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Repeated quantum error correction on a continuously encoded qubit by real-time feedback

Julia Cramer, Norbert Kalb, M. Adriaan Rol, Bas Hensen, Machiel S. Blok, Matthew Markham, Daniel J. Twitchen, Ronald Hanson, Tim H. Taminiau

arXiv:1508.01388v2quant-phcond-mat.mes-hall

TL;DR

Reliable active correction must keep quantum information encoded while repeated errors are detected and corrected. This paper implements repeated non-destructive syndrome measurements and real-time feedback, achieving robust correction of single-qubit errors and preventing errors from accumulating across rounds.

  • Problem

    The paper addresses the challenge of reliably performing repeated active error correction while quantum information remains continuously encoded.

  • Method

    The authors encode a logical qubit in three nuclear spins, repeatedly measure phase-error syndromes with an ancilla electron spin, and apply real-time feedback corrections.

  • Results

    The protocol achieves an average single-qubit error-correction probability of 0.94(1), while multiple correction rounds prevent errors from accumulating.

  • Takeaways & Limitations

    Repeated active correction establishes a platform for investigating quantum error correction under different noise types and advances fault-tolerant quantum information processing.

  • Takeaways & Limitations

    Uncertainty in readout calibration can introduce a small systematic error by rescaling all reported y-axes.

Abstract

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Reliable quantum information processing in the face of errors is a major fundamental and technological challenge. Quantum error correction protects quantum states by encoding a logical quantum bit (qubit) in multiple physical qubits. To be compatible with universal fault-tolerant computations, it is essential that the states remain encoded at all times and that errors are actively corrected. Here we demonstrate such active error correction on a continuously protected qubit using a diamond quantum processor. We encode a logical qubit in three long-lived nuclear spins, repeatedly detect phase errors by non-destructive measurements using an ancilla electron spin, and apply corrections on the encoded state by real-time feedback. The actively error-corrected qubit is robust against errors and multiple rounds of error correction prevent errors from accumulating. Moreover, by correcting correlated phase errors naturally induced by the environment, we demonstrate that encoded quantum superposition states are preserved beyond the dephasing time of the best physical qubit used in the encoding. These results establish a powerful platform for the fundamental investigation of error correction under different types of noise and mark an important step towards fault-tolerant quantum information processing.

METHODS

The methods combine a cryogenic diamond NV processor with controlled nuclear-spin qubits, ancilla-mediated initialization/readout, real-time feedback, and fidelity-based analysis of error-correction performance.

  • Sample and Setup: The processor uses a naturally occurring NV centre in high-purity diamond at approximately 4.2 K with a 403.553(3) G magnetic field.A solid-immersion lens and aluminium-oxide anti-reflection coating enhance photon collection.
  • Nuclear spin qubit control: Nuclear-spin gates use electron-spin π-pulse sequences, with pulse number N setting the rotation angle and inter-pulse delay 2τ selecting the controlled qubit and conditionality.The three nuclear spins are initialized by swapping with the ancilla electron and read out by mapping correlations to the ancilla.
  • Feedback: A programmable ADwin Pro II microprocessor applies detected phase-error corrections immediately after stabilizer measurements during the experimental sequence.The implementation does not perform real-time correction of ancilla readout errors through multi-round error analysis.
  • Feedback: Feedback adapts subsequent qubit bases or is absorbed into the next gate, avoiding unnecessary physical operations that could introduce additional errors.This directly modifies the physical control sequence according to measurement outcomes for logical-qubit and ancilla corrections.
  • Quantum error correction analysis: Process fidelity with the identity is calculated from six final-state fidelities and analyzed by fitting quantum-error-correction and linear-error models.The models use FQEC(pe) = O + A(1 − 3pe^2) and Flinear(pe) = O + A(1 − pe), with A and O accounting for experimental fidelities.
  • Quantum error correction analysis: For incoherent errors divided into n equal rounds, the per-round error probability is calculated as pn = (1 − n√(1 − 2pe))/2 for pe ≤ 0.5.The relation follows from equating the total expectation-value reduction, (1 − 2pe), with (1 − 2pn)^n.

SUPPLEMENTARY INFORMATION · Supplementary Note 1: Theoretical analysis: state and process fidelities

The supplementary analysis models state and process fidelities under ideal and imperfect quantum error correction. It compares error-corrected and uncorrected behavior through a weighted fit governed by experimental offsets, amplitudes, and relative curve weights.

