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Realization of real-time fault-tolerant quantum error correction
C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti, N. C. Brown, T. M. Gatterman, S. K. Halit, K. Gilmore, J. Gerber, B. Neyenhuis, D. Hayes, R. P. Stutz
TL;DR
Real-time fault-tolerant quantum error correction requires repeated quantum–classical interactions to diagnose and correct errors during computation. This work demonstrates those capabilities for a single logical qubit in a ten-qubit QCCD trapped-ion processor, while simulations identify hardware improvements needed to approach the pseudo-threshold.
Problem
Reliable large-scale quantum computation requires error suppression beyond current physical-control precision and real-time quantum–classical processing for diagnosing and correcting errors.
Method
The authors encode, control, repeatedly error-correct, and decode a single logical qubit with the smallest color code on a ten-qubit QCCD trapped-ion quantum computer, complemented by system-level simulations.
Results
The experiments demonstrate logical state preparation and readout, single-qubit gates, repeatable error-correction cycles, magic-state encoding, and dynamically protected single-qubit memory.
Takeaways & Limitations
The demonstrated hardware capabilities and simulations support co-design of QCCD systems and QEC software, with multiple-logical-qubit operations and operation below the pseudo-threshold remaining milestones.
Takeaways & Limitations
The color-code circuits are fault-tolerant to qubit errors but not leakage errors, which can eventually corrupt logical information if untreated.
Abstract
from arXiv · showhide
Correcting errors in real time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, mid-circuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a ten qubit QCCD trapped-ion quantum computer to encode a single logical qubit using the $[[7,1,3]]$ color code, first proposed by Steane~\cite{steane1996error}. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical SPAM error of $1.7(6) \times 10^{-3}$, compared to the average physical SPAM error $2.4(8) \times 10^{-3}$ of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections that are done either as software updates to the Pauli frame or as physically applied gates. Moreover, these procedures are done repeatedly while maintaining coherence, demonstrating a dynamically protected logical qubit memory. Additionally, we demonstrate non-Clifford qubit operations by encoding a logical magic state with an error rate below the threshold required for magic state distillation. Finally, we present system-level simulations that allow us to identify key hardware upgrades that may enable the system to reach the pseudo-threshold.
INTRODUCTION
Reliable large-scale quantum computation requires fault-tolerant error correction that combines precise physical operations with real-time classical diagnosis and correction. This work demonstrates these capabilities for a single logical qubit encoded in a ten-qubit trapped-ion QCCD processor.
- Motivation: Quantum error correction redundantly encodes information into a protected subspace to enable extremely low error rates despite imperfect physical control.The motivation is that physical control precision alone is unlikely to meet large-scale computation requirements.
- Requirements: Fault-tolerant QEC requires below-threshold initialization, gates, and measurements plus repeated real-time interaction with a classical processor.The classical processor diagnoses errors and determines corrections during computation.
- Prior work: Prior work demonstrated individual fault-tolerant components across several architectures, but not repeated correction of all single-qubit errors in one complete implementation.The paper identifies this missing full demonstration as the central gap.
- This work: The experiment encodes, controls, and repeatedly error-corrects one logical qubit with ten physical qubits on a trapped-ion QCCD computer.The system uses high-fidelity gates, mid-circuit measurement and reset, real-time decoding, and corrections through software or physical operations.
- This work: The demonstrated procedures include fault-tolerant initialization, logical Clifford operations, syndrome measurement, and repeated correction while preserving data-qubit coherence.Low crosstalk during ancilla measurement helps avoid decohering the encoded data qubits.
Background
The experiment uses the [[7, 1, 3]] color code, implemented with seven data qubits and three syndrome-measurement ancillas in a reconfigurable ion-trap processor. Stabilizer syndromes are decoded classically to determine corrections during QEC cycles.
- Code: In a [[7, 1, 3]] code, seven physical qubits encode one logical qubit, and distance three protects against one arbitrary single-qubit error.The code is the smallest topological color-code instance and is also called the Steane code.
- QEC cycle: A QEC cycle measures stabilizers, processes the resulting syndrome with a classical decoder, and applies corrections, including repeated measurements to address measurement errors.Real-time decoding is required for non-trivial logical computation.
- Hardware: The ten-qubit processor uses seven data qubits for the logical state and three ancillas for syndrome measurements.The available qubits are arranged through interleaved initialization, gating, measurement, and transport operations.
- Code layout: The color-code layout uses polygon vertices for data qubits and off-to-the-side ancillas, with each polygon supporting both X-type and Z-type stabilizer measurements.Logical Z is implemented with Z operations on qubits 5, 6, and 7, while logical Z measurement combines their individual outcomes.
