Source-linked AI summary
State preservation by repetitive error detection in a superconducting quantum circuit
J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, I. -C. Hoi, C. Neill, P. J. J. O'Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, John M. Martinis
TL;DR
Quantum error correction must preserve quantum states against environmental errors while scaling to larger systems. This paper uses repetitive QND parity measurements and graph-based processing in a nine-qubit superconducting repetition code, reporting reduced logical error rates as device information is incorporated into decoding.
Problem
Preserving quantum information from environmental errors and extending qubit lifetime remain outstanding challenges for scalable quantum computing.
Method
A nine-qubit superconducting repetition code uses repeated parity measurements and minimum-weight matching, with physical error information incorporated into postprocessing.
Results
Including weighted edges reduces the 9-qubit 8-cycle logical error rate from 3.29% to 2.897%, and including leakage intelligence reduces it further to 2.414%.
Takeaways & Limitations
The study motivates comparing fault-tolerant systems by how rapidly errors are suppressed as qubits are added.
Abstract
from arXiv · showhide
Quantum computing becomes viable when a quantum state can be preserved from environmentally-induced error. If quantum bits (qubits) are sufficiently reliable, errors are sparse and quantum error correction (QEC) is capable of identifying and correcting them. Adding more qubits improves the preservation by guaranteeing increasingly larger clusters of errors will not cause logical failure - a key requirement for large-scale systems. Using QEC to extend the qubit lifetime remains one of the outstanding experimental challenges in quantum computing. Here, we report the protection of classical states from environmental bit-flip errors and demonstrate the suppression of these errors with increasing system size. We use a linear array of nine qubits, which is a natural precursor of the two-dimensional surface code QEC scheme, and track errors as they occur by repeatedly performing projective quantum non-demolition (QND) parity measurements. Relative to a single physical qubit, we reduce the failure rate in retrieving an input state by a factor of 2.7 for five qubits and a factor of 8.5 for nine qubits after eight cycles. Additionally, we tomographically verify preservation of the non-classical Greenberger-Horne-Zeilinger (GHZ) state. The successful suppression of environmentally-induced errors strongly motivates further research into the many exciting challenges associated with building a large-scale superconducting quantum computer.
Supplementary Information for: “State preservation by repetitive error detection in a
The classical repetition code protects a bit by redundantly storing it and correcting sparse independent flips through majority voting. Increasing the code size produces exponential suppression of storage failure when the physical flip probability is below one-half.
- Independent bit flips with probability p are modeled while correlated m-bit errors are neglected.
- A trusted supervisor checks each bit and replaces minority values with the majority value.The method succeeds when fewer than half of the n stored bits have flipped.
- p < 1/2 ensures the average number of errors remains below n/2 as the code size increases.
- Exponential suppression of pfail follows from increasing the separation between the average error count and n/2.To first order in p, pfail ∼ p^⌈n/2⌉.
B. Secret data
Parity-only access protects secret data by converting changes in neighboring-bit parity into a graph-decoding problem. With imperfect measurements, decoding must include time so measurement errors are distinguished from data errors.
- Neighboring-bit XOR measurements reveal parity without directly exposing the stored bit values.A single interior bit flip changes the parities of the two neighboring pairs.
- Parity changes are represented as colored graph vertices, and error events are decoded by connecting vertices into error chains.
- Minimum-total-length matching selects the most likely error pattern because independent errors make shorter patterns more probable.
- Parity-only decoding succeeds when fewer than ⌈n/2⌉ errors occur in one time interval, retaining pfail ∼ p^⌈n/2⌉.
- Imperfect parity measurements require space-time decoding because a measurement error produces sequential detection events.Vertical graph edges represent measurement errors and do not require data-bit corrections.
- Final-round parity information from direct data measurements completes the graph and removes the future time boundary.
E. Two possible corrected outputs
Quantum parity operators detect error information without measuring individual qubits, and decoding maps the observed events to either the intended state or its bit-inverse. Logical failure occurs when the net error chain crosses code boundaries.
- Decoding yields the proper state or its logical bit-inverse, and a logical error occurs only when a net error chain crosses boundaries.
- The procedure treats state-preparation and measurement errors to the same order as error-correction errors.
- Multi-qubit parity operators detect bit- and phase-parity information without knowing or collapsing each qubit’s individual state.
