A Regulator for Quantum Gates: How Robotics Controls Qubits ✛
Context
Optimal control of quantum systems is about finding pulses of external fields that drive the system to a desired state in minimal time and with maximal accuracy. For superconducting qubits (transmons), analytical methods already exist, such as DRAG for single-qubit operations and cross-resonance for two-qubit gates, but they require fine-tuning to specific hardware. Numerical methods like GRAPE and Krotov's method offer more flexibility but often need a good initial guess. This work stands alongside other attempts to apply classical control theory to quantum problems: PRONTO uses a projected Newton method, PICO uses direct collocation, and this paper brings iLQR, known in robotics, to gate synthesis.
Methods
The authors represented the unitary matrix in an isomorphic real form (vectorized it), and approximated the matrix exponential with an eighth-order Padé approximant. The optimal control problem was formulated as an iLQR problem: minimizing a quadratic cost function subject to dynamics constraints. To obtain smooth pulses, they used the derivatives of the envelopes as control variables and added the amplitudes themselves to the state. This allowed penalizing the rate of change and the final values of the pulses. Hyperparameters (cost matrices) were tuned manually and via grid search. Simulations were performed for single- and two-qubit transmons with parameters from the IBM Brisbane processor, with a time step of 0.5 ns and durations of 40 ns (X gate) and 240 ns (CR gate).
Results
For the single-qubit X gate on a two-level model, iLQR without smoothing reproduced the analytical bang-bang solution with an accuracy of 1.3·$10^{-13}$, and with smoothing, 4·$10^{-9}$. For the three-level model, the accuracy was 2.1·$10^{-7}$, with leakage to the |2⟩ state being negligible. For the two-qubit cross-resonance gate on the two-level model, the accuracy reached 1.1·$10^{-8}$, and on the three-level model, 5.9·$10^{-5}$. Comparison with L-BFGS-B GRAPE showed that GRAPE can achieve higher accuracies (up to 1.9·$10^{-13}$ for the single-qubit case), but its result strongly depends on the initial pulse, whereas iLQR finds a solution from random initialization.
Implications
This work shows that iLQR, a method from robotics, is applicable to quantum optimal control and can find high-accuracy solutions without an initial guess. Unlike related works that use other optimization methods (projected Newton, collocation), the key idea here is the use of the cost-to-go function, which allows accounting for future costs when updating controls. This opens the way to more versatile and automated tools for gate synthesis.
Future development
In the future, iLQR could be adapted for shorter pulses by using more complex cost functions with time-dependent weights. It is also possible to include fidelity directly in the cost function. The method could be applied to other architectures, such as transmons with tunable couplers for iSWAP or CZ gates. An important direction is integrating hardware feedback to compensate for model inaccuracies.
Impact
The method could speed up the calibration of quantum processors and improve gate quality, especially for systems where the analytical form of the pulses is unknown. It could also be useful in quantum metrology and sensing, where precise control is required.
Next steps
Next steps include optimizing shorter pulses under realistic AWG constraints, incorporating fidelity into the cost function, and testing on real hardware with feedback.
Key open problems
In quantum optimal control, the problem of scalability remains open: methods that work well for one or two qubits may struggle as the Hilbert space dimension grows. This work does not solve that problem directly, but it offers a computationally efficient method that could potentially be extended to more qubits by leveraging Hessian approximations and parallel computation of derivatives.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
The study adapts the iLQR method from robotics to optimize control pulses in superconducting transmon qubits. The authors introduced constraints on signal derivatives to ensure pulse smoothness and achieved fidelities up to 4·10^-9 for single-qubit X gates and 1.1·10^-8 for two-qubit cross-resonance gates in simulations. The method requires no initial guess and can handle different architectures.
Where its strength is
Using the cost-to-go function in iLQR allows future costs to be accounted for when updating controls, making the method robust to random initialization and potentially generalizable to various quantum systems.
Where it could go next
- It might be interesting to see how iLQR handles real hardware constraints, such as limited bandwidth of control electronics. There is a related paper on cryoelectronics that addresses qubit control at low temperatures.based on: Cryoelectronics: Controlling Qubits at 4 Kelvin arXiv:2602.14165
- One could consider adapting the method to optimize more complex multilevel systems or systems with multiple qubits, where minimizing crosstalk is crucial. A preprint on four qubits on a single bridge could provide context for scaling.based on: Four Qubits on a Single Bridge arXiv:2602.06812
- It would be interesting to compare iLQR with reinforcement learning approaches for tuning quantum gates, for example, with a work where a quantum computer learns to play without cheating. This could reveal the strengths and limitations of each method.based on: Quantum Computer Learns to Play Without a False Note arXiv:2607.01422
Worth trying
- Test iLQR on problems with tighter constraints on pulse amplitude and duration, for instance, for short gates.
- Use iLQR to initialize gradient-based methods like GRAPE, combining iLQR's global convergence with GRAPE's high precision.
- Apply iLQR to optimize multiple gates simultaneously in a sequence to account for memory effects and crosstalk.
Why this may matter
Bringing methods from robotics into quantum control could help create more automated and versatile tools for gate synthesis, which is important for scaling up quantum processors.
Nearby works in our archive
- Cryoelectronics: Controlling Qubits at 4 Kelvinsimilarity 0.54
- A Quantum Remote That Barely Generates Heatsimilarity 0.53
- Smart Pushes for Quantum Swingssimilarity 0.53
- Robot Tuner for Quantum Processorssimilarity 0.53
- Four Qubits on a Single Bridgesimilarity 0.52
- Universal Light Control in a Quantum Cavitysimilarity 0.52
- Quantum Computer Learns to Play Without a False Notesimilarity 0.51
The archive is sparse around this work: the closest review is only 0.54, against a usual 0.64. It looks like few have drilled nearby.
The iLQR method was originally developed for controlling robot motions, such as humanoid walking. Now it controls quantum states, from robot steps to the evolution of the wave function.
Key numbers
- accuracy of single-qubit X gate (2 levels): 4·10-9
- accuracy of two-qubit CR gate (2 levels): 1.1·10-8
- X gate duration: 40 ns
- CR gate duration: 240 ns
- time step: 0.5 ns