High-fidelity and robust controlled-Z gates for superconducting transmon qubits. II. System-parameter uncertainties and overall errors

In the race to build a machine that can solve problems beyond the reach of today's most powerful computers, scientists are turning to a technology that relies on the strange rules of quantum mechanics. At the heart of this effort are tiny circuits made of superconducting metal, cooled to temperatures colder than deep space. These circuits act as artificial atoms, known as qubits, which can hold information in a state of superposition, existing in multiple possibilities at once. To make these machines useful, the qubits must talk to one another, performing logical operations that link their states together. One of the most critical operations is a gate that flips the state of one qubit only if its partner is in a specific condition, a process known as a controlled-Z gate. For a quantum computer to work reliably, these gates must be incredibly fast and incredibly accurate. However, the physical world is messy. The materials used to build these circuits are never perfect, and the environment is never perfectly still. Tiny variations in how the qubits are made, or slight drifts in their frequencies over time, can cause the gate to fail. If a computer requires perfect knowledge of every single component to function, it will be impossible to scale up to the thousands of qubits needed for real-world applications. The challenge, therefore, is not just to build a fast gate, but to build one that is robust enough to keep working even when the hardware is slightly imperfect or the conditions change.

A team of researchers has developed a new strategy to solve this problem, creating a method to design these quantum gates so they can withstand the inevitable uncertainties of the physical world. In a companion study, they had already figured out how to make these gates fast and resistant to random noise from the environment. In this new work, they turned their attention to the other side of the problem: the uncertainty in the hardware itself. They asked what happens if the strength of the connection between two qubits is not exactly what the engineers thought it was, or if the natural frequencies of the qubits are slightly off due to manufacturing differences. They found that these errors fall into two distinct categories, each requiring a different kind of defense. One type of error comes from the strength of the link between the qubits, while the other comes from errors in the energy levels of the qubits themselves. By treating these two problems separately, the team designed control pulses—precise sequences of magnetic signals—that could neutralize both types of errors simultaneously.

The researchers discovered that the errors related to the connection strength could be suppressed by rapidly modulating the frequency of the qubits. Imagine shaking a heavy object back and forth very quickly to keep it from settling into a position where it might tip over; similarly, by oscillating the qubit frequencies, the system effectively averages out the errors caused by an imprecise connection strength. This technique, known as dynamical decoupling, allowed them to reduce the error caused by coupling uncertainties to a level of 6.21 × 10⁻⁷ for a gate that takes 30 nanoseconds to complete. This is a massive improvement over standard methods, which would fail under the same conditions. With this strategy, the gate can tolerate variations in the connection strength of up to 2π × 4 MHz while still maintaining a high level of accuracy. This means that engineers do not need to measure the connection between every pair of qubits with extreme precision, nor do they need to recalibrate the system constantly as the hardware drifts.

The second type of error, arising from uncertainties in the qubits' energy levels, required a different approach. Instead of shaking the system, the researchers found a specific "sweet spot" where the gate operates. They identified a point where the detuning, or the difference in frequency between the qubits, is balanced in a way that makes the gate insensitive to small shifts in energy. By operating near this symmetric point, the gate becomes naturally resistant to errors in the anharmonicity and zero-flux frequencies of the qubits. For a gate lasting 24 nanoseconds, this method allows the system to tolerate individual uncertainties of up to 2π × 5 MHz in these parameters. This finding is crucial because it shows that by choosing the right operating point, the gate can remain accurate even if the hardware is not perfectly characterized.

The team did not stop at looking at these errors in isolation. They combined their new robustness strategies with their previous work on noise and decoherence to simulate how these gates would perform across six different stages of hardware improvement. They imagined a future where the materials get better, the noise gets quieter, and the uncertainties shrink. In the earliest stages, where the hardware is noisier and less stable, the best strategy is to make the gate as fast as possible, even if it is slightly less robust, because the time spent operating exposes the system to too much environmental noise. However, as the hardware improves and the noise decreases, the balance shifts. The researchers found that in more advanced stages, it becomes better to take a little more time to run a longer, more robust gate. At the most advanced stage they considered, where the hardware is highly refined, their optimized gate achieved an intrinsic error of 1.93 × 10⁻⁸ and a total error of 4.55 × 10⁻⁶. At this level, the remaining errors are limited primarily by the natural energy loss of the qubits, a fundamental physical limit.

This work provides a practical roadmap for building large-scale quantum processors. It demonstrates that by engineering the control pulses to be intrinsically less sensitive to parameter variations, scientists can reduce the burden of calibration. Instead of needing to measure and adjust every single component with extreme precision, which becomes impossible as the number of qubits grows into the thousands, the system can be designed to work well across a range of conditions. The resulting gates are faster, more accurate, and far more forgiving of the imperfections that come with real-world manufacturing. By balancing speed, fidelity, and robustness, this framework offers a path forward for making quantum computers that are not just theoretical possibilities, but practical tools that can be maintained and operated reliably.