⚛️ quantum physics

Geodesic-based optimal control for leakage suppression in superconducting qubits

This paper presents a geodesic-based optimal control method for superconducting qubits that significantly improves gate fidelity and suppresses leakage errors for single-qubit operations, while achieving higher fidelities for two-qubit $i$ SWAP gates at the cost of slightly longer evolution times.

Original authors: Adonai Hilário da Silva, Tanmoy Pandit, Francesco Cosco

Published 2026-10-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Adonai Hilário da Silva, Tanmoy Pandit, Francesco Cosco

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Quantum computers promise to solve problems that are currently impossible for classical machines, but they are notoriously fragile. The basic building blocks of these machines, called qubits, are delicate systems that can easily be disturbed by their surroundings. In many of the most advanced designs, these qubits are built from tiny electrical circuits that behave like oscillating springs. To make them work as bits of information, scientists use only the two lowest energy levels of these circuits, treating them as a zero and a one. However, because these circuits are not perfectly isolated, energy can accidentally push the system up to a third, higher energy level that was never meant to be used. This unwanted jump is known as leakage. When a qubit leaks into this higher state, it stops behaving like a normal bit, and the information it holds is lost or corrupted. Fixing this problem is critical because if qubits leak too often, the complex error-correction codes needed to run a useful quantum computer will fail.

A team of researchers at the VTT Technical Research Centre of Finland has developed a new way to steer these qubits that significantly reduces this leakage. They focused on a method called sub-Riemannian geodesic search, which is essentially a mathematical technique for finding the most efficient path between two points on a curved surface. In the context of quantum control, the "surface" is the space of all possible states a quantum system can be in, and the "path" is the sequence of control signals used to move the system from its starting point to a desired operation. The researchers applied this method to a realistic model of two superconducting qubits connected by a tunable coupler, a device that allows the qubits to interact when needed. By carefully shaping the electrical pulses that drive the qubits, they found a way to perform logic operations with much higher accuracy and far less leakage than previous methods, particularly when the operations need to be completed very quickly.

The researchers tested their approach on two specific types of operations. The first was a simple rotation of a single qubit, similar to flipping a coin, and the second was a two-qubit swap operation, where the states of two qubits are exchanged. In the single-qubit tests, they compared their new pulses against standard techniques currently used in the field, such as Gaussian pulses and a method known as DRAG, which is designed to correct for leakage. The results were striking. For very short pulses, lasting around 17 nanoseconds, their new method produced gate fidelities that were nearly perfect, reaching a level of 0.9996. More importantly, it reduced the rate of leakage errors to a level that is essentially limited only by the unavoidable thermal noise of the environment. This means that for these fast operations, the new method is as good as it can possibly get, outperforming the standard DRAG pulses which struggled to maintain accuracy at such short durations.

When the team moved to the more complex two-qubit swap operation, the results were equally promising, though with a slight trade-off. Without any external control, the natural interaction between the qubits could perform the swap in about 79.5 nanoseconds, but with a fidelity of only about 0.95. By applying their optimized pulses, the researchers were able to boost the fidelity to 0.9942. This improvement came at the cost of extending the operation time by just 6 nanoseconds, bringing the total duration to 86 nanoseconds. While the leakage rate increased slightly compared to the uncontrolled version, the gain in overall accuracy was substantial. The key insight here is that the control pulses act like real-time corrections, fine-tuning the frequencies of the qubits as the swap happens to ensure the exchange is clean and precise.

A crucial part of the researchers' success was how they shaped the control signals. In many quantum control methods, the electrical pulses turn on and off very abruptly. This sudden change creates a broad spread of frequencies that can accidentally excite the qubits into those unwanted higher energy states. The new method forces the pulses to rise and fall smoothly, starting and ending at zero amplitude. This smoothness prevents the broad frequency spread, keeping the qubits focused on the intended transition. The researchers achieved this by modifying their mathematical search to include a specific constraint that ensures the pulses are gentle at the beginning and end. They found that this smoothness was essential for minimizing errors, especially when the operations had to be completed in a fraction of a microsecond.

The study was conducted through detailed numerical simulations based on the physical parameters of real superconducting quantum computers, specifically those developed by IQM. The team modeled the qubits as having three energy levels to account for the possibility of leakage, and they included the effects of thermal dissipation, which is the natural loss of energy to the environment. While the results are currently limited to these simulations, they provide a strong theoretical foundation for future experiments. The researchers noted that the main challenge for practical implementation would be the ability of current electronic equipment to generate these precise, smooth pulses with the required accuracy. However, the findings suggest that if hardware can keep up, this geodesic-based approach could become a standard tool for building more reliable quantum computers, offering a way to perform fast, high-fidelity operations without the penalty of information loss to leakage.

Technical Summary: Geodesic-based Optimal Control for Leakage Suppression in Superconducting Qubits

Problem Statement
Superconducting qubits, typically modeled as the first two energy levels of an anharmonic LC oscillator, are susceptible to leakage errors. Because the anharmonicity is often small relative to energy gaps, population can inadvertently leak into higher energy states (e.g., $|2\rangle$ ). This leakage is detrimental to quantum error correction (QEC) as it cannot be described by Pauli operators. While time-optimal control protocols can suppress general decoherence by executing gates rapidly, they often induce significant leakage due to the broad frequency spectrum caused by abrupt turn-on and turn-off of control fields. Conversely, standard control methods like Derivative Removal by Adiabatic Gate (DRAG) pulses attempt to mitigate leakage but may not simultaneously optimize for gate fidelity and speed, particularly for short-duration pulses.

