Quantum control papers on arXiv, selected and classified automatically by QOC. https://llorebaga.github.io/QOC/topics/second-order-methods/ 2026-07-11T00:00:00Z QOC https://arxiv.org/abs/2603.13020 2026-03-13T00:00:00Z 2026-07-11T00:00:00Z Ziwen Song

Quantum-control pulses are often optimised for nominal fidelity before waveform constraints are imposed. This sequence can conceal the fidelity cost of producing smooth, band-limited, and amplitude-admissible controls. Here, we evaluate a constraint-native alternative based on inexact proximal alternating-direction updates. The formulation combines gate-infidelity minimisation with amplitude bounds, Fourier-domain bandwidth projection, amplitude sparsity, and total-variation regularisation. We compare it with GRAPE, standard Krotov optimisation, and L-BFGS-B on a single-qubit gate, a leakage-prone qutrit gate, and a two-qubit entangler without a directly controlled target generator. Random seeds are paired across methods, and qutrit computational-subspace fidelity is reported alongside leakage. PADMM-Warm reached mean qutrit and two-qubit fidelities of $0.6363$ and $0.9541$, respectively, while reducing total variation by factors of $13.2$ and $10.7$ relative to L-BFGS-B. These results define a reproducible fidelity--complexity trade-off, not a universal fidelity advantage. The method is therefore a numerical tool for exploring low-complexity control frontiers rather than a replacement for unconstrained high-fidelity solvers.

https://arxiv.org/abs/2602.08526 2026-02-09T00:00:00Z 2026-02-09T00:00:00Z Duc-Kha VuMinh T. P. NguyenÖzgür E. MüstecaplıoğluFatih Ozaydin

Designing scalable, noise-tolerant control protocols for multipartite entanglement is a central challenge for quantum technologies, and it naturally calls for \emph{algorithmic} synthesis of interaction parameters rather than handcrafted gate sequences. Here we introduce an intelligent, constraint-aware control framework for deterministic generation of symmetric Dicke states $|D_n^{(m)}\rangle$ in repeated-interaction (collision-model) architectures. The protocol employs excitation-preserving partial-SWAP collisions between two disjoint qubit registers, mediated by $m$ ancillary ``shuttle'' qubits, and poses Dicke-state preparation as a \emph{closed-loop design} problem: given the target $(n,m)$, automatically infer collision strengths that maximize fidelity under practical constraints. Concretely, we formulate a two-parameter, bound-constrained optimization over intra-register and shuttle--register collision angles and solve it using a multi-start strategy with L-BFGS-B, yielding a reproducible controller prescription (optimized $γ_{\mathrm{in}}$, $γ_{\mathrm{sh}}$, and minimal-round convergence points) for each target. This removes the need for projective measurements and extends collisional entanglement generation beyond the single-excitation (W-state) sector to arbitrary $m$. Crucially, we optimize \emph{within} imperfect collisional dynamics where errors act throughout the sequence, including stochastic interaction dropouts (missing collisions) and standard decoherence channels. Strikingly, across wide error ranges the optimized controller preserves high preparation fidelity; imperfections manifest primarily as a modest increase in the required number of collision rounds. This behavior reflects a tunable competition in which noise suppresses correlations while properly chosen collisions continuously replenish them, allowing the control algorithm to trade time for fidelity.

https://arxiv.org/abs/2505.00529 2025-05-01T00:00:00Z 2025-05-01T00:00:00Z Harish S. Bhat

We derive and implement a second-order adjoint method to compute exact gradients and Hessians for a prototypical quantum optimal control problem, that of solving for the minimal energy applied electric field that drives a molecule from a given initial state to a desired target state. For small to moderately sized systems, we demonstrate a vectorized GPU implementation of a second-order adjoint method that computes both Hessians and gradients with wall times only marginally more than those required to compute gradients via commonly used first-order adjoint methods. Pairing our second-order adjoint method with a trust region optimizer (a type of Newton method), we show that it outperforms a first-order method, requiring significantly fewer iterations and wall time to find optimal controls for four molecular systems. Our derivation of the second-order adjoint method allows for arbitrary parameterizations of the controls.

https://arxiv.org/abs/2501.12564 2025-01-22T00:00:00Z 2025-01-22T00:00:00Z Carrie A. WeidnerSean Patrick O'NeilEdmond JonckheereFrank C. LangbeinS. G. Schirmer

