The quest for building large-scale, fault-tolerant quantum computers hinges on the ability to perform high-fidelity quantum operations consistently across multiple qubits. Recent breakthroughs in quantum computation, such as demonstrations of quantum supremacy using programmable superconducting processors (Arute et al., 2019), suppressing quantum errors by scaling a surface code logical qubit (Acharya et al., 2023), and quantum computational advantage using photons (Zhong et al., 2020) and fault-tolerant universal quantum gate operations (Postler et al., 2022), highlight the immense potential of quantum technologies. However, achieving these feats requires overcoming significant challenges, particularly concerning the consistency and fidelity of quantum gates. Solid-state platforms are especially vulnerable to errors stemming from material imperfections and variations between individual qubits. This inherent variability in physical characteristics leads to inconsistencies in qubit performance and poses a major obstacle for scalability. Spin qubits, owing to their nanometric size and nanosecond operational timescales, are particularly susceptible to this type of noise. In silicon MOS quantum dot qubits, this sensitivity is amplified by the close proximity of the qubits to the amorphous Si-SiO2 interface, where material disorder is significant. Maintaining high-fidelity operation across multiple qubits and over extended periods is essential for performing complex computations and implementing effective error correction schemes. Exchange-based entangling gates, while recently showing promise in achieving high fidelity (Mądzik et al., 2022; Noiri et al., 2022; Xue et al., 2022; Mills et al., 2022; Weinstein et al., 2023), still face challenges related to consistency and scalability. The current study aims to analyze the statistical characteristics and temporal stability of two-qubit gates in silicon MOS quantum dots, identifying the underlying physical error mechanisms and proposing strategies to improve fidelity and consistency for future scaling.

Significant advancements have been made in the field of silicon-based quantum computing, with notable progress in achieving high-fidelity single- and two-qubit gates. Early work demonstrated the potential of addressable quantum dot qubits with fault-tolerant control fidelity (Veldhorst et al., 2014) and implemented a two-qubit logic gate in silicon (Veldhorst et al., 2015). Subsequent research showcased programmable two-qubit quantum processors (Watson et al., 2018) and investigated the limitations imposed by electrostatic crosstalk on spin qubits in dense CMOS quantum dot arrays (Cifuentes et al., in press). The development of tunable coupling and isolation of single electrons in silicon MOS quantum dots (Eenink et al., 2019) and the operation of a silicon quantum processor unit cell above one kelvin (Yang et al., 2020) are important steps. Recent studies have demonstrated increasingly high fidelities, exceeding the fault-tolerance threshold in silicon (Mądzik et al., 2022; Noiri et al., 2022; Xue et al., 2022; Mills et al., 2022; Weinstein et al., 2023), paving the way for more complex quantum computations. However, a comprehensive understanding of the various error sources and their impact on gate fidelity remains crucial for achieving scalable and robust quantum computation.

This research employs three state-of-the-art techniques for characterizing quantum gate errors: randomized benchmarking (RB), fast Bayesian tomography (FBT), and gate set tomography (GST). Experiments were conducted on three different silicon MOS quantum dot devices (A, B, and C). Devices A and B were nominally identical three-dot chains fabricated in the same batch, while device C had four dots but a similar material stack. The isotopic purity of the silicon substrate differed between devices (800 ppm 29Si for A and B, 50 ppm for C). Two entangling gate strategies were used: a simple square voltage pulse implementing a controlled phase (CZ) gate and a decoupled controlled phase (DCZ) gate, incorporating dynamical decoupling. RB, a relatively simple technique, provided an initial estimate of gate fidelity. FBT allowed for analysis of any arbitrary gate sequence, offering a robust way to assess fidelities and identify error sources. GST, providing detailed information about the process matrix, offered a deeper understanding of individual error components. In the GST experiments, special attention was paid to the impact of the parity readout method on the results, leveraging two approaches: incorporating projections of single-qubit states into the parity analysis and adapting the pyGSTi analysis tool. The researchers also carefully analyzed the coherence times (T1 and T2) of the qubits, characterizing noise spectra and applying frequency feedback to mitigate low-frequency noise components. Specific attention was given to studying the effects of crosstalk and contextual errors. The impact of various experimental parameters, such as magnetic field orientation, was also investigated to pinpoint the physical origins of decoherence.

