Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.03656

Iterative linear quadratic regulator on SU(N) for multi-qubit gate synthesis

Dirk Heimann, Felix Wiebe, Elie Mounzer, Shivesh Kumar

quant-phSubmitted Aug 4, 2026• Announced Aug 5, 2026• 0 citations
arXivHTMLPDF

Abstract

In quantum optimal control theory, gradient-based trajectory optimization techniques have proven versatile in designing multi-qubit quantum gates. Furthermore, incorporating the underlying Lie-group structure can accelerate the optimization process. In this work, we adapt the Lie-group formulation of the iterative linear quadratic regulator (iLQR) to the special unitary group SU(N) and apply it to quantum gate synthesis, systematically comparing it against the standard Euclidean iLQR formulation across multiple two- to five-qubit gates. We find that in the idealized, unconstrained setting, where all Lie-algebra basis elements are available as drive Hamiltonian terms, the Lie-group formulation converges faster than the Euclidean iLQR formulation. If drive terms are constrained to 2-local Hamiltonian terms, the Lie-group variant converges faster in early optimization iterations, but exhibits greater sensitivity to initialization and a stronger tendency towards local minima. These results demonstrate that incorporating Lie-group geometry into iLQR substantially improves convergence and highlight important next steps for improvements in constrained control settings.

Comments
21 pages, 11 figures

Related papers

Iterative Linear Quadratic Regulator for Quantum Optimal Control

Dirk Heimann, Felix Wiebe, Tahereh Abad +4 more

Quantum optimal control for gate optimization aims to provide accurate, robust, and fast pulse sequences to achieve gate fidelities on quantum systems below the error correction threshold. Many methods have been developed and successfully applied in simulation and on quantum hardware. In this paper, we establish a connection between the iterative linear quadratic regulator and quantum optimal control by adapting it to gate optimization of quantum systems. We include constraints on the controls and their derivatives to enable smoother pulses. We achieve high-fidelity simulation results for$\mathbf{X}$and cross-resonance gates on one- and two-qubit fixed-frequency transmons simulated with two and three levels.

arXiv:2504.10938•IEEE International Conference on Quantum Computing and Engineering (QCE)•Aug 30, 2025•0 citations

Quantum Circuit Optimization through Iteratively Pre-Conditioned Gradient Descent

Dhruv Srinivasan, Kushal Chakrabarti, Nikhil Chopra +1 more

Gate-based quantum algorithms are typically implemented by circuits consisting of many single-qubit and multi-qubit gates operating on a quantum input state, followed by measurements of the output state. For certain quantum subroutines, such as initial state preparation and quantum Fourier transforms, explicit decompositions of the circuit in terms of single-qubit and two-qubit maximally entangling gates may exist. However, they often lead to large-depth circuits that are challenging for noisy intermediate-scale quantum (NISQ) hardware. Additionally, exact decompositions might only exist for some modular quantum circuits. Therefore, it is essential to find gate combinations that approximate these circuits to high fidelity with potentially low depth. Gradient-based optimization has been used to find such approximate decompositions. Still, these traditional optimizers often run into problems of slow convergence requiring many iterations, and performing poorly in the presence of noise, a factor that is especially relevant for NISQ hardware. Here we present iteratively preconditioned gradient descent (IPG) for optimizing quantum circuits and demonstrate performance speedups for state preparation and implementation of quantum algorithmic subroutines. IPG is a noise-resilient, higher-order algorithm that has shown promising gains in convergence speed for classical optimizations, converging locally at a linear rate for convex problems and superlinearly when the solution is unique. Specifically, we show an improvement in fidelity by a factor of 104for preparing a 4-qubit W state and a maximally entangled 5-qubit GHZ state with compared to other commonly used classical optimizers tuning the same ansatz. Such faster convergence with promise for noise-resilience could provide advantages for quantum algorithms on NISQ hardware, especially since the cost of running each iteration on a quantum computer is substantially higher than the classical optimizer step.

arXiv:2309.09957•IEEE International Conference on Quantum Computing and Engineering (QCE)•Sep 17, 2023•2 citations