Robust quantum control is crucial for realizing practical quantum technologies. Energy landscape shaping offers an alternative to conventional dynamic control, providing theoretically enhanced robustness and simplifying implementation for certain applications. This work demonstrates the feasibility of robust energy landscape control in a practical implementation with ultracold atoms. We leverage a digital mirror device (DMD) to shape optical potentials, creating complex energy landscapes. To achieve a desired objective, such as efficient quantum state transfer, we formulate a novel hybrid optimization approach that effectively handles both continuous (laser power) and discrete (DMD pixel activation) control parameters. This approach combines constrained quasi-Newton methods with surrogate models for efficient exploration of the vast parameter space. Furthermore, we introduce a framework for analyzing the robustness of the resulting control schemes against experimental uncertainties. By modeling uncertainties as structured perturbations, we systematically assess controller performance and identify robust solutions. We apply these techniques to maximize spin transfer in a chain of trapped atoms, achieving high-fidelity control while maintaining robustness. Our findings provide insights into the experimental viability of controlled spin transfer in cold atom systems. More broadly, the presented optimization and robustness analysis methods apply to a wide range of quantum control problems, offering a toolkit for designing and evaluating robust controllers in complex experimental settings.

https://arxiv.org/abs/2207.09882 2022-07-20T00:00:00Z 2022-08-18T00:00:00Z David L. GoodwinMads Sloth Vinding

A Hessian based optimal control method is presented in Liouville space to mitigate previously undesirable polynomial scaling of computation time. This new method, an improvement to the state-of-the-art Newton-Raphson GRAPE method, is derived with respect to two exact time-propagator derivative techniques: auxiliary matrix and ESCALADE methods. We observed that compared to the best current implementation of Newton-Raphson GRAPE method, for an ensemble of 2-level systems, with realistic conditions, the new auxiliary matrix and ESCALADE Hessians can be 4-200 and 70-600 times faster, respectively.

https://arxiv.org/abs/2204.13671 2022-04-28T00:00:00Z 2023-12-04T00:00:00Z Boris VolkovAlexander Pechen

In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proven on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in the case of saddle the numbers of negative and positive eigenvalues of Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of Hessian at this saddle point and moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian [Theorem~3 in B.O.~Volkov, O.V.~Morzhin, A.N.~Pechen, J.~Phys.~A: Math. Theor. {\bf 54}, 215303 (2021)].

https://arxiv.org/abs/2111.08795 2021-11-16T00:00:00Z 2022-10-26T00:00:00Z Jieqiu ShaoJoshua CombesJohn HauserMarco M. Nicotra

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 new 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 Schrödinger equation directly into the cost function, which is then minimized using a quasi-Newton 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 proximity of the solution. To highlight the potential of PRONTO, we present an numerical example that employs it to solve the optimal state-to-state mapping problem for a qubit and compares its performance to a state-of-the-art quadratic optimal control method.

https://arxiv.org/abs/2109.07226 2021-09-15T00:00:00Z 2021-09-15T00:00:00Z Irtaza KhalidCarrie A. WeidnerEdmond JonckheereS. G. SchirmerFrank C. Langbein

We explore the use of policy gradient methods in reinforcement learning for quantum control via energy landscape shaping of XX-Heisenberg spin chains in a model agnostic fashion. Their performance is compared to finding controllers using gradient-based L-BFGS optimisation with restarts, with full access to an analytical model. Hamiltonian noise and coarse-graining of fidelity measurements are considered. Reinforcement learning is able to tackle challenging, noisy quantum control problems where L-BFGS optimization algorithms struggle to perform well. Robustness analysis under different levels of Hamiltonian noise indicates that controllers found by reinforcement learning appear to be less affected by noise than those found with L-BFGS.

https://arxiv.org/abs/2106.14310 2021-06-27T00:00:00Z 2022-08-19T00:00:00Z N. Anders PeterssonFortino Garcia

We consider the optimal control problem of determining electromagnetic pulses for implementing logical gates in a closed quantum system, where the Hamiltonian models the dynamics of coupled superconducting qudits. The quantum state is governed by Schrödinger's equation, which we formulate in terms of the real and imaginary parts of the state vector and solve by the Störmer-Verlet scheme, which is a symplectic partitioned Runge-Kutta method. A novel parameterization of the control functions based on B-splines with carrier waves is introduced. The carrier waves are used to trigger the resonant frequencies in the system Hamiltonian, and the B-spline functions specify their amplitude and phase. This approach allows the number of control parameters to be independent of, and significantly smaller than, the number of time steps for integrating Schrödinger's equation. We present numerical examples of how the proposed technique can be combined with an interior point L-BFGS algorithm for realizing quantum gates, and generalize our approach to calculate risk-neutral controls that are resilient to noise in the Hamiltonian model. The proposed method is also shown to compare favorably with QuTiP/pulse\_optim and Grape-Tensorflow.

