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Training Schrödinger's cat: quantum optimal control
Stefffen J. Glaser, Ugo Boscain, Tommaso Calarco, Christiane P. Koch, Walter Köckenberger, Ronnie Kosloff, Ilya Kuprov, Burkard Luy, Sophie Schirmer, Thomas Schulte-Herbrüggen, D. Sugny, Frank K. Wilhelm
TL;DR
Quantum optimal control addresses how to drive quantum systems toward useful tasks with limited resources and experimental imperfections. The paper reviews mathematical methods and applications across spectroscopy, imaging, and quantum technologies, while identifying challenges in open-system control, computation, and theory–experiment integration. It concludes that the field provides a framework for determining achievable precision under decoherence and practical constraints, but important control-design and modeling problems remain.
Problem
Quantum technologies require precise control despite limits on amplitudes, power, timing, instrument accuracy, and environmental interactions.
Method
The paper reviews optimal-control theory and its applications to spectroscopy, imaging, and quantum dynamics in closed and open systems.
Results
The review identifies established control techniques and applications across sensing, state engineering, spectroscopy, imaging, quantum information, and simulation.
Takeaways & Limitations
Optimal control provides a framework for identifying which quantum tasks can be accomplished with what precision under decoherence and experimental limitations.
Takeaways & Limitations
Efficient control design for increasingly complex open-system dynamics and robustness to stochastic parameters remain open challenges.
Abstract
from arXiv · showhide
It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a system from a given initial state into a desired target state with minimized expenditure of energy and resources -- as famously applied in the Apollo programme. As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantum-enhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. --- Here state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium uniting expertise in optimal control theory and applications to spectroscopy, imaging, quantum dynamics of closed and open systems. We address key challenges and sketch a roadmap to future developments.
1 Introduction
Quantum optimal control designs external-field pulses or pulse sequences to manipulate quantum systems toward desired tasks under practical constraints. The field spans established applications and emerging quantum technologies, motivating cross-disciplinary collaboration and a shared research agenda.
- Quantum optimal control devises pulse shapes or pulse sequences that drive a quantum system toward a task while minimizing costs such as energy or duration.
- Applications range from magnetic resonance imaging and spectroscopy to precise chemical-reaction control and quantum technologies based on superposition, entanglement, and many-body systems.
- Recent techniques have achieved efficient and robust control in examples including NV-center magnetometry, Bose-Einstein-condensate state engineering, and high-fidelity superconducting quantum gates.
- Although physical implementations differ from ultrafast lasers to radio waves, quantum-control problems share commonalities while remaining distinct from classical control problems.
- The paper reviews mathematical optimal control theory, application state of the art, future perspectives, and prospects for commercial exploitation.
2 General aspects and mathematics of optimal control
Quantum control combines mathematical control theory with quantum-specific concepts such as entanglement and measurement. Its central design question is how to reach a target, using either open-loop or measurement-aware closed-loop approaches.
- Recent quantum-control advances draw on mathematical control theory, while quantum features such as entanglement and measurement have driven further mathematical development.
- Mathematical quantum control theory has emerged as a distinct research domain at the interface of mathematics and physics.
- Quantum control theory asks which targets are accessible and how an accessible target can be reached.
- Control-design approaches may be open-loop or closed-loop, with closed-loop methods requiring explicit consideration of quantum measurements.
2.1 Controllability and simulability
Controllability determines which quantum states or evolutions can be reached, while simulability concerns generating another system’s effective dynamics. Open-system, infinite-dimensional, and non-Markovian cases remain mathematically challenging.
- Controllability analysis asks whether a system can connect specified initial and target states or sets of states.
- Finite-dimensional closed systems admit a rigorous Lie-framework for controllability, whereas infinite-dimensional results are more limited and mathematically intricate.
- Ensemble controllability uses a few control fields to control a continuum of finite-dimensional systems and supports robustness to experimental imperfections.
- For open systems, dissipation can prevent full compensation, and controllability for non-Markovian dynamics remains largely uncharted.
- A system need not be fully controllable to simulate another system if its system algebra encompasses the other system’s algebra.
- Major open problems include controllability of open systems, especially non-Markovian systems, and rigorous treatment of mixed or continuous spectra.
- Reachable sets and universal or minimal control times remain largely unknown.
2.2 Control design
Quantum control design combines analytical and numerical optimization, adiabatic and shortcut methods, and feedback-oriented strategies. Progress is substantial, but computational cost, open-system complexity, experimental integration, and landscape constraints remain challenges.
- 2.2 Control design: Optimal control formulates pulse-sequence design as finding trajectories that satisfy dynamics and boundary conditions while minimizing a cost such as energy or duration.
- 2.2 Control design: Analytical solutions can provide optimal structures, global-optimality proofs, and physical limits when systems are sufficiently simple.
