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Fault-tolerant detection of a quantum error
S. Rosenblum, P. Reinhold, M. Mirrahimi, Liang Jiang, L. Frunzio, R. J. Schoelkopf
TL;DR
Uncorrected thermal excitation during the parity map and decay remain important error sources. The demonstrated parity-check measurement protects against first-order ancilla errors, yielding a fault-tolerance gain of five for a hardware-efficient architecture.
Problem
Uncorrected errors, particularly thermal excitation during the parity map and decay, remain important error sources.
Method
The approach uses a parity-check syndrome measurement of a logical qubit protected against first-order ancilla errors, with a hardware-efficient architecture and engineered sideband interaction.
Results
A fault-tolerance gain of five was achieved while protecting the parity-check syndrome measurement against first-order ancilla errors.
Takeaways & Limitations
Fault-tolerant syndrome measurements are a necessary step toward architectures supporting additional fault-tolerant computing operations.
Takeaways & Limitations
Additional drives may ultimately degrade system coherence, making protection challenging.
Abstract
from arXiv · showhide
A critical component of any quantum error-correcting scheme is detection of errors by using an ancilla system. However, errors occurring in the ancilla can propagate onto the logical qubit, irreversibly corrupting the encoded information. We demonstrate a fault-tolerant error-detection scheme that suppresses spreading of ancilla errors by a factor of 5, while maintaining the assignment fidelity. The same method is used to prevent propagation of ancilla excitations, increasing the logical qubit dephasing time by an order of magnitude. Our approach is hardware-efficient, as it uses a single multilevel transmon ancilla and a cavity-encoded logical qubit, whose interaction is engineered in situ by using an off-resonant sideband drive. The results demonstrate that hardware-efficient approaches that exploit system-specific error models can yield advances toward fault-tolerant quantum computation.
5.1 . We can compare the observed cavity dephasing rates with predictions for residual
The demonstrated parity-check syndrome measurement protects against first-order ancilla errors and reduces their propagation relative to non-fault-tolerant measurement. The approach also supports photon-loss transparency, higher-order extensions, and hardware-efficient fault-tolerant operations, although additional protection drives may degrade coherence.
- Implications: The results provide a necessary step toward fault-tolerant architectures, which require fault-tolerant syndrome measurements.The paper distinguishes fault-tolerant operation assessment from architecture-level error-threshold claims.
- Advantages: The scheme prevents ancilla-induced errors from propagating, eliminating the need for a subsequent correction round.The measurement is also transparent to photon loss and compatible with cat-code error correction.
- Extensions: Four ancilla levels could protect against relaxation errors up to second order, or against relaxation and thermal excitations to first order.Higher-order protection is presented as an extension of the three-level implementation.
- Limitation: Additional drives required for higher-order protection may ultimately degrade system coherence.This is identified as a challenge for extending the protection order.
- Results: Fivefold fault-tolerance gain protects the logical qubit against all first-order ancilla errors compared with non-FT measurement.The authors summarize the parity-check syndrome measurement as protected against first-order ancilla errors, with a gain of five.
Supplementary Materials
The supplementary setup uses FPGA-based control and signal-processing hardware to generate, gate, detect, and demodulate the experiment’s microwave signals.
- Experimental Setup: An FPGA controller uses two Innovative Integration X6-1000M boards in a VPXI-ePC chassis.The FPGA generates four pairs of I/Q waveforms using 500 Msample/s DACs.
- Drive Control: An RF generator is gated by an FPGA pulse to enable or disable the detuned sideband drive.Single-sideband modulation and local-oscillator offsets are used to reduce mixer-leakage problems.
- Filtering: Attenuation and low-pass filtering at temperature stages thermalize blackbody radiation and suppress spurious high-frequency components.These components protect the sample from unwanted environmental and electronic signals.
- Readout: The FPGA samples the amplified signal at 1 Gsample/s, demodulates and integrates it, and identifies whether the transmon occupies g, e, or f.The output chain includes amplification, isolation, frequency conversion, and ADC sampling.
2. Deriving the Sideband Hamiltonian
The sideband Hamiltonian is derived from a driven cavity–transmon system by moving to rotating and displaced frames and retaining the desired effective interaction. The resulting transition is selective to transmon state but unselective to cavity state, with an experimentally observed strong sideband coupling.