https://arxiv.org/abs/2006.00935 2020-06-01T00:00:00Z 2020-09-16T00:00:00Z Mogens DalgaardFelix MotzoiJesper Hasseriis Mohr JensenJacob Sherson

Efficient optimization of quantum systems is a necessity for reaching fault tolerant thresholds. A standard tool for optimizing simulated quantum dynamics is the gradient-based \textsc{grape} algorithm, which has been successfully applied in a wide range of different branches of quantum physics. In this work, we derive and implement exact $2^{\mathrm{nd}}$ order analytical derivatives of the coherent dynamics and find improvements compared to the standard of optimizing with the approximate $2^{\mathrm{nd}}$ order \textsc{bfgs}. We demonstrate performance improvements for both the best and average errors of constrained unitary gate synthesis on a circuit-\textsc{qed} system over a broad range of different gate durations.

https://arxiv.org/abs/2001.05941 2020-01-16T00:00:00Z 2020-01-16T00:00:00Z Martin LaroccaEsteban A. CalzettaDiego Wisniacki

Quantum Optimal Control (QOC) is the field devoted to the production of external control protocols that actively guide quantum dynamics. Solutions to QOC problems were shown to constitute continuous submanifolds of control space. A solution navigation method exploiting this property to achieve secondary features in the control protocols was proposed [Larocca \textit{et al}, arXiv:1911.07105]. Originally, the technique involved the computation of the exact Hessian matrix. In this paper, we show that the navigation can be alternatively performed with a finite-difference approximation scheme, thus enabling the application of this procedure in systems where computing the exact Hessian is out of reach.

https://arxiv.org/abs/2001.01013 2020-01-04T00:00:00Z 2020-11-19T00:00:00Z N. Anders PeterssonFortino GarciaAustin E. CopelandYlva L. RydinJonathan L DuBois

This paper considers the optimal control problem for realizing logical gates in a closed quantum system. The quantum state is governed by Schrodinger's equation, which we formulate as a time-dependent Hamiltonian system in terms of the real and imaginary parts of the state vector. The system is discretized with the Stormer-Verlet scheme, which is a symplectic partitioned Runge-Kutta method. Our main theoretical contribution is the derivation of a compatible time-discretization of the adjoint state equation, such that the gradient of the discrete objective function can be calculated exactly, at a computational cost of solving two Schrödinger systems, independently of the number of parameters in the control functions. A parameterization of the control functions based on B-splines with built-in carrier waves is also introduced. The carrier waves are used to specify the frequency spectra of the control functions, while the B-spline functions specify their envelope and phase. This approach allows the number of control parameters to be independent of, and significantly smaller than, the number of time steps for integrating Schrodinger's equation. We consider Hamiltonians that model the dynamics of a superconducting multi-level qudit and present numerical examples of how the proposed technique can be combined with the interior point L-BFGS algorithm from the IPOPT package for realizing quantum gates. In a set of test cases, the proposed algorithm is shown to compare favorably with QuTiP/pulse_optim and Grape-Tensorflow.

https://arxiv.org/abs/1510.02420 2015-10-08T00:00:00Z 2016-04-28T00:00:00Z David L. GoodwinIlya Kuprov

Quadratic convergence throughout the active space is achieved for the gradient ascent pulse engineering (GRAPE) family of quantum optimal control algorithms. We demonstrate in this communication that the Hessian of the GRAPE fidelity functional is unusually cheap, having the same asymptotic complexity scaling as the functional itself. This leads to the possibility of using very efficient numerical optimization techniques. In particular, the Newton-Raphson method with a rational function optimization (RFO) regularized Hessian is shown in this work to require fewer system trajectory evaluations than any other algorithm in the GRAPE family. This communication describes algebraic and numerical implementation aspects (matrix exponential recycling, Hessian regularization, etc.) for the RFO Newton-Raphson version of GRAPE and reports benchmarks for common spin state control problems in magnetic resonance spectroscopy.

https://arxiv.org/abs/1502.04045 2015-02-13T00:00:00Z 2015-02-13T00:00:00Z Qiuyang SunIstván PelczerGregory RivielloRe-Bing WuHerschel Rabitz