- Hardware: The trap’s transport operations rearrange ions during circuits, allowing a one-dimensional trap geometry to implement the color code’s two-dimensional connectivity.The processor supports parallel operations across three zones.
- Control: The control language was extended beyond OpenQASM 2.0 to support classical assignment, classical operations, and conditional logical operations required for QEC.These extensions provide programming support for measurement-dependent computation.
EXPERIMENTS
The experiments demonstrate the quantum-error-correction operations needed for universal QEC computation restricted to a single logical qubit.
- EXPERIMENTS: The experimental program targets universal QEC computation restricted to a single logical qubit.The section introduces the demonstrations of the required QEC operations.
QEC cycles
The central experiment repeatedly executes adaptive fault-tolerant QEC cycles while decoding syndromes, tracking corrections, and updating them in real time. It begins with fault-tolerant logical-state encoding and ends with logical-basis measurement.
- QEC cycles: The experiment repeatedly runs adaptive fault-tolerant QEC cycles with real-time correction determination during computation.This is identified as the paper’s main result.
- QEC cycles: Each protocol initializes a logical Pauli-basis state, measures syndromes, decodes them, tracks corrections, and performs final measurement in the appropriate logical basis.Corrections are updated after every QEC cycle.
Logical state preparation
A fault-tolerant encoding circuit prepares six logical Pauli-basis states by first preparing and verifying |0⟩L, repeating initialization when verification fails.
- The circuit prepares {|0⟩L, |1⟩L, |+⟩L, |−⟩L, |+i⟩L, |−i⟩L} as the target logical basis states.
- Verification measures Z using three CNOTs with an ancilla after preparing |0⟩L.
- If the ancilla is measured in |0⟩, preparation proceeds; if it is measured in |1⟩, all qubits are reinitialized.
- The verification procedure repeats until successful, for up to three iterations.
Adaptive syndrome extraction protocol
The protocol adaptively extracts repeated syndromes and decodes them with lookup tables to protect against both hook and measurement errors during QEC cycles.
- Adaptive syndrome extraction protocol: Syndrome extraction repeats measurements within each QEC cycle because noisy gates make single syndrome measurements vulnerable to measurement errors.
- Adaptive syndrome extraction protocol: Hook errors are flagged because they can spread to higher-weight errors that cause logical errors.
- Adaptive syndrome extraction protocol: The protocol measures two flagged stabilizer sets sequentially and triggers a final unflagged round when either flagged circuit indicates an error.
- Adaptive syndrome extraction protocol: A two-stage decoder uses syndrome changes between QEC cycles and between extraction rounds within a cycle.
- Adaptive syndrome extraction protocol: Conditional extraction rounds and a second lookup table make the QEC cycle fault tolerant to hook and measurement errors.
Pauli frame update
The experiment tracks logical corrections in a Pauli frame while measuring logical performance across repeated QEC cycles, revealing logical error rates, timing costs, and remaining benchmarking gaps.
- Pauli frame update: Pauli-frame corrections are stored as two bits for {I, X, Y, Z} and updated after each QEC cycle by Pauli multiplication.
- Pauli frame update: 1.7(6)×10^-3 average logical SPAM error was measured versus 2.4(8)×10^-3 physical SPAM error.
- Pauli frame update: Approximately 2.70(6)% logical error per QEC cycle was obtained from an exponential decay fit.
- Pauli frame update: Logical SPAM experiments took < 60 ms, while each QEC cycle took < 200 ms.
- Pauli frame update: Complete QEC overhead remains uncharacterized until logical-qubit entangling operations are measured.
Active vs software corrections
The experiment compares physically applied corrections with software-tracked Pauli-frame corrections for a logical S gate, demonstrating real-time correction handling.
- Active vs software corrections: Pauli-frame updates cannot always replace physical corrections because Pauli operators transform nontrivially under non-Clifford gates.
- Active vs software corrections: The logical S gate serves as a stand-in for the non-Clifford T gate when demonstrating physical correction application.
- Active vs software corrections: 92(1)% logical fidelity was achieved when corrections were handled by rotating the Pauli frame instead of applying a physical correction.
- Active vs software corrections: The physical- and software-correction error rates were not significantly different in this experiment.
Preparing a magic state
Universal quantum computing needs non-Clifford gates, which the color code cannot construct using simple transversal operations. The paper therefore prepares a logical magic state using a non-fault-tolerant encoding circuit and obtains an error rate below the magic-state-distillation threshold.
- Non-Clifford gates are required for universal quantum computing but cannot be built by simple transversal operations in the color code.
- The logical T|+⟩L state is prepared with a non-fault-tolerant color-code encoding circuit because fault-tolerant verification would collapse the state.The prepared state can be used for T gates through gate teleportation in a system supporting logical two-qubit gates.
