A. Steane code: The [[7, 1, 3]] Steane code is used to prepare a high-fidelity logical Bell state with relatively low space-time overhead and transversal Clifford gates.Its CSS structure supports simple preparation and measurement protocols.
  • A. Steane code: Each logical qubit uses seven data qubits and three ancilla qubits, producing experiments with 20 physical qubits.The logical program includes encoding, transversal gates, flagged syndrome extraction, and destructive logical measurements.
  • A. Steane code: Fault-tolerant |0⟩ preparation combines an initial encoding circuit with ancilla-based verification and conditional retry or pre-selection after failure.The verification measures the logical Z operator before accepting the preparation.
  • A. Steane code: After transversal Hadamard and CNOT operations, flagged syndrome extraction and destructive measurements in logical X, Y, and Z bases verify the Bell resource state.Destructive data-qubit measurements provide logical outcomes and syndrome information.
  • A. Steane code: Exz is the fraction of incorrect X- and Z-parity results, while Steane-code Y-basis measurements additionally support state-fidelity estimates.For the [[12, 2, 4]] code, only X and Z parities are used because it lacks a transversal fault-tolerant Y eigenbasis measurement.
  • A. Steane code: Error-detection analysis excludes runs with non-trivial destructive-measurement syndromes, whereas physical baselines omit syndrome extraction, pre-selection, and post-selection.The physical circuits otherwise match the logical-program structure.

2. Experimental results

The experiments compare physical and fault-tolerantly encoded Bell-state preparation using the Steane and Carbon codes. Encoded experiments show lower observed error rates, with post-selection providing further suppression.

  • Steane code: 0.50%+0.03% −0.03% was the physical Bell-correlation error rate across 411,600 unencoded experiments.The incorrect parity counts were 572, 530, and 795 for X, Y, and Z correlations, respectively.
  • Steane code: Approximately 75% of Steane error-correction experiments were pre-accepted, with non-trivial syndromes triggering decoder corrections.Among pre-accepted runs, incorrect parity occurred 9 times for X, 8 times for Y, and 0 times for Z.
  • Steane code: Post-selection reduced acceptance from approximately 75% to 72%, with zero incorrect X or Z parity outcomes and two incorrect Y outcomes.The accepted sample sizes were 8,688 for X, 8,701 for Z, and 8,665 for Y.
  • Steane code: 9.8-fold and 500-fold reductions were observed for Steane logical error correction and error detection, respectively.The corresponding logical error rates were 0.001%+0.013% −0.001% for error correction and 0.001%+0.013% −0.001% for error detection as reported in the comparison passage.
  • Carbon code: The Carbon experiment used a [[12, 2, 4]] CSS code to entangle two pairs of logical qubits and measure Bell correlations.Its effective logical circuit uses separate blocks, physical-level pre-selection, and optional syndrome-based post-selection.

1. Circuits

The circuits implement logical Bell-state preparation and measurement with error correction, pre-selection, and optional post-selection. Their design exploits Carbon’s distance-four properties while recognizing acceptance and scalability trade-offs.

  • Carbon circuit: The Carbon Bell-state circuit requires 30 physical qubits: 24 data qubits across two blocks and 6 ancillas.Verification measurements are distributed throughout preparation rather than confined to separately prepared blocks.
  • Verification: The preparation criteria detect correlated error propagation in unitary stabilizer-state circuits but do not measure Bell correlations nondestructively or all stabilizer generators.This defines the scope of the verification procedure used in the experiments.
  • Carbon circuit: Carbon corrects all weight-1 errors in each block and detects all errors up to weight 2, but cannot correct every weight-2 error.The two operating modes are error correction alone and error correction combined with rejection.
  • Operating modes: In correction mode, all pre-accepted runs produce logical outcomes, with syndrome-based decoding performed by table lookup.Post-selection on syndrome outcomes is not used in this mode.
  • Operating modes: Post-selection can effectively increase an even-distance code’s distance from d to d + 1 by rejecting errors of weight d/2 that would otherwise cause logical errors.The passage states that post-rejection is second order in the physical error rate because single faults are corrected rather than rejected.
  • Overheads: Pre-rejection is limiting because its rate is first order in the physical error rate, whereas post-selection becomes the dominant overhead in larger systems with state-preparation factories.The text describes pre-selection as scalable in that larger-system setting.

2. Experimental results

The [[12, 2, 4]] Carbon code reduced Bell-correlation error rates below the unencoded physical baseline, with stronger suppression when post-selection was applied. The physical-versus-logical difference was statistically significant.

  • 0.8%+0.1%−0.1% was the estimated error rate for the unencoded circuit.
  • 15,483 of 22,000 encoded runs were pre-accepted, while post-selection accepted 15,409 runs and rejected about 0.5%.
  • 0.17%+0.07%−0.06% was the estimated encoded error rate with pre-selection, a 4.7-fold reduction.
  • 0.001%+0.015%−0.001% was the estimated encoded error rate with pre-selection and post-selection, corresponding to an 800-fold reduction.
  • The physical and logical error rates differed statistically significantly, as indicated by separated 95% confidence intervals.