  • Supplementary Note 1: Theoretical analysis: state and process fidelities: For ideal error correction, process fidelity to the identity is analyzed as a function of single-round error probability p_e.The theoretical analysis specifically considers one round of quantum error correction.
  • Supplementary Note 1: Theoretical analysis: state and process fidelities: The experimental state fidelities are represented by an offset O and amplitude A.These parameters account for finite experimental state fidelities.
  • Supplementary Note 1: Theoretical analysis: state and process fidelities: At p_e = 0.5, the quantum-error-correction fidelity satisfies F_QEC(p_e = 0.5) = O + A/2.This value is determined by the fidelities of the logical states |0⟩_L and |1⟩_L.
  • Supplementary Note 1: Theoretical analysis: state and process fidelities: The logical states |0⟩_L and |1⟩_L are insensitive to phase errors.Their fidelities therefore determine the quantum-error-correction fidelity at p_e = 0.5.
  • Supplementary Note 1: Theoretical analysis: state and process fidelities: Without error correction, the expected fidelity dependence on error probability is linear.This provides the uncorrected reference function for the experimental analysis.
  • Supplementary Note 1: Theoretical analysis: state and process fidelities: Experimental data are fitted to a weighted sum of the ideal-error-correction and no-error-correction functions.The fit combines the two theoretical behaviors rather than selecting only one.
  • Supplementary Note 1: Theoretical analysis: state and process fidelities: The parameter w sets the curve shape by determining the relative weights of ideal error correction and no error correction.Thus, w controls the balance between the two component equations.

Assignment of ancilla states to the error syndrome: effective measurement fidelity FM

The experiment’s asymmetric ancilla readout makes effective measurement fidelity depend on how ancilla states are assigned to stabilizer outcomes. Assigning |1⟩a to the most likely +1 outcome is optimal and yields high single-qubit error-correction performance.

  • Readout asymmetry: The ancilla readout fidelities are asymmetric: F1 = 0.988(2) for |1⟩a and F0 = 0.890(4) for |0⟩a.Effective measurement fidelity FM therefore depends on stabilizer-outcome assignments and error probabilities.
  • Symmetric assignment: 0.939(2) is the effective measurement fidelity FM after symmetrizing ancilla readout across all four assignments, independent of pe.The symmetrized data are fit using a constant A.
  • Symmetric assignment: 0.94(1) is the average probability ⟨Pn⟩ of successfully correcting single-qubit errors for the symmetrized assignment, with w = 0.81(3).The fit also gives A = 0.557(2) and O = 0.086(1).
  • Optimal assignment: 0.93(3) is the average single-qubit error-correction probability when |1⟩a is assigned to +1 for every stabilizer measurement, with w = 0.8(1).This assignment is optimal because +1,+1, indicating no error, is the most likely outcome and receives the highest-fidelity readout state.

Multiple rounds of error correction (incoherent errors), Fig. 4b

Multiple-round error-correction data were fit using weighted models combining ideal correction with linear no-correction behavior. The fits yielded distinct parameters for two and three rounds, while a single-round model could not accurately describe the data.

  • Multiple rounds of error correction: For two rounds of error correction, the fit gives w = 0.66(4) and A′ = 0.850(9).The analysis uses the optimal ancilla state assignment and a weighted sum of ideal-correction and linear no-correction equations.
  • Multiple rounds of error correction: For three rounds of error correction, the fit gives w = 0.71(2) and A′ = 0.810(5).The three-round model is presented as an extension of the weighted fitting approach used for multiple rounds.
  • Multiple rounds of error correction: The multiple-round data cannot be accurately described by the expected shape for a single round of error correction.This conclusion is stated after comparing the fitted multiple-round behavior with the single-round model in Eq. 3.

Naturally occurring decoherence (coherent errors), Fig. 4d

Under naturally occurring decoherence, the best unencoded qubit and a majority-vote encoded qubit are fitted with general exponentially decaying functions, yielding distinct decay parameters. Numerical Monte Carlo simulations examine the interplay between quantum error correction and error projection in the Fig. 4d stabilizer-measurement experiments.