- The Z1 Z2 parity operator distinguishes even from odd computational-basis parity but cannot identify which qubit flipped.
- A one-dimensional array of nearest-neighbor parity operators supplies spatial information needed to locate bit-flip errors.
III. QUANTUM REPETITION CODE: EXPERIMENT
The experiment implements a nine-qubit repetition code with five data qubits and four measurement qubits. It runs up to eight parity-measurement cycles and postprocesses the outcomes to identify logical errors.
- The experimental repetition code contains nine qubits: five data qubits and four measure qubits.
- The parity-measurement circuit uses an ancilla and quantum gates to detect bit flips while preserving the measured parity information.
- Control-electronics memory limits the experiment to a maximum of eight parity-measurement cycles.
- Each run initializes the qubits, executes one to eight gate-sequence cycles, measures the data qubits, and postprocesses the results for logical errors.
- Postprocessing incorporates increasing levels of physical-device information to suggest lower-error corrections.The reported results use the most detailed level described.
A. Basic processing
The basic processing pipeline converts raw QND measurement data into detection events, places them on an error-connectivity graph, and uses matching to infer corrections. For the 9-qubit, 8-cycle dataset, this procedure produced a 3.35% failed-run rate.
- A. Basic processing: Raw data are supplemented with simulated initial and final parity rounds so the first measurements can detect changes and final data-qubit outcomes can be inferred.The initial rounds use the intended logical state, while the final round is computed from the terminating data-qubit measurements.
- A. Basic processing: QND parity measurements produce alternating or constant patterns, and changes between these patterns identify bit-flip detection events.For three measurements, the detection-event test reduces to mt−2 ⊕ mt.
- A. Basic processing: Detection events are placed on a graph whose vertices represent possible event locations and whose edges encode how errors can connect them.The graph supports classical postprocessing of parity-measurement and data-qubit errors between repetition-code cycles.
- A. Basic processing: Minimum-weight perfect matching selects a minimal set of edges reproducing the observed detection events, thereby generating corrections for the final measurement results.A run succeeds when the corrected output matches the input.
- A. Basic processing: 3.35% of 9-qubit 8-cycle runs failed after matching-based postprocessing.In the illustrated instance, the selected correction restored the observed output to the input.
B. Data errors during the repetition code cycle
The basic graph model is extended to represent data-qubit errors occurring during repetition-code cycles. Including these additional error paths slightly improves the measured logical error rate.
- B. Data errors during the repetition code cycle: Data-qubit errors can occur at any time during a repetition-code cycle, creating detection-event patterns that require additional graph edges.The timing of an error determines when neighboring measure qubits detect it.
- B. Data errors during the repetition code cycle: 3.35% to 3.29%: the 9-qubit 8-cycle logical error rate after adding edges for within-cycle data-qubit errors.The improvement is small because the extra edges address rare errors occurring in precise time windows.
- B. Data errors during the repetition code cycle: A real-data example changes from failure to success when within-cycle data-error edges make a unique, better graph solution available.The basic postprocessing is shown in Fig. S9b, while the expanded graph accounts for data errors during the cycle.
C. Weighted edges
The decoder is improved by weighting graph edges according to the modeled probabilities of the corresponding errors. This resolves degeneracies that occur when all edges are treated equally.
- C. Weighted edges: Edge weights are set as w = −ln p, so low-probability connections, especially diagonal edges, receive larger penalties.The probabilities are calculated from an error model for the gates used in the repetition-code cycle.
- C. Weighted edges: 2.897% from 3.29%: the 9-qubit 8-cycle logical error rate after probability-weighted edges replace equally weighted edges.Weighting favors the correct solution in the illustrated run and significantly improves the measured rate.
- C. Weighted edges: Equal edge weights can leave multiple equally acceptable matchings, whereas probability weighting favors the solution more consistent with the physical error model.In the example, only one of the equally weighted solutions leads to successful correction.
D. Leakage
The decoder incorporates leakage information from measure-qubit outputs in the non-computational |2⟩ state. Reweighting subsequent edges helps match leakage-related detection events and lowers the logical error rate.
- D. Leakage: A measure-qubit output of 2 makes the next measurement unreliable, so the decoder reweights the next vertical edge to reflect a 0.5 error probability.The output 2 is treated as a 1 together with the reweighting rule.
