⚛️ quantum physics

Iterative Linear Quadratic Regulator for Quantum Optimal Control

This paper adapts the Iterative Linear Quadratic Regulator (ILQR) to quantum gate optimization by incorporating constraints on controls and their derivatives to generate smoother, high-fidelity pulse sequences for X and cross-resonance gates on transmon qubits.

Original authors: Dirk Heimann, Felix Wiebe, Tahereh Abad, Elie Mounzer, Tangyou Huang, Frank Kirchner, Shivesh Kumar

Published 2026-08-10

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CC BY 4.0

Original authors: Dirk Heimann, Felix Wiebe, Tahereh Abad, Elie Mounzer, Tangyou Huang, Frank Kirchner, Shivesh Kumar

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

Imagine you are trying to teach a tiny, invisible dancer to perform a perfect pirouette. This dancer isn't made of flesh and bone, but of pure energy and probability, living in a world where the rules of physics are as slippery as a bar of soap in a bathtub. This is the world of quantum computing. In this realm, "dancers" are called qubits, and the "pirouettes" are logic gates—the fundamental instructions that make a computer think. But here's the catch: these dancers are incredibly sensitive. If you push them too hard, too fast, or with the wrong rhythm, they stumble, lose their balance, and the whole performance turns into a chaotic mess. This is why scientists need "Quantum Optimal Control." It's the art of designing the perfect sequence of invisible pushes (called pulses) to guide these qubits exactly where they need to go, without tripping them up. The goal is to make these moves so precise and reliable that we can build computers capable of solving problems that would take today's supercomputers thousands of years to crack.

Now, enter the team of researchers from Germany and Sweden who decided to try a new dance instructor for these quantum dancers. They took a method called the "Iterative Linear Quadratic Regulator" (or iLQR), which is already a superstar in the world of robotics for teaching complex machines how to walk or fly, and taught it how to choreograph quantum gates. Instead of just guessing and checking, their new approach treats the problem like a game of "hot and cold" played in reverse. It starts at the finish line, figures out what the perfect move should have been to land there, and then works backward step-by-step to adjust the controls. But they didn't just copy-paste the robot method; they tweaked it to make the "dance moves" smoother. In the real world, the machines that generate these pulses (called Arbitrary Waveform Generators) can't handle sudden, jerky changes in speed. So, the researchers added a rule that penalizes abrupt jumps, forcing the algorithm to find a path that flows like a river rather than crashing like a waterfall.

What they found in their simulations is quite promising. By using this new, smoothed-out version of iLQR, they were able to design pulse sequences for one-qubit and two-qubit gates that are incredibly accurate. For a single qubit performing an "X-gate" (a basic flip), they achieved a level of precision where the error was so small it was almost invisible—only about 4 errors in a billion attempts. For a more complex two-qubit "cross-resonance" gate, which entangles two dancers together, they also found high-fidelity solutions, though the error rate was slightly higher at about 6 errors in 100,000 attempts. The best part? They didn't need to know the "right" answer beforehand. Unlike other methods that often get stuck if you don't give them a good starting guess, this new method seems to find the perfect path even when it starts with random, messy movements.

However, there is a "but" to this story. These results are currently just simulations—digital rehearsals on a computer, not a performance on a real quantum machine. The researchers admit that the pulses they found are still a bit long, taking up to 240 nanoseconds to complete, which is like a slow-motion dance compared to what might be possible. They also noted that while their method produces very smooth, gentle pulses that are friendly to the hardware, other existing methods (like L-BFGS-B GRAPE) can sometimes find even more precise solutions if you give them a very specific, well-chosen starting point. But if you don't know the perfect starting point, this new iLQR method is like a reliable coach that can figure out the routine from scratch.

The team's work suggests that this "robotics-style" approach could be a powerful new tool for the future of quantum computing. They showed that by treating the control problem as a trajectory optimization task and smoothing out the edges, they can get very close to the perfect gate without needing a crystal ball to see the solution first. While they haven't solved every problem yet—real-world hardware is messy and full of surprises that simulations can't always predict—they have opened a new door. Their next step is to take this digital choreography and test it on actual quantum chips, hoping to see if the dancers can perform just as beautifully in the real world as they did in the simulation.

Technical Summary: Iterative Linear Quadratic Regulator for Quantum Optimal Control

Problem Statement
Quantum optimal control (QOC) seeks to design precise external field pulse sequences to achieve high-fidelity gate operations on quantum systems, specifically aiming for fidelities below the error correction threshold. While established methods like GRAPE (Gradient Ascent Pulse Engineering) and Krotov's method are widely used, they often rely on specific initialization strategies or have limitations regarding convergence guarantees and handling of complex constraints. The authors identify a gap in applying the Iterative Linear Quadratic Regulator (iLQR)—a state-of-the-art trajectory optimization method in robotics—to the domain of quantum gate synthesis. The challenge lies in adapting iLQR to the non-linear dynamics of quantum systems (governed by the Schrödinger equation) while incorporating hardware-specific constraints, such as smooth pulse envelopes and zero-boundary conditions required by Arbitrary Waveform Generators (AWGs).