IV. REPEATED FAULT-TOLERANT ERROR CORRECTION

The paper demonstrates repeated fault-tolerant error correction with the [[12, 2, 4]] Carbon code and evaluates it against physical circuit baselines. The experiments target logical circuits whose error rates can improve on physical operations while addressing the need for repeated correction.

  • IV. REPEATED FAULT-TOLERANT ERROR CORRECTION: 3 rounds of error correction were demonstrated for the [[12, 2, 4]] Carbon code using error correction and detection.
  • IV. REPEATED FAULT-TOLERANT ERROR CORRECTION: The error rate accumulated per correction round was comparable to that accumulated with two physical CNOTs in series.
  • IV. REPEATED FAULT-TOLERANT ERROR CORRECTION: The Carbon code’s high threshold was accompanied by a requirement for three code blocks when using teleportation-based syndrome extraction.
  • IV. REPEATED FAULT-TOLERANT ERROR CORRECTION: Logical circuits combined gate teleportation and permutation-based gates within a single round of error correction.
  • IV. REPEATED FAULT-TOLERANT ERROR CORRECTION: The comparison used sequences of two physical 1-bit teleportations and two physical CNOTs as baselines.
  • IV. REPEATED FAULT-TOLERANT ERROR CORRECTION: The two-CNOT baseline was chosen because CNOTs had the highest physical error rates in the system.

C. Compiler optimizations

Compiler and scheduling optimizations were used to reduce memory and transport errors in structured Carbon-code circuits. Across repeated error-correction experiments, logical error rates remained below physical baselines, with fitted per-round rates near the two-CNOT baseline.

  • C. Compiler optimizations: Compiler optimization and dynamical decoupling were introduced to mitigate coherent memory noise from shuttling, cooling, and qubit idling.The transport optimizer was given a new cost function and allowed longer runtimes to increase parallelism and reduce gating steps.
  • C. Compiler optimizations: 0.017% logical error rate was observed for one error-correction round, more than an order-of-magnitude below the physical baselines.The reported logical result is paired with physical baseline rates of 1.0% for two 1-bit teleportations and 0.43% for two CNOTs.
  • C. Compiler optimizations: 0.5% ± 0.3% was the logical error rate for three rounds, versus physical baselines of 2.5% ± 0.2% and 1.2% ± 0.1%.This still represented a gain of more than two over the physical baselines.
  • C. Compiler optimizations: 0.3% ± 0.1% was the fitted error rate per error-correction round, compared with 0.7% ± 0.1% for 1-bit teleportations and 0.42% ± 0.02% for the CNOT baseline.The differences in fitted per-round rates were not statistically significant given the uncertainty in the error-correction fit.
  • C. Compiler optimizations: The apparent nonlinear gain trend resulted from different y-intercepts rather than nonlinear per-round error growth.The logical circuit model uses pL(2r −1) + pSPAM,L, with memory and transport errors dominating the experiments.

V. SUMMARY AND OUTLOOK

The experiments demonstrate fault-tolerant circuits whose logical error rates outperform physical counterparts in a trapped-ion QCCD system. The Carbon-code experiments also extend repeated error correction while comparing against physical teleportation and CNOT baselines.

  • V. SUMMARY AND OUTLOOK: 4.7 to 800 times lower error rates were demonstrated in Bell correlation experiments using the [[7, 1, 3]] and [[12, 2, 4]] codes.The comparison is against physical error rates and uses different codes and protocols.
  • V. SUMMARY AND OUTLOOK: Up to ten rounds of error correction were demonstrated with the [[12, 2, 4]] Carbon code by combining error correction and error detection.The supplied figure caption reports observed error rates for one to three rounds and physical baselines for comparison.
  • V. SUMMARY AND OUTLOOK: Figure 7 compares one to three rounds of Carbon-code error correction with physical baselines formed from pairs of 1-bit teleportations and pairs of CNOTs.Results are offset along the x-axis, and linear fits are obtained by maximum-likelihood estimation.

Appendix A: Carbon code from Knill’s C4/C6 scheme

The Carbon code is a first-level, monolithic adaptation of Knill’s C4/C6 scheme designed to reduce the original scheme’s space and time overheads. Its preparation and error-correction circuits use fault-tolerant detection, teleportation, tailored decoding, and post-rejection.