Lie Group Diffusion Models for Hardware-Aware Quantum Circuit Synthesis

Jyotirmai Singh

An important task in quantum computing is unitary circuit synthesis compatible with physical hardware constraints. This problem has a natural hybrid structure as local single-qubit gates are continuous variables on the Lie group $SU(2)$ while the entangling circuit structure is discrete and hardware-dependent. In this work, we use generative models to perform quantum circuit synthesis incorporating both the natural $SU(2)$ manifold geometry of quantum gates and hardware constraints that determine the overall circuit structure. Our model comprises two components: a circuit skeleton selector that chooses an entangling circuit and a diffusion model that generates quantum gates on the given circuit template by performing diffusion on the curved manifold $\mathrm{SU(2)} \simeq S^3$ itself. We demonstrate this approach with unitary compilation of physically motivated three-qubit Hamiltonian simulation targets such as the Transverse Field Ising Model and the Heisenberg-XXZ Model and show that Lie group diffusion outperforms comparable baselines. The synthesised circuits can also be customised subject to constraints, which we demonstrate by producing circuits with large and small gate rotation angles for the same target unitary evolution. We also investigate the fidelity-complexity frontier of the synthesised gates to demonstrate that the circuit selector learns to select circuits that balance fidelity with complexity rather than collapsing onto the most expansive entangling template. These results demonstrate that Lie group diffusion provides a natural generative framework for hardware-aware quantum circuit synthesis.

quant-pharXiv:2606.29636•Jun 30, 2026•0 citations

GRADIENT FLOWS FOR OPTIMIZATION IN QUANTUM INFORMATION AND QUANTUM DYNAMICS: FOUNDATIONS AND APPLICATIONS

THOMAS SCHULTE-HERBRÜGGEN, Steffen J. Glaser, Gunther Dirr +1 more

Many challenges in quantum information and quantum control root in constrained optimization problems on finite-dimensional quantum systems. The constraints often arise from two facts: (i) quantum dynamic state spaces are naturally smooth manifolds (orbits of the respective initial states) rather than being Hilbert spaces; (ii) the dynamics of the respective quantum system may be restricted to a proper subset of the entire state space. Mathematically, either case can be treated by constrained optimization over the reachable set of an underlying control system. Thus, whenever the reachable set takes the form a smooth manifold, Riemannian optimization methods apply. Here, we give a comprehensive account on the foundations of gradient flows on Riemannian manifolds including new applications in quantum information and quantum dynamics. Yet, we do not pursue the problem of designing explicit controls for the underlying control systems. The framework is sufficiently general for setting up gradient flows on (sub)manifolds, Lie (sub)groups, and (reductive) homogeneous spaces. Relevant convergence conditions are discussed, in particular for gradient flows on compact and analytic manifolds. This is meant to serve as foundation for new achievements and further research. Illustrative examples and new applications are given: we extend former results on unitary groups to closed subgroups with tensor-product structure, where the finest product partitioning relates to SUloc(2n) := SU(2) ⊗ ⋯ ⊗ SU(2) — known as (qubit-wise) local unitary operations. Such applications include, e.g., optimizing figures of merit on SUloc(2n) relating to distance measures of pure-state entanglement as well as to best rank-1 approximations of higher-order tensors. In quantum information, our gradient flows provide a numerically favorable alternative to standard tensor-SVD techniques.

arXiv:0802.4195•Reviews in Mathematical Physics•Jul 1, 2010•49 citations

Projection-operator-based Newton method for the trajectory optimization of closed quantum systems

Jieqiu Shao, Joshua Combes, John Hauser +1 more

Quantum optimal control is an important technology that enables fast state preparation and gate design. In the absence of an analytic solution, most quantum optimal control methods rely on an iterative scheme to update the solution estimate. At present, the convergence rate of existing solvers is, at most, superlinear. This paper develops a general-purpose solver for quantum optimal control based on the projection operator Newton method for trajectory optimization, or PRONTO. Specifically, the proposed approach uses a projection operator to incorporate the Schr\"odinger equation directly into the cost function, which is then minimized using Newton's descent method. At each iteration, the descent direction is obtained by computing the analytic solution to a linear-quadratic trajectory-optimization problem. The resulting method guarantees monotonic convergence at every iteration and quadratic convergence in the proximity of the solution. The potential of PRONTO is showcased by solving the optimal state-to-state mapping problem for a qubit and providing comparisons to a state-of-the-art quantum optimal control method.