Methodology
The paper proposes an adaptation of the iLQR algorithm for quantum gate optimization, specifically targeting fixed-frequency transmon qubits. The methodology involves several key technical components:

Isomorphic Representation: To handle the complex-valued unitary matrices of quantum evolution, the authors vectorize the state space. Complex matrices $U \in \mathbb{C}^{d \times d}$ are mapped to real-valued vectors and matrices of dimension $2d^2$ , allowing the use of standard real-valued optimization techniques.
Padé Approximation: Instead of computing exact matrix exponentials (which can be computationally expensive for higher-level systems), the authors utilize a Padé approximation (specifically an eighth-order power series) to approximate the matrix exponential $\exp(-iH(u_k)\Delta t)$ required for forward propagation.
iLQR Formulation: The QOC problem is cast as a trajectory optimization problem minimizing a cost functional $J$ composed of a running cost and a final cost. The algorithm employs a backward pass to compute value function derivatives and feedback gains ( $\kappa_k, K_k$ ) using a Gauss-Newton approximation (ignoring second-order derivatives of the dynamics), followed by a forward pass to update the control trajectory. Levenberg-Marquardt regularization is used to ensure the invertibility of the Hessian approximation.
Smooth Control Constraints: To address hardware limitations where abrupt amplitude changes are undesirable, the authors modify the control variables. Instead of optimizing the pulse amplitudes $u_k$ directly, they optimize the derivatives of the pulse envelopes ( $\dot{u}_k$ ). The actual controls are updated via $u_{k+1} = u_k + \dot{u}_k \Delta t$ . This formulation allows the cost function to penalize the rate of change, pulse amplitudes, and final values, ensuring pulses start and end at zero.
System Models: The method is tested on:
- One-qubit systems: Modeled as Duffing oscillators with two and three energy levels.
- Two-qubit systems: Modeled as coupled transmons in the dispersive regime using an effective Hamiltonian derived via Schrieffer-Wolf transformation. Both two-level and three-level models are simulated.

Key Contributions

Algorithm Adaptation: The successful adaptation of iLQR to quantum gate optimization, establishing a connection between robotics trajectory optimization and QOC.
Smooth Pulse Synthesis: The introduction of a derivative-based control formulation that inherently enforces smooth pulse shapes and zero-boundary conditions without requiring complex inequality constraints.
Comprehensive Simulation: High-fidelity simulation results for $X$ gates (single-qubit) and Cross-Resonance (CR) gates (two-qubit) on fixed-frequency transmons, considering both two-level and three-level (leakage) models.
Comparative Analysis: A direct comparison with L-BFGS-B GRAPE, highlighting differences in convergence behavior and sensitivity to initial pulse shapes.

Results
The authors simulated gate synthesis using parameters from the IBM Eagle processor (ibm_brisbane):

One-Qubit (2-Level): For the $X$ -gate, the iLQR method with derivative controls achieved an infidelity of $4.0 \times 10^{-9}$ with smooth, parabola-like pulse shapes. This was compared to a L-BFGS-B GRAPE solution with an infidelity of $1.9 \times 10^{-13}$ using a Gaussian initial guess.
One-Qubit (3-Level): Including leakage to the $|2\rangle$ state, iLQR achieved an infidelity of $2.1 \times 10^{-7}$ . The resulting pulses naturally exhibited characteristics similar to the DRAG (Derivative Removal by Adiabatic Gate) procedure, where the off-phase quadrature is proportional to the derivative of the in-phase quadrature.
Two-Qubit (2-Level): For the Cross-Resonance gate (240 ns duration), iLQR achieved an infidelity of $1.1 \times 10^{-8}$ .
Two-Qubit (3-Level): With leakage considered, the infidelity was $5.9 \times 10^{-5}$ .
Hyperparameter Sensitivity: The study found that the design of the quadratic cost matrices ( $Q_f, R_d, R_c, R_f$ ) had a more significant impact on the final fidelity and smoothness than the initial pulse shapes.

Significance and Claims
The paper claims that iLQR offers a robust alternative to existing QOC methods, particularly for scenarios where no analytical solution is known a priori.

Independence from Initialization: Unlike L-BFGS-B GRAPE, which tends to produce solutions that remain close to the initial pulse shape (e.g., Gaussian), the iLQR approach finds smooth, high-fidelity solutions regardless of the random initialization, provided the cost function is well-tuned.
Hardware Compatibility: By optimizing derivatives, the method naturally produces pulses that respect AWG constraints (smoothness, zero start/end), which is critical for experimental implementation.
Future Directions: The authors modestly note that while the current gates are high-fidelity, they are relatively long (e.g., 240 ns). Future work aims to optimize for shorter pulses using time-dependent cost functions and to integrate hardware feedback to mitigate model inaccuracies inherent in superconducting systems. The method is presented as applicable to various superconducting architectures, including those with tunable transmons.

Technical Summary: Iterative Linear Quadratic Regulator for Quantum Optimal Control

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