- Hamiltonian: The driven cavity–transmon Hamiltonian contains cavity, transmon, Josephson, and drive terms, with the drive applied at frequency ω_d.The cavity and transmon modes are represented by â and q̂.
- Derivation: Moving to a displaced frame removes the quasi-static drive term, and the interaction frame leaves effective nonlinear terms after high-frequency terms are dropped.The derivation uses rotating-frame transformations and retains diagonal terms plus the desired sideband interaction.
- Interaction: The induced sideband transitions are unselective with respect to cavity state but selective with respect to transmon state.The sideband strength and detuning are set by the drive amplitude and mode frequencies.
- Measurement: Observed Rabi oscillations correspond to the |e,n⟩ ↔ |g,n−1⟩ sideband transition, with a sideband Rabi rate of Ω/2π = 1.7 MHz for n = 1.The interaction is much larger than χ_fe^0 and χ_eg^0, a precondition for its use as a χ-cancelling drive.
3. Deriving the Effective Hamiltonian
A sufficiently detuned sideband drive can be replaced by a static effective Hamiltonian whose leading terms produce a dispersive shift and whose next resonant term produces a nonlinear dispersive shift. The e–f transition becomes nearly photon-number independent, while a small residual nonlinear term remains.
- Effective Hamiltonian: For a detuned interaction, the time-dependent Hamiltonian is replaced by the static effective Hamiltonian H_eff = g^2[A,A†]/Δ.This form can be derived using operator-based Floquet theory or the second-order rotating-wave approximation.
- Dispersive Shift: The leading effective-Hamiltonian term produces the dispersive shift used in the main-text model.The effective description follows from the detuned sideband drive derived previously.
- Nonlinear Shift: The next resonant term produces a nonlinear dispersive shift measured in Fig. S3.The nonlinear contribution is associated with a photon-number-dependent term in the e–f transition.
- Measurement: The e–f transition becomes nearly independent of photon number, but a small residual nonlinear term remains.The residual term is extracted by fitting measured transition frequency against photon number with a quadratic fit.
4. Analytic Model for Cavity Dephasing due to Thermal Excitation
Thermal ancilla excitation causes cavity dephasing through frequency fluctuations in the dispersive regime. The analytic model captures the measured data and identifies distinct behavior in the large- and small-coupling limits.
- Thermal ancilla population produces cavity frequency fluctuations and therefore dephasing in the dispersive interaction regime.
- In the large χ_e/Γ_e limit, long-lived thermal excitations can completely dephase the cavity, yielding a χ_e-independent dephasing time.
- In the small χ_e/Γ_e limit, each ancilla excitation produces a small random cavity phase rotation.
- The cavity coherence model combines thermal-excitation and residual dephasing contributions and accurately describes the measured data.
- The measured residual dephasing time is T_φ,res = 14 ± 1 ms and is partially accounted for by double ancilla excitations from g to f.
5. Cat State Preparation Protocol
Cat states are prepared by displacement, repeated parity measurement, and postselection on even parity. Four parity measurements improve preparation fidelity but yield a 33% preparation success rate.
- The experiment prepares cat states by displacing the cavity, measuring parity, and postselecting on even parity.
- Four parity measurements using the Π_gf protocol maximize preparation fidelity.
- 33% is the resulting preparation success rate, while the final photon-number parity reaches 99%.
- The remaining odd-parity probability partly reflects finite parity-assignment fidelity and a 0.4% photon-jump probability per parity measurement.
6. Wigner Tomography and State Reconstruction
Wigner tomography reconstructs the postselected cavity state and evaluates its fidelity after parity-based photon-loss filtering. Repeated parity measurements use the outcome trajectory to identify shots consistent with no photon loss.
- Repeated parity measurements use the trajectory of outcomes to filter photon loss rather than relying on final-state parity.
- Wigner tomography is performed after postselection to determine the fidelity of the final cavity state.
- For single parity measurements, two successive even-parity checks retain the data, discarding approximately 30% because assignment fidelity is 85%.
- Shots with a no-photon-loss record probability below 20% are discarded, removing between 10% and 50% of data as N varies.
- State reconstruction obtains a positive-semidefinite, unit-trace maximum-likelihood density matrix, then optimizes a phase rotation before calculating fidelity.