  • Naturally occurring decoherence (coherent errors), Fig. 4d: The best qubit has T = 17.3(2) ms and n = 2.09(7), whereas the majority-vote encoded qubit has T = 13.7(1) ms and n = 2.37(8).Both experiments are fitted to a general exponentially decaying function.
  • Naturally occurring decoherence (coherent errors), Fig. 4d: Numerical Monte Carlo simulations analyze the interplay between quantum error correction and error projection in Fig. 4d experiments with stabilizer measurements at half the free-evolution time.The simulations are presented with details and results in Fig. 10.

Supplementary Note 2: Theoretical analysis: error probabilities

The theoretical analysis derives error-detection probabilities for a three-qubit code, incorporating finite input errors and imperfect ancilla readout. Encoded-state tomography provides measured error-outcome probabilities that determine expected QEC measurement outcomes as applied errors vary.

  • Error probabilities: The no-error probability P(0) includes either no qubits flipped or all three qubits flipped.The probability to detect an error on qubit i similarly includes an error on qubit i or errors on both other qubits.
  • Error probabilities: The model extends total-error calculations to finite input error probability p_in(i) and applied error probability pe.This accounts for errors already present in the initially prepared state.
  • Error probabilities: Imperfect ancilla readout is incorporated into the probabilities D for each detected error outcome as functions of applied error pe.The resulting expressions combine the ideal outcome probabilities P(i) with readout fidelity F and 1−F terms.
  • Tomography-derived errors: Encoded-state tomography yields outcome probabilities of 0.785(2), 0.060(2), 0.083(2), and 0.071(2) for the four XX-stabilizer combinations.These values are uncorrected for qubit readout and are translated into input-error estimates.
  • Predicted QEC outcomes: The estimated detection probabilities D as a function of applied error probability pe generate the expected QEC measurement outcomes shown as solid inset lines in Fig. 3b.These expectations are calculated from Eqs. 10–16.

Error syndrome assignment

Ancilla readout asymmetry makes QEC measurement fidelity dependent on the error probability and complicates error-detection curves. Four error assignments are modeled with corresponding equations and plotted in Fig. 8.

  • Readout asymmetry: Ancilla readout asymmetry causes QEC measurement fidelity to depend on the error probability, complicating error-detection curves.The section illustrates this dependence by considering an assignment in which both +1 stabilizer outcomes map to {|1⟩a, |1⟩a}.
  • Assignment models: Equations 17–20 describe four error assignments using combinations of P(0), P(1), P(2), P(3), F0, and F1.The expressions combine error probabilities with ancilla-readout fidelities and their complements.
  • Assignment comparison: All error-detection curves for the four assignments are plotted in Fig. 8.The curves are generated using similar equations for each assignment.

Multiple rounds of error correction, Fig. 4b

For multiple rounds of error correction, the analysis derives an average input error from the detection probability with no additional applied error and reports p(avg) in = 0.086(1) for round 1. The resulting curves are shown in the inset of Fig. 4b.

  • Multiple rounds of error correction, Fig. 4b: 0.086(1) is the average input error p(avg) in obtained for round 1.This value is obtained using Eq. 17.
  • Multiple rounds of error correction, Fig. 4b: The resulting curves according to Eq. 17 are shown in the inset of Fig. 4b.

Supplementary Note 3: Qubit readout calibration

Qubit readout calibration corrects tomography results for gate and initialization fidelities across single-, two-, and three-qubit expectation values. The calibration assumes uncorrelated non-nitrogen initialization errors, while its uncertainty can cause a small systematic rescaling of the y-axes.

  • Calibration procedure: Tomography results are corrected for the fidelity of gates used in final readout, distinguishing single-, two-, and three-qubit expectation values.These corrections provide estimates of the actual states.
  • Single-qubit calibration: For immediate single-qubit initialization and readout, ⟨Zi⟩ depends on nitrogen initialization fidelity FN = 0.94(3) and initialization and readout factors.Because the same gates are used, the calibration assumes Cinit,Qi = CQi and determines 1/CQi.
  • Multi-qubit calibration: Multi-qubit readouts are calibrated by initializing the three qubits in separable states and relating measured expectation values to nitrogen, individual initialization, and joint readout factors.The procedure calibrates three-qubit readout factors and uses single-qubit expectation values and CQi to obtain initialization fidelities.
  • Calibration limitation: Readout-calibration uncertainty can introduce a small systematic error by rescaling all y-axes, so raw uncalibrated error-correction data are also provided.The raw data are shown in Fig. 11.

