Transmon circuits, the foundation of many quantum processors, are increasingly susceptible to errors from unwanted transitions to higher energy levels as control drives become more precise. Researchers have demonstrated that after suppressing single-photon leakage, transitions between the |0⟩ and |2⟩ states and between |1⟩ and |3⟩ become the dominant sources of qubit error, an issue previously unaddressed by existing error mitigation strategies.

To counter this, the team constructed recursive DRAG-type corrections that simultaneously suppress both single- and two-photon leakage, achieving gate fidelity of 99.999% for an 11.8 nanosecond pulse. This work extends beyond analytical construction to provide a systematic understanding of calibration parameters, reducing gate time to 6.8 nanoseconds while maintaining fidelity.

Transmon Qubit Weak Anharmonicity Limits Gate Fidelity

Analytical construction of quantum gate pulses must account for transitions beyond the primary computational states to achieve high fidelity. Transmon circuits, favored for their scalability, are inherently weakly anharmonic oscillators, and previously unnoticed two-photon transitions now dominate error rates after single-photon leakage is suppressed. The research details how conventional error mitigation strategies falter as quantum processors become more sophisticated and gate times shrink.

The standard derivative removal by adiabatic gate, or DRAG, protocol relies on a three-level description of the qubit, but this simplification becomes insufficient when multiple leakage channels activate simultaneously. This breakdown occurs because decreasing gate times and increasing drive amplitude bring the system closer to the point where perturbative models are no longer valid; the team demonstrated that a minimum four-level description is required to accurately capture the relevant dynamics.

The paper states how higher energy levels become increasingly important as gate fidelity improves. This approach utilizes a sequence of frame transformations to create fully analytical pulse families that eliminate leading error terms even in strong-driving regimes, where the pulse amplitude is comparable to the detuning. An 8 nanosecond pulse was used. Beyond the analytical construction, the work provides a systematic understanding of calibration parameters often introduced empirically to improve performance.

Using a combination of toggling-frame transformations and a Magnus expansion, the researchers explained observed correlations between these parameters, clarifying the relationship between the DRAG prefactor and detuning. They found that a constant detuning can outperform time-dependent detuning in relevant regimes.

By incorporating these additional prefactors into R2D pulses, the team reduced the gate time from 11.8 to 6.8 nanoseconds, achieving gate fidelity of 99.999% at 6.75 nanoseconds and beyond. Simulations revealed that the primary source of error in standard DRAG protocols could be addressed by employing the recursive R1D correction, which suppresses two-photon transitions to the |2⟩ state. However, leakage to the |3⟩ state remained a significant error source with R1D, a problem further mitigated by more than an order of magnitude using the full R2D correction.

The simulations modeled a transmon with an anharmonicity of −2π × 0.225 GHz and Δ3 = 3Δ2, aiming to implement an on-resonant interaction between the lowest two levels while preventing population transfer to higher levels. The parameterized R2D pulse consistently outperformed all others across all gate times considered, achieving 99.999% fidelity at 6.75 nanoseconds and beyond.

Standard DRAG Protocol Mitigates Single-Photon Leakage

Superconducting quantum circuits increasingly rely on precise control to manage qubit behavior, but the nature of transmon circuits, weakly anharmonic oscillators, introduces challenges as technology advances. Control drives can unintentionally induce transitions to higher energy levels, causing population leakage and phase errors that degrade gate fidelity. However, recent work demonstrates that conventional DRAG methods become insufficient as gate times shorten and drive amplitudes approach the qubit’s anharmonicity.

The limitations of standard DRAG arise from its reliance on a simplified, three-level description of the transmon circuit. Researchers developed a framework that explicitly accounts for these multi-photon processes and the population of higher energy levels, moving beyond the perturbative regime where simpler models break down. To counter these increasingly complex dynamics, the team constructed recursive DRAG-type corrections.

Further refinement through optimized parameters reduced the gate time to 6.8 nanoseconds while achieving gate fidelity of 99.999% at 6.75 nanoseconds and beyond. A systematic understanding of calibration parameters commonly used with DRAG protocols was also achieved. While prefactors are often added empirically to improve performance, their underlying roles and interdependencies have remained unclear. The work provides a pathway toward more robust and reliable quantum computations by addressing error sources previously overlooked in standard DRAG implementations.

Multi-Photon Transitions Dominate Errors Beyond Perturbative Regime

Researchers are refining methods to mitigate errors in transmon qubits by explicitly modeling multi-photon transitions, a previously overlooked source of decoherence. This finding challenges the effectiveness of conventional error mitigation strategies that rely on simplified, perturbative models. The inadequacy of the three-level description arises because transmon circuits, as weakly anharmonic oscillators, are susceptible to inducing unwanted transitions to higher energy levels when driven.

To address this, the researchers constructed a recursive DRAG-type correction, termed R2D, which simultaneously suppresses both single- and two-photon leakage through a sequence of frame transformations. The team’s work also clarifies the often-empirically determined relationships between calibration parameters within DRAG protocols.

Recursive DRAG (R1D) Suppresses |0⟩ ⇄ |2⟩ Two-Photon Errors

Simulations reveal that the parameterized R2D pulse, building upon R1D, outperforms all other tested methods, achieving a fidelity of 99.999% at 6.75 nanoseconds and beyond. Detailed analysis of gate performance using the R2D pulse revealed specific error patterns.

The simulations, performed with a transmon circuit exhibiting an anharmonicity of Δ2 = −2π × 0.225 GHz and Δ3 = 3Δ2, showed that while R1D effectively suppresses two-photon transitions to |2⟩, leakage to the |3⟩ state becomes a major source of error. This refined control over qubit behavior represents a step toward building larger, more stable quantum processors capable of tackling complex computational problems.

R2D Frame Transformations Correct Higher Excited State Errors

Conventional approaches to mitigating qubit errors often focus on suppressing single-photon leakage, yet a recent analysis reveals this strategy inadvertently amplifies errors stemming from transitions to higher energy levels. This finding challenges the efficacy of established error correction methods reliant on simplified, perturbative models. Researchers discovered that the resulting complex dynamics are not adequately captured by conventional DRAG leakage modeling, necessitating a more sophisticated approach. This analytical approach requires a minimum four-level description to accurately model the qubit’s behavior.

Analytical Pulse Families Eliminate Leading Errors in Strong Driving

Analytical calculations of pulse sequences now account for previously overlooked quantum errors, significantly improving qubit stability. Simulations reveal that two-photon transitions, specifically between the |0⟩ and |2⟩, and |1⟩ and |3⟩ energy levels, emerge as the primary source of error once single-photon leakage is minimized, a challenge not adequately addressed by existing methods. Conventional derivative removal by adiabatic gate (DRAG) protocols, while widely used to suppress off-resonant transitions, falter as gate times shorten and drive amplitude nears the qubit’s anharmonicity.

The team’s approach moves beyond simply suppressing single-photon leakage, a common focus of earlier work, to address the more subtle but increasingly dominant two-photon errors. An 8 nanosecond pulse demonstrated a substantial improvement in qubit fidelity, and further optimization, including the addition of prefactors to the R2D pulses, reduced the gate time to 6.8 nanoseconds, achieving gate fidelity errors of 10−5.

👉 More information
🗞 Analytical Blueprint for 99.999% Fidelity X-Gates on Present Superconducting Hardware Under Strong Driving
✍️ José Diogo Da Costa Jesus, Boxi Li, Yuan Gao, Rami Barends, Francisco Andrés Cárdenas-López and Felix Motzoi
🧠 DOI: http://link.aps.org/doi/10.1103/sdxb-v39h

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