The growing successes in performing quantum control experiments motivated the development of control landscape analysis as a basis to explain these findings.When a quantum system is controlled by an electromagnetic field, the observable as a functional of the control field forms a landscape. Theoretical analyses have revealed many properties of control landscapes, especially regarding their slopes, curvatures, and topologies. A full experimental assessment of the landscape predictions is important for future consideration of controlling quantum phenomena. Nuclear magnetic resonance (NMR) is exploited here as an ideal laboratory setting for quantitative testing of the landscape principles. The experiments are performed on a simple two-level proton system in a H$_2$O-D$_2$O sample. We report a variety of NMR experiments roving over the control landscape based on estimation of the gradient and Hessian, including ascent or descent of the landscape, level set exploration, and an assessment of the theoretical predictions on the structure of the Hessian. The experimental results are fully consistent with the theoretical predictions. The procedures employed in this study provide the basis for future multispin control landscape exploration where additional features are predicted to exist.

https://arxiv.org/abs/1112.4454 2011-12-19T00:00:00Z 2011-12-19T00:00:00Z Ofer M. ShirJonathan RoslundDarrell WhitleyHerschel Rabitz

The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) has been the most successful Evolution Strategy at exploiting covariance information; it uses a form of Principle Component Analysis which, under certain conditions, is suggested to converge to the correct covariance matrix, formulated as the inverse of the mathematically well-defined Hessian matrix. However, in practice, there exist conditions where CMA-ES converges to the global optimum (accomplishing its primary goal) while it does not learn the true covariance matrix (missing an auxiliary objective), likely due to step-size deficiency. These circumstances can involve high-dimensional landscapes with large condition numbers. This paper introduces a novel technique entitled Forced Optimal Covariance Adaptive Learning (FOCAL), with the explicit goal of determining the Hessian at the global basin of attraction. It begins by introducing theoretical foundations to the inverse relationship between the learned covariance and the Hessian matrices. FOCAL is then introduced and demonstrated to retrieve the Hessian matrix with high fidelity on both model landscapes and experimental Quantum Control systems, which are observed to possess a non-separable, non-quadratic search landscape. The recovered Hessian forms are corroborated by physical knowledge of the systems. This study constitutes an example for Natural Computing successfully serving other branches of natural sciences, and introducing at the same time a powerful generic method for any high-dimensional continuous search seeking landscape information.

https://arxiv.org/abs/1108.3806 2011-08-18T00:00:00Z 2013-04-29T00:00:00Z Carlee Joe-WongTak-San HoRuixing LongHerschel RabitzRe-Bing Wu

Optimal control of molecular dynamics is commonly expressed from a quantum mechanical perspective. However, in most contexts the preponderance of molecular dynamics studies utilize classical mechanical models. This paper treats laser-driven optimal control of molecular dynamics in a classical framework. We consider the objective of steering a molecular system from an initial point in phase space to a target point, subject to the dynamic constraint of Hamilton's equations. The classical control landscape corresponding to this objective is a functional of the control field, and the topology of the landscape is analyzed through its gradient and Hessian with respect to the control. Under specific assumptions on the regularity of the control fields, the classical control landscape is found to be free of traps that could hinder reaching the objective. The Hessian associated with an optimal control field is shown to have finite rank, indicating the presence of an inherent degree of robustness to control noise. Extensive numerical simulations are performed to illustrate the theoretical principles on a) a model diatomic molecule, b) two coupled Morse oscillators, and c) a chaotic system with a coupled quartic oscillator, confirming the absence of traps in the classical control landscape. We compare the classical formulation with the mathematically analogous state-to-state transition probability control landscape of N-level quantum systems. The absence of traps in both circumstances provides a broader basis to understand the growing number of successful control experiments with complex molecules, which can have dynamics that transcend the classical and quantum regimes.

https://arxiv.org/abs/1102.4096 2011-02-20T00:00:00Z 2011-06-23T00:00:00Z Pierre de FouquieresS. G. SchirmerSteffen J. GlaserIlya Kuprov

We report some improvements to the gradient ascent pulse engineering (GRAPE) algorithm for optimal control of quantum systems. These include more accurate gradients, convergence acceleration using the BFGS quasi-Newton algorithm as well as faster control derivative calculation algorithms. In all test systems, the wall clock time and the convergence rates show a considerable improvement over the approximate gradient ascent.