arXiv:2111.08795•Physical Review A•Mar 9, 2022•12 citations

Optimal Control of a Fluxonium Circuit with the Q-PRONTO Numerical Method

John Barta, Brian D. Isakov, Pablo Aramburu Sanchez +5 more

Optimizing the control of large quantum systems is a resource-intensive task, where the convergence rate of the underlying iterative schemes is crucial. The recently developed Quantum Projection Operator-Based Newton Method for Trajectory Optimization (Q-PRONTO) offers a unique quadratic convergence for controlling quantum states, while most model-based solvers guarantee only an approximately linear convergence rate. As a proof of concept, here, we design and benchmark single-qubit gates optimized by Q-PRONTO based on the circuit Hamiltonian of a fluxonium superconducting qubit. To ensure that we can apply a long string of arbitrary pulses to the circuit, we modify the firmware associated with the Quantum Instrumentation Control Kit (QICK) by altering its memory allocation. We characterize the performance of the optimal solutions using standard randomized benchmarking and achieve a total error per gate of $6.6\times10^{-4}$.

quant-pharXiv:2610.31885•Oct 26, 2026•0 citations
In quantum optimal control theory, gradient-based trajectory optimization techniques have proven versatile in designing multi-qubit quantum gates. Furthermore, incorporating the underlying Lie-group structure can accelerate the optimization process. In this work, we adapt the Lie-group formulation of the iterative linear quadratic regulator (iLQR) to the special unitary group SU(N) and apply it to quantum gate synthesis, systematically comparing it against the standard Euclidean iLQR formulation across multiple two- to five-qubit gates. We find that in the idealized, unconstrained setting, where all Lie-algebra basis elements are available as drive Hamiltonian terms, the Lie-group formulation converges faster than the Euclidean iLQR formulation. If drive terms are constrained to 2-local Hamiltonian terms, the Lie-group variant converges faster in early optimization iterations, but exhibits greater sensitivity to initialization and a stronger tendency towards local minima. These results demonstrate that incorporating Lie-group geometry into iLQR substantially improves convergence and highlight important next steps for improvements in constrained control settings.
Quantum optimal control for gate optimization aims to provide accurate, robust, and fast pulse sequences to achieve gate fidelities on quantum systems below the error correction threshold. Many methods have been developed and successfully applied in simulation and on quantum hardware. In this paper, we establish a connection between the iterative linear quadratic regulator and quantum optimal control by adapting it to gate optimization of quantum systems. We include constraints on the controls and their derivatives to enable smoother pulses. We achieve high-fidelity simulation results forX\mathbf{X}Xand cross-resonance gates on one- and two-qubit fixed-frequency transmons simulated with two and three levels.
Gate-based quantum algorithms are typically implemented by circuits consisting of many single-qubit and multi-qubit gates operating on a quantum input state, followed by measurements of the output state. For certain quantum subroutines, such as initial state preparation and quantum Fourier transforms, explicit decompositions of the circuit in terms of single-qubit and two-qubit maximally entangling gates may exist. However, they often lead to large-depth circuits that are challenging for noisy intermediate-scale quantum (NISQ) hardware. Additionally, exact decompositions might only exist for some modular quantum circuits. Therefore, it is essential to find gate combinations that approximate these circuits to high fidelity with potentially low depth. Gradient-based optimization has been used to find such approximate decompositions. Still, these traditional optimizers often run into problems of slow convergence requiring many iterations, and performing poorly in the presence of noise, a factor that is especially relevant for NISQ hardware. Here we present iteratively preconditioned gradient descent (IPG) for optimizing quantum circuits and demonstrate performance speedups for state preparation and implementation of quantum algorithmic subroutines. IPG is a noise-resilient, higher-order algorithm that has shown promising gains in convergence speed for classical optimizations, converging locally at a linear rate for convex problems and superlinearly when the solution is unique. Specifically, we show an improvement in fidelity by a factor of 104for preparing a 4-qubit W state and a maximally entangled 5-qubit GHZ state with compared to other commonly used classical optimizers tuning the same ansatz. Such faster convergence with promise for noise-resilience could provide advantages for quantum algorithms on NISQ hardware, especially since the cost of running each iteration on a quantum computer is substantially higher than the classical optimizer step.
