Assessment of the errors of high-fidelity two-qubit gates in silicon quantum dots

The study addresses the need for consistent, high-fidelity two-qubit entangling operations in solid-state spin qubit platforms, where device-to-device variability and materials-induced noise can degrade performance. In silicon MOS quantum dot qubits, nanometric size and nanosecond operation times make them particularly sensitive to atomic-scale disorder and the amorphous Si/SiO2 interface, challenging the reproducibility of high-fidelity gates. The work focuses on analyzing statistical characteristics and temporal stability of primitive two-qubit gates based on Heisenberg exchange in silicon quantum dots. The authors aim to identify and attribute dominant error sources, evaluate gate performance across multiple devices and over time, and assess strategies—such as robust gate designs and active feedback—to achieve consistent fidelities exceeding 99% suitable for scalable quantum processors.

Entangling mechanisms and gate fidelities critically impact algorithmic performance and error correction. Recent advances have enabled high-fidelity exchange-based two-qubit gates in silicon spin qubits, crossing error-correction thresholds (e.g., Nature 2022 reports, Sci. Adv. 2022). Prior demonstrations include CZ-style exchange gates and encoded spin logic in silicon. Foundational randomized benchmarking and interleaved RB methods, gate set tomography (GST), and fast Bayesian tomography (FBT) provide frameworks to characterize errors and process matrices. Earlier work has identified decoherence stemming from charge noise, hyperfine interactions from residual 29Si, and control-induced shifts (e.g., AC Stark, off-resonant driving). This study builds on these results to systematically compare CZ versus dynamically decoupled CZ (DCZ) implementations, incorporate parity readout into GST, and benchmark across devices with differing isotopic purification and fabrication batches.

- Devices: Three silicon MOS multi-gate quantum dot devices were studied. Devices A and B are nominally identical three-dot chains fabricated in the same batch on 28Si substrates with 800 ppm 29Si; device C is a four-dot device on 28Si with 50 ppm 29Si. Two dots were used at a time for two-qubit operation. Exchange is controlled via an interstitial J-gate between plunger gates P1 and P2. Spin readout uses Pauli spin blockade sensed by a SET. - Gate implementations: Two entangling strategies were implemented: (1) CZ via a single square voltage pulse producing an effective Ising interaction, with z-phase corrections for Stark shifts; (2) DCZ, a composite sequence of two exchange pulses separated by a microwave X rotation that dynamically decouples single-qubit dephasing during the entangling operation. - Operating points and calibration: Charge stability maps in isolated/reservoir modes were used to select symmetric exchange-on points to minimize detuning noise. d.c. biases positioned the exchange-on near the high end of the dynamic range to support blockade readout. Qubit resonance frequencies, coherence times (T1, T2*, Hahn echo T2), and Rabi rates were measured. Frequency feedback loops (e.g., on ESR frequency and other parameters) were applied in devices A and B to mitigate 1/f noise and slow hyperfine-induced drifts. - Error characterization toolchain: Three complementary methods were used: • Interleaved randomized benchmarking (IRB) to estimate average two-qubit gate fidelities by interleaving the target entangling gate within random two-qubit Clifford sequences. • Fast Bayesian tomography (FBT) to infer process matrices directly from arbitrary input circuits (using IRB data decomposed into primitive gates), enabling time-resolved Bayesian estimates of fidelity during long runs. • Gate set tomography (GST) with tailored workflows that explicitly incorporate parity readout (via measurement fiducials or analysis adaptation in pyGSTi), yielding decompositions into Hamiltonian and stochastic error generators and enabling error taxonomy. - Non-Markovian/contextual analysis: Additional experiments probed contextual drifts, notably Larmor frequency shifts depending on microwave pre-pulse duration, consistent with two-level fluctuator dynamics in the oxide. These effects were related to biases in calibrations for gates later in a sequence and contrasted with Markovian assumptions in standard analyses. - Comparative study across devices: The same characterization methods were applied to devices A (CZ), B (DCZ), and C (DCZ) to evaluate reproducibility, impact of isotopic purification, and required feedback overheads. - Reported parameters: Extended Data include device parameters (T1, T2*, Hahn echo T2, Rabi times, exchange tunability), GST error-channel decompositions, and IRB/FBT/GST fidelity summaries.