7. Step-by-step Analysis of Parity Measurements
Parity-measurement outcomes reveal photon loss and can correlate with specific ancilla errors. The detailed analysis distinguishes some error signatures while showing that relaxation and unfiltered photon loss can be ambiguous with dephasing.
- Parity-measurement outcomes provide information about photon loss and may correlate with specific ancilla errors.
- Tables S1 and S2 analyze ancilla-state evolution for multiple event types, assuming an even cavity-parity state.
- Relaxation errors cannot be singled out because they produce g and e measurements with equal probabilities.
- Unfiltered photon loss is indistinguishable from a dephasing error.
- Other events can produce different final ancilla states, allowing some errors to be distinguished, while photon loss remains ambiguous if unfiltered.
8. Monte-Carlo Simulation
The Monte-Carlo model simulates cavity-state evolution over repeated parity measurements by sampling error events, their phase shifts, and resulting fidelity decay. An accompanying per-channel analysis estimates how each error contributes to dephasing.
- Fidelity simulation: The model accounts for the listed error channels when simulating the cavity state after repeated parity measurements.The Monte-Carlo simulation is described as the most accurate method for predicting the fidelity-decay curve.
- Monte-Carlo trajectory: The simulation samples the number of each error event from a multinomial distribution over N parity measurements.Each event is assigned an occurrence probability and associated cavity-frequency shift.
- Monte-Carlo trajectory: Each sampled event contributes a cavity-phase change drawn from a uniform distribution over that event’s active interval.The individual phases are then summed to obtain the final cavity phase.
- Fidelity simulation: 10,000 repetitions are averaged to compute fidelity as a function of N.This produces the simulated fidelity-decay curve after a sequence of parity measurements.
- Error-channel analysis: Each error channel is assigned an effective dephasing per occurrence between 0 and 1, representing the dephasing induced by that event.The dephasing probability is calculated as occurrence probability multiplied by dephasing per occurrence.
- Error-channel analysis: The effective dephasing per occurrence is obtained by comparing the modeled fidelity with the fidelity of a completely dephased state.The model uses a normalization factor in its fidelity expression.
9. Error Budget
The error budget catalogs dominant logical errors from higher-order ancilla errors and quantifies their occurrence, timing, assignment, and dephasing contributions. It uses protocol durations and final ancilla-state probabilities to parameterize these contributions.
- Error sources: The error budget identifies dominant logical errors caused by higher-order ancilla errors.These errors are summarized in Table S3.
- Protocol parameters: 2.1 μs is the time required for parity mapping, while 1.2 μs is the time required for ancilla readout.These durations define the protocol timing used in the error model.
- Ancilla outcomes: The probabilities of ending the protocol in g, e, and f are approximately 0.8, 0.12, and 0.08, respectively.These final ancilla-state probabilities parameterize the error budget.
- Readout and dephasing: Ancilla assignment error is estimated from Gaussian-readout histogram overlap together with the prior probability of each measured state.Dephasing per occurrence is calculated from the modeled fidelity expression defined in supplementary material 8.
10. System Parameters
The system-parameter table specifies the cavity, transmon, readout, and interaction properties used in the model. It includes frequencies, anharmonicity, Kerr terms, cross-Kerr couplings, and transmon relaxation.
- Frequencies: The cavity frequency is approximately 4.5 GHz, the transmon g-e frequency 6.5 GHz, and the readout-resonator frequency 9.3 GHz.These frequencies define the principal hardware modes.
- Transmon parameters: The transmon anharmonicity is approximately -210 MHz.This parameter characterizes the separation of the transmon’s higher levels.
- Kerr interactions: The cavity self-Kerr is approximately -10 Hz, while the cavity-readout cross-Kerr is approximately -236 kHz.These nonlinearities enter the cavity and readout dynamics.
- Cross-Kerr interactions: The transmon-readout cross-Kerr is approximately -1.3 MHz, and the transmon-cavity cross-Kerr is approximately -93 kHz.These couplings describe dispersive interactions between the transmon and the two resonator modes.
- Relaxation: The transmon |e⟩-to-|g⟩ relaxation time is approximately 1.07 ms.The relaxation channel is included among the system parameters.