An important task in quantum computing is unitary circuit synthesis compatible with physical hardware constraints. This problem has a natural hybrid structure as local single-qubit gates are continuous variables on the Lie group SU(2)SU(2)SU(2) while the entangling circuit structure is discrete and hardware-dependent. In this work, we use generative models to perform quantum circuit synthesis incorporating both the natural SU(2)SU(2)SU(2) manifold geometry of quantum gates and hardware constraints that determine the overall circuit structure. Our model comprises two components: a circuit skeleton selector that chooses an entangling circuit and a diffusion model that generates quantum gates on the given circuit template by performing diffusion on the curved manifold SU(2)≃S3\mathrm{SU(2)} \simeq S^3SU(2)≃S3 itself. We demonstrate this approach with unitary compilation of physically motivated three-qubit Hamiltonian simulation targets such as the Transverse Field Ising Model and the Heisenberg-XXZ Model and show that Lie group diffusion outperforms comparable baselines. The synthesised circuits can also be customised subject to constraints, which we demonstrate by producing circuits with large and small gate rotation angles for the same target unitary evolution. We also investigate the fidelity-complexity frontier of the synthesised gates to demonstrate that the circuit selector learns to select circuits that balance fidelity with complexity rather than collapsing onto the most expansive entangling template. These results demonstrate that Lie group diffusion provides a natural generative framework for hardware-aware quantum circuit synthesis.
Many challenges in quantum information and quantum control root in constrained optimization problems on finite-dimensional quantum systems. The constraints often arise from two facts: (i) quantum dynamic state spaces are naturally smooth manifolds (orbits of the respective initial states) rather than being Hilbert spaces; (ii) the dynamics of the respective quantum system may be restricted to a proper subset of the entire state space. Mathematically, either case can be treated by constrained optimization over the reachable set of an underlying control system. Thus, whenever the reachable set takes the form a smooth manifold, Riemannian optimization methods apply. Here, we give a comprehensive account on the foundations of gradient flows on Riemannian manifolds including new applications in quantum information and quantum dynamics. Yet, we do not pursue the problem of designing explicit controls for the underlying control systems. The framework is sufficiently general for setting up gradient flows on (sub)manifolds, Lie (sub)groups, and (reductive) homogeneous spaces. Relevant convergence conditions are discussed, in particular for gradient flows on compact and analytic manifolds. This is meant to serve as foundation for new achievements and further research. Illustrative examples and new applications are given: we extend former results on unitary groups to closed subgroups with tensor-product structure, where the finest product partitioning relates to SUloc(2n) := SU(2) ⊗ ⋯ ⊗ SU(2) — known as (qubit-wise) local unitary operations. Such applications include, e.g., optimizing figures of merit on SUloc(2n) relating to distance measures of pure-state entanglement as well as to best rank-1 approximations of higher-order tensors. In quantum information, our gradient flows provide a numerically favorable alternative to standard tensor-SVD techniques.
Quantum optimal control is an important technology that enables fast state preparation and gate design. In the absence of an analytic solution, most quantum optimal control methods rely on an iterative scheme to update the solution estimate. At present, the convergence rate of existing solvers is, at most, superlinear. This paper develops a general-purpose solver for quantum optimal control based on the projection operator Newton method for trajectory optimization, or PRONTO. Specifically, the proposed approach uses a projection operator to incorporate the Schr\"odinger equation directly into the cost function, which is then minimized using Newton's descent method. At each iteration, the descent direction is obtained by computing the analytic solution to a linear-quadratic trajectory-optimization problem. The resulting method guarantees monotonic convergence at every iteration and quadratic convergence in the proximity of the solution. The potential of PRONTO is showcased by solving the optimal state-to-state mapping problem for a qubit and providing comparisons to a state-of-the-art quantum optimal control method.
Optimizing the control of large quantum systems is a resource-intensive task, where the convergence rate of the underlying iterative schemes is crucial. The recently developed Quantum Projection Operator-Based Newton Method for Trajectory Optimization (Q-PRONTO) offers a unique quadratic convergence for controlling quantum states, while most model-based solvers guarantee only an approximately linear convergence rate. As a proof of concept, here, we design and benchmark single-qubit gates optimized by Q-PRONTO based on the circuit Hamiltonian of a fluxonium superconducting qubit. To ensure that we can apply a long string of arbitrary pulses to the circuit, we modify the firmware associated with the Quantum Instrumentation Control Kit (QICK) by altering its memory allocation. We characterize the performance of the optimal solutions using standard randomized benchmarking and achieve a total error per gate of 6.6×10−46.6\times10^{-4}6.6×10−4.