- High and consistent two-qubit fidelities across devices and methods: • IRB average two-qubit entangling gate fidelities: 98.4% (CZ, device A), 99.37% (DCZ, device B), 99.76% (DCZ, device C). Overall average across three devices: 99.17% with 0.56% standard deviation. • FBT-based two-qubit fidelities: A ≈ 98.35 ± 0.54%, B ≈ 99.03 ± 0.91%, C ≈ 99.04 ± 0.51% (95% CIs from Bayesian re-sampling), with time-resolved analyses showing calibration drift and fidelity degradation over hours. • GST single-qubit gate fidelities (example): device A Xπ/2 ≈ 96.8 ± 0.2%, device B Xπ/2 ≈ 98.0 ± 0.1% (log-likelihood fit errors). - Error taxonomy and physical attributions (from GST and FBT): • DCZ substantially suppresses both stochastic and Hamiltonian IZ/ZI error channels (order-of-magnitude reduction versus CZ) by echoing quasi-static Larmor shifts and mitigating Stark-shift-induced phase errors. • Dominant stochastic errors are dephasing-like (T2/Hahn-limited) on both target and spectator qubits; idling qubits accrue significant dephasing and crosstalk-induced errors, making single-qubit gates the lowest-fidelity operations in multi-qubit contexts. • Hamiltonian (systematic) errors arise from AC Stark shifts, off-resonant driving (frequency crosstalk, ΔEz comparable to Ω tails), residual exchange during single-qubit control, and calibration inaccuracies (under/overrotations, exchange level errors), including contextual discrepancies between calibration and circuit conditions. • Unexplained but consistent error channels (e.g., strong IY Hamiltonian terms in CZ-only runs; stochastic XZ terms) suggest additional microscopic mechanisms. - Non-Markovian/contextual noise: • Measurable Larmor frequency shifts depend on microwave pre-pulse duration, consistent with oxide two-level fluctuator excitation, causing calibration biases for gates later in long sequences. • Such drifts can lead to IRB overestimation, occasionally producing unphysical fidelities >100%, while FBT captures time-varying fidelity and larger uncertainties. - Device/material insights: • Device C (50 ppm 29Si) operated at high fidelity with reduced feedback overhead compared to A/B (800 ppm 29Si), indicating benefits of isotopic purification; across all devices, charge noise more strongly limits T2(Hahn) than hyperfine noise. - Practical thresholds and scaling: • Reported fidelities exceed 99% and are compatible with sustainable error correction; error structure is biased toward dephasing rather than depolarizing, which can be advantageous for tailored error-correction codes.

The results directly address the central question of achieving consistent, high-fidelity two-qubit entangling operations in silicon spin qubits and identifying the underlying error mechanisms. By deploying CZ and DCZ gates across multiple devices and applying IRB, FBT, and GST, the study shows that fidelities above 99% are reproducible and that careful gate design (DCZ) significantly reduces key phase-error channels (IZ/ZI). The analyses disentangle stochastic dephasing from Hamiltonian systematics and highlight the impact of spectator-qubit dephasing and crosstalk, explaining why single-qubit gates embedded in two-qubit contexts can appear as fidelity bottlenecks. The identification of contextual, non-Markovian drifts (e.g., Larmor shifts due to microwave history) clarifies discrepancies between methods and underscores the need for adaptive calibration and robust control to maintain performance over long experiments. These insights inform scalable control strategies by prioritizing materials improvements, active feedback, and pulse engineering to suppress dominant errors, thereby enhancing prospects for fault-tolerant quantum computing with silicon spin qubits.

This work demonstrates consistent, repeatable two-qubit gate operation in silicon MOS quantum dots with fidelities exceeding 99% across three devices and extended operating times. A comprehensive error taxonomy reveals that DCZ gates suppress dominant phase-error channels and that residual infidelity is largely stochastic dephasing, with systematic Hamiltonian errors mitigable through calibration and pulse strategies. Non-Markovian, contextual effects are identified and quantified, guiding the development of feedback and robust control techniques. The study establishes that current fidelities are not fundamentally limited and can be improved via better materials (lower charge and hyperfine noise), active recalibration leveraging tomographic diagnostics, and pulse engineering (robust, dynamically decoupled control, and scalable global-field approaches). Future research should target automated, scalable control stacks that reduce calibration overhead, explore pulse shapes less sensitive to device idiosyncrasies, and further investigate unexplained error channels to push fidelities and consistency to the next level.

- Analysis methods (IRB, GST, FBT) largely assume Markovian errors; identified contextual, non-Markovian drifts (e.g., Larmor shifts from microwave history, low-frequency 1/f and nuclear noise) can bias estimates and inflate IRB fidelities beyond 100%. - Gate calibrations can drift over long experiments, leading to time-dependent fidelities; environmental conditions during calibration differ from those during long circuits, introducing contextual errors. - While device C suggests benefits from higher isotopic purification, the study does not present a controlled, quantitative comparison of purification levels. - Single-qubit gate errors in multi-qubit contexts (due to spectator dephasing/crosstalk) remain a performance limiter and complicate direct comparisons to isolated single-qubit metrics. - Some consistent error channels (e.g., IY Hamiltonian in CZ, stochastic XZ) lack clear microscopic attribution, indicating incomplete physical understanding.