The study achieved average two-qubit gate fidelities exceeding 99% using the DCZ gate strategy. Specifically, average fidelities of 98.4% (CZ), 99.37% (DCZ), and 99.76% (DCZ) were achieved in devices A, B, and C, respectively, as determined by interleaved randomized benchmarking (IRB). These fidelities demonstrate sufficient operational fidelity for sustainable error correction. However, non-Markovian effects, such as contextual Larmor frequency shifts dependent on the duration of microwave pulses, were observed, leading to biases in gate calibration and posing challenges to the Markovian assumptions underlying the analysis methods. Gate set tomography (GST) and FBT revealed the dominant error channels. The DCZ gate implementation showed significantly lower stochastic and Hamiltonian IZ and ZI errors compared to the CZ gate, largely due to its inherent ability to suppress phase accumulation caused by quasistatic shifts. The main sources of errors identified were dephasing errors, calibration errors, unintended driving errors (from frequency crosstalk), and other unexplained errors. Interestingly, single-qubit gates showed lower overall fidelity than two-qubit gates. A more detailed GST analysis revealed that this was primarily due to strong dephasing of the idling qubit and crosstalk during single-qubit operations, while the on-target fidelity of the active qubit was very high. The analysis revealed a strong bias towards dephasing errors, offering potential advantages for error correction code performance. Analysis of the data using Bayesian techniques highlighted the impact of the experimental runtime on the gate fidelities, suggesting a drift in gate performance over time. These drift and other non-Markovian effects are not fully captured by simpler methods like IRB, which is seen in cases of unphysical fidelity estimates exceeding 100%. This highlights the importance of utilizing more sophisticated techniques such as GST and FBT, and employing Bayesian approaches for a more accurate representation of the experimental reality.

The findings of this study address the critical need for understanding and mitigating errors in quantum gate operations. The demonstration of consistently high fidelities exceeding 99% for two-qubit gates in silicon MOS quantum dots represents a significant advance in the field. The detailed error analysis using multiple tomographic methods provides valuable insights into the physical origins of these errors, paving the way for targeted strategies to improve gate performance. The identification of dominant error channels such as dephasing, calibration errors, and unintended driving errors allows for informed engineering design choices, including improved materials to reduce noise, active feedback to correct Hamiltonian errors, and pulse engineering to reduce stochastic errors. The bias towards dephasing errors is particularly promising, suggesting that certain error correction codes could be particularly effective in this system. The observed non-Markovian effects underscore the complexity of the quantum system, underscoring the need for more sophisticated analysis techniques and modeling beyond simple Markovian assumptions. Further investigation is needed to fully understand the microscopic origin of some of the unexplained errors identified. This study serves as a foundation for future research aimed at developing scalable and fault-tolerant quantum processors based on silicon spin qubits.

This research demonstrates high-fidelity, consistent two-qubit gate operations in silicon MOS quantum dots, exceeding 99% fidelity on average. Detailed analysis revealed dominant error mechanisms and highlighted the presence of non-Markovian effects. Strategies for enhancing fidelity include improved materials, active feedback, and sophisticated pulse engineering. The strong bias towards dephasing errors opens opportunities for tailored error correction. Future research should focus on understanding unexplained error sources and developing scalable control strategies that mitigate non-Markovian effects.

The study relies on several assumptions, particularly the Markovian nature of the gates and errors, which may not perfectly capture the observed behavior. The observed non-Markovian effects, such as contextual Larmor frequency shifts, represent a limitation of the current analysis and suggest the need for more sophisticated models incorporating time-dependent error characteristics. The number of devices studied (three) may limit the generalizability of the findings. Future work could benefit from testing a larger sample size to better characterize device-to-device variability.