- 2.2 Control design: Geometric methods, including Cartan decomposition, can reduce system dimensionality and enable analytical control-field descriptions.
- 2.2 Control design: Numerical methods include gradient-ascent, Newton-type, and Krotov-type algorithms for cases where maximum-principle equations cannot be solved analytically.
- 2.2 Control design: Repeated equation-of-motion propagations can hinder numerical optimization for high-dimensional systems, motivating state compression, gradient-free methods, and local control.
- 2.2 Control design: Control landscapes may lack local traps in some unconstrained systems, but constraints such as finite amplitudes can induce traps, and robustness effects remain open.
- 2.2 Control design: Adiabatic methods offer robustness through slow, intense pulse sequences but require more time and energy than fast optimal-control approaches.
- 2.2 Control design: Open-system control faces unresolved extensions of adiabatic methods, incomplete system knowledge, efficient stochastic modeling, and the theory–experiment integration gap.
2.3 Quantum feedback control theory
Quantum feedback control uses closed-loop information or direct quantum-controller connections to improve robustness, but quantum measurement backaction prevents straightforward transfer of classical feedback concepts. Key open issues include controller advantages, broader dynamics, and model uncertainty.
- Closed-loop control can address unpredictable disturbances by feeding system information back to correct the applied field.
- Measurement-based feedback estimates the quantum state from measurements and uses real-time manipulation based on the results.
- Coherent feedback directly connects a quantum system to a controller without measurement, reducing control-process noise and enabling fast operation.
- Measurement backaction requires quantum-specific feedback techniques and complicates the balance between open-loop and closed-loop control.
- No general theory yet establishes when quantum controllers outperform classical counterparts or identifies the relevant experimental conditions.
- Open directions include weak measurements, non-Markovian dynamics, model uncertainty, programmable quantum processors, superconducting circuits, and quantum transport.
2.4 Long-term vision
Quantum optimal control is mature enough to address systems affected by environmental couplings, imperfect controls, and incomplete characterization. The long-term aim is a rigorous account of controllability limits and control design under real-life conditions.
- Quantum optimal control now targets systems that are never fully quantum because of environmental couplings, control imperfections, or incomplete system characterization.
- Its ultimate goal is to understand fundamental limits and opportunities for quantum control in both controllability and control design.
3 Atomic, molecular, and chemical physics
Quantum control in atomic, molecular, and chemical physics spans coherent reaction control, cooling, spectroscopy, electron dynamics, and emerging single-molecule applications. Progress is substantial, but complete reaction control and experimentally suitable pulse shaping remain important challenges.
- 3.1 State of the art: Coherent control originated in chemical reactions, using tailored laser fields to enhance desired outcomes and suppress others through matter-wave interference.
- 3.1 State of the art: Femtosecond lasers and pulse shaping enabled experimental control of unimolecular dissociation and fragmentation, with feedback loops often determining pulse shapes.
- 3.1 State of the art: Weak-field control exploits wavepacket dynamics and optical interference, whereas strong-field control modifies energy levels or molecular potential landscapes during the pulse.
- 3.1 State of the art: For isolated systems, phase-only control is impossible for objectives commuting with the free Hamiltonian under weak-field conditions.
- 3.1 State of the art: Complete control of a binary reaction from scattering reactants to selected-state products remains an open goal, especially because bond formation begins from an incoherent thermal ensemble.
- 3.1 State of the art: Spatial averaging blurs coherent effects because gas-phase particles experience different intensities across the beam profile.
- 3.2 Mid-term prospects: goals and challenges: Optimal control addresses cooling restrictions, but accurate potential energy surfaces remain necessary for designing pulses and interpreting experimentally optimized mechanisms.
- 3.2 Mid-term prospects: goals and challenges: Control is shifting from nuclear to electron dynamics, supported by attosecond x-ray sources and tools such as laser-induced electron diffraction and high-harmonic spectroscopy.
4 Magnetic resonance
Optimal control has produced robust, flexible pulse sequences across magnetic-resonance spectroscopy and imaging, while also providing analytical insight into how optimal pulses work. Current and prospective work targets faster, more adaptive, sensitive, and cost-effective experiments, but large coupled systems and algorithm usability remain challenges.
- State of the art: Pulse optimization enables broader offset bandwidth and robustness to control-amplitude variation, with pulse duration typically scaling linearly with the desired offset range.These properties support high-fidelity applications such as quantitative NMR and quantum error correction.
- State of the art: Optimal control has improved performance and design flexibility in heteronuclear decoupling, while cooperative optimization of pulses yields gains in single- and multiple-scan experiments.UR pulses are longer than PP pulses for comparable error resilience, motivating mixed sequence strategies.