  • Appendix A: Carbon code from Knill’s C4/C6 scheme: The [[12, 2, 4]] Carbon code is obtained by concatenating C4 into C6 at Knill’s first level and substantially reducing space and time overheads.Knill’s scheme has a stated circuit noise threshold of 3%, but incurs large resource overheads.
  • Appendix A: Carbon code from Knill’s C4/C6 scheme: Even-distance C4/C6 codes can detect but not correct some errors, requiring multiple circuit copies for high-probability execution in those cases.This is a scope limitation of the code family rather than a claim that every detected error is correctable.
  • Appendix A: Carbon code from Knill’s C4/C6 scheme: 19 qubits and 37 CNOTs prepare logical |00⟩ in the adapted circuit, versus 144 qubits and 216 CNOTs in Knill’s preparation.Bell-state preparation requires as few as 36 qubits.
  • Appendix A: Carbon code from Knill’s C4/C6 scheme: Teleportation detects and corrects errors while implementing logical operations, with transversal gates and within-block CNOTs or SWAPs realized through permutations.The experiments focus on operations needed for error correction and Bell-state demonstrations, primarily logical |00⟩ and |++⟩ preparation.
  • Appendix A: Carbon code from Knill’s C4/C6 scheme: Carbon circuits are designed so logical errors require at least three circuit faults, while post-rejection requires at least two faults.These conditions support the code’s effective error-detection and correction behavior.
  • Appendix A: Carbon code from Knill’s C4/C6 scheme: Carbon preparation uses flags and stabilizer measurements so that single faults leave residual CSS weight at most one after accepted preparation.Negative stabilizer outcomes cause pre-rejection before interaction with other encoded blocks.
  • Appendix A: Carbon code from Knill’s C4/C6 scheme: Early flag measurements detect high-weight errors before they spread, simplifying fault-tolerant detection circuitry.A weight-two error can otherwise spread to a higher-weight error during preparation.

b. Bell states

Logical Bell states are prepared by applying transversal CNOTs between fault-tolerantly prepared Carbon blocks, with stabilizer measurements either performed before or after entanglement. Bell-state teleportation then supplies syndrome information for logical error correction.

  • b. Bell states: Transversal CNOTs between fault-tolerantly prepared |++⟩ and |00⟩ Carbon blocks create logical Bell states.Stabilizer measurements may be postponed until after the transversal CNOTs to detect additional errors.
  • b. Bell states: Bell-state teleportation transfers an input Carbon data block into the remaining Bell half while measuring a Carbon syndrome.The syndrome is used to detect or correct errors on the teleported data.
  • b. Bell states: The Carbon decoder post-rejects uncorrectable syndromes to achieve effective distance five, so logical errors require at least three circuit faults.Distinct weight-two Pauli errors can share syndromes in the distance-four code.
  • b. Bell states: Postponed measurements of X⊗4 on top blocks and Z⊗4 on bottom blocks complete the alternative Bell-state preparation circuit.The circuit pairs corresponding C4 blocks before applying transversal CNOTs.
  • b. Bell states: Simultaneous X⊗4 and Z⊗4 measurements on paired C4 blocks are fault tolerant: one fault is detected or leaves an error of at most weight one.The data-qubit CNOTs can be reordered or parallelized.
  • b. Bell states: The lookup decoder is tailored to the circuit’s fault structure because an independent-qubit phenomenological error model is insufficient for these circuits.The tailored approach considers single and double faults in the state-preparation circuits.
  • b. Bell states: Keeping only syndromes consistent with CSS weight-one errors preserves an O(p^2) post-rejection rate while substantially improving logical error rates.The more aggressive decoder discards some configurations that might otherwise be correctable and affects post-rejection only marginally.

3. Scaling to larger sizes

The section discusses extending the Carbon scheme through concatenation and explains the statistical procedures used to estimate error rates and fit scaling trends. It also cautions that the fitted average accumulated error rates per round cannot be conclusively distinguished.

  • Scaling to larger sizes: Eliding lower-level error correction when encoding C6 with Carbon saves a factor of three in qubit overhead.
  • Scaling to larger sizes: Concatenating Carbon with higher-level C6 can increase code distance and convert lower-level post-selection into higher-level pre-selection.At higher encoding levels, Carbon syndromes can be corrected as erasures, and Carbon gate gadgets can be contained in state-preparation circuits.
  • Statistical analysis: Failure-rate error bars use beta posteriors updated from binomially distributed experimental outcomes.The analysis assumes independent, identically distributed Bernoulli trials for each circuit.
  • Statistical analysis: 95% credible intervals use posterior medians as point estimates and the 2.5% and 97.5% posterior quantiles as bounds.The estimates remain non-zero even when no failures are observed because of the chosen prior.
  • Statistical analysis: The fitted average accumulated error rates per round cannot be compared conclusively because the fit-parameter uncertainty is too large.The observed gap between the rates is nevertheless described as not large.

Appendix C: Rejection rates in the Carbon code

Carbon rejection rates provide diagnostics of fault-tolerant circuit behavior: they grow approximately linearly with repetition count, while pre- and post-rejection rates remain widely separated. The section also contrasts parity-based error-rate measurements with full state-fidelity measurements for logical and physical Bell preparations.

  • Rejection-rate diagnostics: Pre-selection and post-selection rejection behavior helps assess whether the fault-tolerant circuits perform as expected.The experimental rejection data are highly consistent with the predicted linear dependence.
  • Rejection-rate scaling: Pre- and post-rejection rates increase roughly linearly with the number of error correction rounds.Here, logical volume is proportional to the number of error correction rounds, and Fig. 12 shows the expected linear trend.
  • Rejection-rate scaling: The large separation between pre- and post-rejection rates is consistent with first- and second-order events.
  • Bell-state measurements: The Steane-code experiments report full Bell-state infidelity and individual X⊗X, Y⊗Y, and Z⊗Z error rates for physical and logical preparations.Logical tables include analyses using quantum error correction and quantum error detection.
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