- D. Leakage: A neighboring data qubit in |1⟩ is essentially randomized after measure-qubit leakage, whereas a neighboring |0⟩ qubit is mostly undisturbed.The decoder models this by setting the two associated horizontal-edge probabilities to 0.25.
- D. Leakage: 2.414% from 2.897%: the 9-qubit 8-cycle logical error rate after incorporating |2⟩-state information into measurement and postprocessing.Without leakage-aware reweighting, the matching can produce unsuccessful corrections in data regions corrupted by leakage.
- D. Leakage: Leakage-related detection events are preferentially matched to one another after edge reweighting, producing successful correction in the illustrated run.The figure describes a large region of data corrupted by leakage.
E. Fine-tuning the error model
The authors fine-tune postprocessing by optimizing error-model edge weights using experimental data, reducing the logical error rate relative to basic processing. Cross-dataset testing indicates the optimized models are nearly equivalent.
- Geometrically- or weight-distinct edges share variables, while cyclic repetition-code structure makes relatively few unique edge weights sufficient.Two leakage-related edge types are dynamically added when leakage is detected.
- Leakage is handled by lowering the weights of nearby edges, encouraging leakage-generated detection events to pair together.In the illustrated example, reweighting produces corrections that return the output to the input.
- 90,000 runs per cycle count and logical-error totals provide the fitness metric for Nelder–Mead optimization of edge variables.The primary file contains 1.44 million runs across one to eight cycles, two logical inputs, and all cycle counts.
- 2.300% and 2.243% cross-dataset logical error rates show similar performance when models are tested on data they did not see.The even-trained model is evaluated on odd data and vice versa.
- 30% reduction: hardware-optimized postprocessing lowers the logical error rate from 3.35% to 2.3%.The authors use this as evidence that better error models yield lower logical error rates.
IV. ERROR SUPPRESSION FACTOR Λ - FIGURE OF MERIT FOR FAULT-TOLERANT QUANTUM ERROR CORRECTION
This section introduces Λ as a system-level measure of how logical errors change with fault-tolerance order and evaluates it using five- and nine-qubit data. The measured ratio decreases with cycle number, leaving scalable behavior unresolved.
- Λ measures how rapidly logical errors are suppressed as qubits and fault-tolerance order increase, rather than requiring an experimentally imposed threshold error rate.Ideally, logical error scales as ϵ ∼1/Λ^(n+1).
- The nine-qubit experiment is 2nd-order fault-tolerant to X errors, while three five-qubit subsets provide 1st-order fault tolerance.The five-qubit subset performance is inferred directly from full nine-qubit data.
- Comparing a bare qubit with first-order fault tolerance alone cannot establish Λ, because gate errors may make the corrected system perform worse despite Λ > 1.At least one system demonstrating both first- and second-order fault tolerance is required.
- Λ is calculated as the ratio of the average five-qubit logical error rate to the nine-qubit logical error rate.Both basic and hardware-optimized processing are compared across cycle number.
- Λ decreases with increasing cycle number because accumulated leakage increases the error rate.The data are insufficient to determine whether a scalable system would approach a constant Λ > 1.
VI. DECOMPOSING THE FAILURE RATE
The authors decompose nine-qubit failure rates by final data-qubit state and identify leakage and measure-qubit energy relaxation as contributors to increasing detection events over cycles.
- 0.33 to 6.9·10^-4: the ratio of final data states decreases exponentially as the number of bit-flips increases.The decomposition uses 90,000 statistics after eight cycles with input state |00000⟩.
- 2·10^-3 to 11·10^-3: final states containing one bit-flip have this error-correction failure-rate range, whereas |00000⟩ has no observed failure.The matching algorithm returns either a correction to the input or its bit-wise inverse.
- 12.5%: for final state |01011⟩, matching selects the inverse operator and fails in this fraction of cases.The final state can conceal measurement errors and multiple bit-flips that cancel.
- Increasing detection events are attributed primarily to state leakage and measure-qubit energy relaxation.These mechanisms are analyzed through cycle-dependent detection-event patterns.
- Removing |2⟩-events significantly reduces detection events and weakens their cycle-number dependence, implicating non-computational leakage.The remaining increase is attributed to measure-qubit relaxation, whose alternating pattern reflects switching between |0⟩ and |1⟩.