SUPPLEMENTARY TABLES

The supplementary tables define the qubit and gate parameters used in the experiment and quantify the operational complexity of representative error-correction sequences.

  • Qubit and gate parameters: Supplementary Table 1 defines the hyperfine components, nuclear precession frequencies, pulse timing, pulse counts, and conditional gate durations.A∥ and A⊥ describe hyperfine interactions parallel and perpendicular to the magnetic field; τ is half the interpulse delay, N the pulse number, and gate time the total conditional ±x-gate duration.
  • Qubit and gate parameters: The table also identifies T∗ as the natural dephasing time and T1 as the longitudinal relaxation time.These parameters are calibrated approximately every 36 hours and therefore vary slightly over the experiment.
  • Experimental complexity: Supplementary Table 2 reports the number of operations from qubit initialization for one-round and three-round QEC examples.The examples measure ⟨Z1Z2Z3⟩ in Fig. 3b and ⟨X1X2X3⟩ in Fig. 4b.
  • Experimental complexity: All 13C-qubit gates comprise ancilla NV-electron-spin refocusing pulses, with repeated ancilla readout and reset.The table presents these operation counts for sequences beginning with qubit initialization.

SUPPLEMENTARY FIGURES · SUPPLEMENTARY REFERENCES

The supplementary material details initialization, tomography, real-time QEC control, entanglement, encoded-state characterization, ancilla-assignment analysis, and naturally occurring error modeling. It also reports supporting simulations, readout-uncorrected fits, and logical-qubit decay parameters.

  • SUPPLEMENTARY FIGURES: Dynamical decoupling with 256 alternating-phase π-pulses produces no significant NV electron-spin decay on the experiments’ relevant timescale.The pulse spacing is chosen as a multiple of the 13C Larmor period to decouple the electron spin from the nuclear-spin bath.
  • SUPPLEMENTARY FIGURES: Spin-selective resonant excitation initializes the ancilla NV electron spin in ms = 0 with fidelity > 0.98.A resonant laser pulse addresses the ms = ±1 ↔ E′ ms = ±1 transitions, while spin-selective resonant excitation is used for readout.
  • SUPPLEMENTARY FIGURES: A reduced SWAP with an ancilla initialized in |0⟩ deterministically initializes each naturally mixed qubit in |0⟩.After each transfer, the ancilla is reinitialized by a 300 µs laser pulse before the process is repeated for the other qubits.
  • SUPPLEMENTARY FIGURES: Three-qubit tomography maps required correlations onto the ancilla before readout, with the final ancilla π/2-pulse phase determined by the number of nonidentity operators.Examples include ⟨X1, I2, I3⟩, ⟨X1, X2, I3⟩, and ⟨−X1, Y2, Z3⟩.
  • SUPPLEMENTARY FIGURES: The QEC sequence uses real-time ADwin microprocessor control for charge-state checks, reset, stabilizer measurements, feedback, and the experimental cycle.The supplementary sequence describes a single correction round and begins by preparing the NV center for resonant readout and reset.
  • SUPPLEMENTARY FIGURES: Deterministic entanglement by stabilizer measurements starts from |00⟩ with fidelities 0.878(6) and 0.910(6) for the two qubit pairs, followed by outcome-dependent feedback.The procedure obtains the same two-qubit state independent of the measurement outcome, with post-selected results also shown.
  • SUPPLEMENTARY FIGURES: Theoretical syndrome probabilities agree well with experimental values, using normalized occurrences from 84000 samples for two assignments and 28000 samples for the other two.The assignments are {|0⟩a, |0⟩a}, {|0⟩a, |1⟩a}, {|1⟩a, |1⟩a}, and {|1⟩a, |0⟩a}; the comparison uses no free parameters.
  • SUPPLEMENTARY FIGURES: Without final-readout-gate correction, fitted unencoded and corrected values of w are −0.02(2), 0.56(6), 0.64(4), and 0.70(2) for zero through three rounds.For the logical qubit, the supplementary fit gives T = 13.7(1) ms and n = 2.35(8).
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