Methodology
The authors adapt a sub-Riemannian geodesic search method within the unitary group to generate optimal control pulses for a realistic two-qubit superconducting system. The system consists of two qubits coupled directly and indirectly via a tunable coupler.

Physical Model: The system is modeled as a pair of Duffing oscillators coupled to a tunable coupler (also a Duffing oscillator). The authors derive an effective Hamiltonian by:
- Truncating the Hilbert space to include the computational subspace ( $|0\rangle, |1\rangle$ ) and the first leakage level ( $|2\rangle$ ) for each qubit, resulting in an $SU(9)$ unitary group (two 3-level systems).
- Applying a Schrieffer-Wolff transformation to treat the coupler in the dispersive regime, effectively decoupling it from the qubits during single-qubit operations or enabling specific exchange interactions for two-qubit gates.
- Utilizing a Rotating Wave Approximation (RWA) for resonant qubits.
Geodesic Control Formalism:
- The control problem is framed as finding a sub-Riemannian geodesic connecting the identity to a target unitary operation. The control Hamiltonian $H_c(t)$ is restricted to a controllable subspace $V$ spanned by specific drive operators.
- The authors minimize an energy functional $E(H) = \frac{1}{2} \int \langle H_c(t), H_c(t) \rangle dt$ subject to the Schrödinger equation constraint.
- Key Modification: To address experimental feasibility and prevent spectral broadening, the authors introduce a smooth envelope function $S(t)$ (based on sigmoid functions) to the geodesic equation. This ensures the control pulses rise and fall smoothly to zero at the beginning and end of the pulse, avoiding the sharp transitions that cause leakage.
- The optimization is solved using a shooting method combined with gradient ascent to find the initial costate $\Lambda(0)$ that yields the target unitary.
Metrics: Performance is evaluated using average gate fidelity ( $F$ ), leakage rate ( $L_1$ , population leaving the computational subspace), and seepage rate ( $L_2$ , population returning from the leakage subspace). These are averaged over the Haar measure of initial states.

Key Contributions and Results

Single-Qubit Operations ( $R_x(\pi/2)$ ):
- The authors benchmarked the geodesic pulses against Gaussian and DRAG pulses (optimized for fidelity, DRAG-F, and leakage, DRAG-L).
- Result: For short pulse durations (specifically $\tau = 17$ ns), the geodesic optimal pulses significantly outperformed both Gaussian and DRAG pulses. The method achieved an average gate fidelity of $F = 0.9996$ , a leakage rate of $L_1 = 1.08 \times 10^{-5}$ , and a seepage rate of $L_2 = 2.12 \times 10^{-3}$ .
- Crucially, these values saturated the limits imposed by simulated thermal dissipation, meaning the control pulses themselves introduced negligible error compared to the physical environment.
- Unlike DRAG pulses, where the quadrature component $\Omega_y(t)$ is proportional to the first derivative of $\Omega_x(t)$ , the geodesic pulses exhibited a more complex relationship, suggesting a potential connection to higher-order derivative corrections.
Two-Qubit Operations (iSWAP):
- The iSWAP gate was analyzed in a regime where the coupler frequency allows for exchange interactions.
- Result: A control-free iSWAP operation achieved a fidelity of $\approx 0.95$ in 79.5 ns. By applying the geodesic optimal pulses, the fidelity was increased to $F = 0.9942$ at the cost of a slightly longer evolution time (86 ns).
- The leakage rate increased negligibly ( $4.55 \times 10^{-4}$ to $6.69 \times 10^{-4}$ ) compared to the substantial gain in fidelity. The control pulses acted as real-time frequency corrections to the qubits to enhance the swap interaction.

Significance and Claims
The paper claims that the sub-Riemannian geodesic search method, when adapted with smooth envelope constraints, provides a robust framework for quantum control that simultaneously maximizes gate fidelity and minimizes leakage errors.

Unified Optimization: The method inherently combines the benefits of DRAG-F and DRAG-L, optimizing for both process fidelity and leakage suppression without requiring separate tuning parameters.
Experimental Relevance: By enforcing smooth turn-on and turn-off transitions, the method addresses a primary source of leakage (spectral broadening) often overlooked in theoretical time-optimal controls.
Scalability: The approach successfully handles the complexity of a two-qubit system with leakage levels, demonstrating that energy-optimal paths in the unitary group can yield practical, high-performance control pulses for superconducting architectures.

The authors conclude that while the work is grounded in rigorous theoretical and numerical simulation, the most natural next step is the practical experimental implementation of these pulses on physical superconducting hardware. They also note an open question regarding the mathematical connection between the derived geodesic pulses and generalized DRAG pulses involving higher-order derivatives.

Technical Summary: Geodesic-based Optimal Control for Leakage Suppression in Superconducting Qubits

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