- State of the art: Optimized broadband and application-specific pulses support liquid-state, solid-state, oriented-system, and ESR experiments, including improved robustness, accuracy, and acquisition speed.ESR implementations also account for transfer-line transients and limited resonator bandwidth.
- Mid-term prospects: goals and challenges: In imaging and chemical analysis, optimal control is being applied to spatially selective excitation, rf-power reduction, rf-inhomogeneity compensation, CEST, reliable quantification, and improved side-product detection.Prospective applications include more sensitive and efficient imaging, shorter scanner examinations, biomolecule analysis, hyperpolarization transfer, and lower instrument or sample-preparation costs.
- State of the art: Geometric and numerical optimal-control tools provide high-quality pulse sequences together with analytical insight, trajectory interpretations, and tools for analyzing complex coupled-spin dynamics.Geometric approaches can prove global optimality in low-dimensional systems and thereby establish physical performance limits.
- Mid-term prospects: goals and challenges: Future adoption depends on faster, easier-to-use, generally applicable algorithms and improved approaches for large coupled spin networks, relaxation, experimental imperfections, and adaptive closed-loop setup.The paper envisions on-the-fly, problem-, sample-, and patient-specific pulse-sequence reoptimization.
5 Quantum information and communication
Quantum optimal control supports quantum information tasks from state preparation and gate implementation to readout, while future progress depends on scalable, experimentally integrated control of complex systems.
- 6.1 State of the art: Quantum optimal control has enabled preparation of useful quantum states and implementation of quantum operations across several experimental platforms.Demonstrated applications include Bose–Einstein condensates, ultracold atoms, trapped ions, NV centers, and superconducting qubits.
- 6.1 State of the art: Optimized control sequences have prepared nonclassical motional states, improved optical-lattice loading, implemented error-resistant gates, and supported nanoscale magnetic-field imaging.In superconducting transmons, optimal control also addressed leakage to non-computational states and frequency crowding.
- 6.1 State of the art: Broader control methods have improved NV-center qubit coherence and stabilized photon-number states through real-time closed-loop feedback.The photon-state experiment incorporated noise back-action using stochastic differential calculus.
- 6.1 State of the art: Optimal-control methodology has been adapted for specific gates, dissipative dynamics, system–bath invariants, local equivalence classes, and robust quantum operations.These adaptations respond to requirements arising in quantum technologies rather than idealized closed-system control alone.
- 6.2 Mid-term prospects: goals and challenges: Near-term goals include robust multi-qubit gates, faster reset and readout, automated surface-code tasks, and robust generation of multi-particle entangled states.All of these milestones require decoherence control.
- 6.2 Mid-term prospects: goals and challenges: A central challenge is converging numerical optimal control with experimentation through accurate models, integrated system identification, tomography, and hybrid open- and closed-loop methods.Numerical control offers versatility, while closed-loop control can be tuned to task-specific parameter uncertainties.
- 6.3 Long-term vision: Scalable quantum control remains a severe long-term challenge, including scalable assembly beyond about 10 qubits and control of many-qubit systems.Tensor-contraction techniques are identified as a possible route for addressing this scaling problem.
6 Prospects for applications and commercial exploitation
Quantum optimal control already underpins magnetic-resonance technologies and is increasingly relevant to commercial quantum computing. Its applications span improved instruments, sensing, imaging, and superconducting-qubit development.
- Prospects for innovation: More sophisticated control may strengthen sensing and imaging technologies, including highly sensitive magnetic detectors, microscopic temperature devices, and molecular imaging.The paper also suggests that better control could reduce instrument costs.
- Established applications: Quantum control has contributed to commercial magnetic-resonance instruments, with optimal-control strategies implemented in NMR and pursued for MRI.The paper identifies magnetic resonance as an established application class alongside emerging quantum technologies.
- Industrial exploitation: IBM, Google, and Microsoft have invested in superconducting-qubit quantum computing, with IBM and Google using optimal-control techniques.The paper links this industrial interest to the technical maturity of these systems in research laboratories.
7 Conclusions
Quantum control provides a framework for precision-limited tasks across spectroscopy, imaging, AMO physics, and emerging quantum technologies. Future progress requires integrated architectures and coordinated interdisciplinary efforts with common terminology, standards, and visions.
- Conclusions: Quantum control facilitates spectroscopy, imaging, AMO physics, computation, simulation, metrology, sensing, and communication under decoherence and experimental limitations.Optimal control theory helps identify which tasks can be achieved and with what precision.
- Conclusions: Future quantum technologies will require control integration across hybrid architectures combining storage, sensing, and communication components.The paper also calls for integration of quantum mechanics in engineering education and engineering in quantum education.
- Conclusions: Because quantum control spans diverse fields, continued advances require close collaboration among basic research, development, and applications.The VF-QC is presented as a common structure for the European quantum-control community and for establishing shared terminology, standards, and visions.