- 2.2(6)% magic-state error is significantly below the 33.5% noise threshold estimate for magic state distillation.The resulting states are therefore described as high quality and distillable for implementing fault-tolerant non-Clifford gates.
SIMULATIONS AND ANALYSIS
The paper combines logical-error modeling, experimental comparisons, and system-level simulations to characterize noise and estimate routes toward the pseudo-threshold. Simulations qualitatively agree with experiment, while gate errors, dephasing, and leakage emerge as central performance considerations.
- Logical error model: The logical channel is modeled as an asymmetric depolarizing channel with fitted Pauli error probabilities p_x, p_y, and p_z.The logical error rate is p_L = p_x + p_y + p_z.
- Logical error model: Logical basis error rates are obtained by fitting exponential decays and inverting the resulting equations to recover p_x, p_y, and p_z.Only Pauli operators that do not commute with a basis state contribute errors for that basis.
- Simulation comparison: 0.26 Hz for coherent simulation and 0.43 Hz for incoherent simulation were required to reproduce the logical QEC cycle error rate.The difference indicates that coherent buildup in the distance-three color code may affect performance.
- Simulation comparison: 1.80(6)×10^-2 for experiment and 1.85(7)×10^-2 for simulation were the average QEC cycle error rates.The simulations and experiments qualitatively agree, although additional known and unknown noise sources remain to be characterized.
- Error budget: Approximately 49% from unitary gates, 45% from dephasing, and 6% from measurement and initialization contribute to the logical QEC cycle error rate.The gate contribution has a large linear dependence on physical gate error, likely associated with leakage.
- Pseudo-threshold analysis: A crude system-level model indicates that reducing dephasing alone is insufficient to reach the pseudo-threshold, whereas substantially reducing leakage is important.The simulations scale physical error rates and compare alternative models with reduced dephasing and leakage.
CONCLUSION
The work demonstrates core fault-tolerant QEC primitives for a single logical qubit and uses experiments and simulations to identify requirements for improving toward the pseudo-threshold.
- The experiments demonstrate high-fidelity logical state preparation and readout, logical single-qubit gates, and repeatable error-correction cycles.
- Two-qubit gate errors are the largest contributor to the physical-level error budget.
- Leakage and dephasing are identified as crucial noise sources for logical performance, even though leakage is not currently dominant physically.
- Reducing two-qubit gate error by approximately a factor of 3, together with substantially reducing leakage and dephasing, could put the pseudo-threshold within reach.
- The experiments and emulation tools support co-design of QCCD hardware and QEC software, with multi-logical-qubit operations and operation below the pseudo-threshold identified as next milestones.
Methods
The system uses trapped-ion hyperfine qubits and Raman-mediated gates in a QCCD architecture, with physical operations characterized through randomized benchmarking and optical state preparation and measurement.
- The physical qubits use 171Yb+ S1/2 hyperfine clock states as the computational basis.
- Single-qubit gates use two-ion {Yb, Ba} crystals, while two-qubit operations use four-ion {Yb, Ba, Ba, Yb} crystals coupled through axial motion.
- The single-qubit axial center-of-mass mode is 1.0 MHz, and the first higher-order mode at 1.74 MHz is used for entangling operations.
- Physical operations are characterized using parallel randomized benchmarking, while initialization and measurement use optical pumping and state-dependent fluorescence.
- The average physical SPAM error is 2.4(9) × 10^-3, with measurement and reset also affected by laser scatter and ion fluorescence absorbed by idle qubits.
Simulations
The simulations reproduce the QEC experiments with coherent and incoherent noise models, while pseudocode specifies repeated state preparation, syndrome cycles, decoding, Pauli-frame tracking, and measurement.
- Simulations: The simulator receives the compiler’s native quantum-gate, classical, and transport instructions and models errors coherently with a state-vector backend or incoherently with a stabilizer simulation.
- Simulations: The error model includes depolarizing gate noise, leakage, and dephasing during transport and cooling, with most parameters taken from experiments or microscopic noise analysis.
- Simulations: Coherent dephasing is modeled as RZ(θ) between ideal gates, where θ equals the dephasing rate multiplied by the elapsed duration.
- Repeated QEC procedure: The repeated-QEC procedure initializes a logical state, executes a chosen number of cycles, tracks the Pauli frame, performs logical measurement, and counts expected outcomes.
- Repeated QEC procedure: The QEC cycle updates syndrome histories and the Pauli frame before rotating the logical measurement basis and performing destructive measurement.
- Active versus software correction: The active-correction experiment compares physically applying logical X or Z corrections with updating those corrections in software.