VIII. SAMPLE FABRICATION
The device uses nine superconducting Xmon qubits with engineered readout, filtering, and control features, while supplementary measurements characterize relaxation, dephasing, leakage, and readout discrimination.
- Nine Xmon qubits are fabricated on sapphire with individual control and readout, using crossovers to suppress parasitic slotline modes.The fabrication process includes multiple deposition and etching steps.
- T1 values are 20–50 µs, while Ramsey 1/e times are 15 µs near the flux-insensitive point and 2–5 µs away from it.The frequency-dependent measurements are reported for all nine qubits.
- Readout resonators couple to a common bandpass filter and an impedance-matched parametric amplifier, with time-multiplexing required for simultaneous nine-qubit readout.The amplifier provides 15–18 dB gain and has saturation power around -100–110 dBm.
- The bandpass filter has roughly 220 MHz bandwidth, accommodating nine readout resonators spaced by 30 MHz.Its geometry and coupling choices are designed to control frequency placement and isolation.
- State discrimination assigns each measured IQ point to the closest ideal |0⟩, |1⟩, or |2⟩ state.The ideal state locations are established from accumulated readout statistics.
XIII. PRESERVATION OF TWO-QUBIT GATE FIDELITY WHEN SCALING UP
The nine-qubit device preserves two-qubit gate performance while integrating high-fidelity measurement and operating across characterized frequency regions. Its repetition-code implementation uses calibrated gates, detuning, and frequency selection to limit unwanted interactions.
- Readout architecture: Nine qubits use individual readout resonators coupled to a shared λ/2 bandpass filter and frequency-multiplexed readout line.The filter isolates the qubits from the 50 Ω environment, while an impedance-matched parametric amplifier supports multiplexed readout.
- Gate fidelity: 0.0191 average error per two-qubit Clifford C2 was measured for Q4 and Q5, close to 0.0189 reported for a five-qubit chip.The result supports maintained gate performance while scaling to nine qubits and integrating high-fidelity measurement.
- Control crosstalk: 0.1–0.9% frequency-control crosstalk was measured, with magnitude decreasing with line distance and sign depending on wire-routing direction.The crosstalk matrix relates actual and ideal SQUID-loop flux through Φactual = (1 + dMΦ)Φideal.
- Component characterization: Single-qubit gate fidelities met or exceeded 0.999, while readout separation fidelities exceeded 0.998 for the characterized measure qubits.The measure qubits also had T1 values in the 20–30 µs range and Ramsey 1/e times of 2–3 µs.
- Frequency selection: A 2% below-median |1⟩-population threshold identifies operable and poor frequency regions, including incoherent and coherent defects.Strongly coupled coherent features are identified through coherent population swapping and are more consequential for operation than narrow, weak features.
- Repetition-code operation: The repetition-code sequence decomposes CNOTs into CZ and π/2 gates and uses detuning pulses to move unused qubits.Frequencies for idling, CZ gates, and readout are selected to avoid poor-coherence regions and mitigate stray interactions, including next-nearest-neighbour coupling.
XVIII. PROTECTING THE GHZ STATE FROM BIT-FLIP ERRORS: CONDITIONAL QUANTUM STATE TOMOGRAPHY
The experiment reconstructs GHZ-state density matrices before and after two repetition-code cycles using conditional quantum state tomography. The tomography separates outcomes by detection-event pattern and reports raw and corrected output states.
- Tomography method: GHZ-state density matrices were reconstructed at the input and after two repetition-code cycles using quantum state tomography.The tomography used gates from {I, X/2, Y/2, X}⊗3 and constrained the reconstructed density matrix to be physical.
- Tomography method: 12 · 10^3 repetitions supported conditional tomography by separating measured data-qubit probabilities according to detection events.A zero-time idle was used to suppress non-idealities from data-qubit measurement and state preparation.
- Tomography results: Raw and corrected output density matrices were reported for the GHZ-state experiment, conditional on detection events.Corrected matrices were reconstructed from the conditional outputs shown in the corresponding figure.
- State preparation: The GHZ-state preparation circuit used for the main experiment is documented separately with detuning pulses indicated in the circuit figure.The figure provides the gate sequence used to generate the input GHZ state.