Resources

Software for designing and simulating controls, what to read first, the field's vocabulary, and how it came to be.

Software

Open-source packages unless marked otherwise. Every link was checked; where a package has a paper, it is cited.

Optimal-control frameworks

  • qutip-qoc Python

    GRAPE, CRAB, GOAT and JAX-based optimization of controls, built on QuTiP.

  • krotov Python

    Reference implementation of Krotov’s method for closed and open systems, built on QuTiP.

    Michael Goerz, Daniel Basilewitsch, Fernando Gago-Encinas, Matthias G. Krauss, Karl P. Horn, et al. Krotov: A Python implementation of Krotov's method for quantum optimal control SciPost Phys. 7, 080 (2019) doi

  • QuantumControl.jl Julia

    Framework for quantum dynamics and control, with GRAPE.jl, Krotov.jl and QuantumPropagators.jl; supports semi-automatic differentiation of arbitrary figures of merit.

    Michael H. Goerz, Sebastián C. Carrasco and Vladimir S. Malinovsky Quantum Optimal Control via Semi-Automatic Differentiation Quantum 6, 871 (2022) doi

  • QuOCS Python

    Quantum optimal control suite (dCRAB, GRAPE, automatic differentiation) designed to talk to experiments.

    Marco Rossignolo, Thomas Reisser, Alastair Marshall, Phila Rembold, Alice Pagano, et al. QuOCS: The quantum optimal control suite Computer Physics Communications 291, 108782 (2023) doi

  • C3 Python

    Toolset that combines open-loop optimization, closed-loop calibration and model learning (TensorFlow).

    Nicolas Wittler, Federico Roy, Kevin Pack, Max Werninghaus, Anurag Saha Roy, et al. Integrated Tool Set for Control, Calibration, and Characterization of Quantum Devices Applied to Superconducting Qubits Phys. Rev. Applied 15, 034080 (2021) doi

  • qopt Python

    Qubit simulation and optimal control with realistic noise models, oriented towards experiments.

    Julian David Teske, Pascal Cerfontaine and Hendrik Bluhm qopt: An Experiment-Oriented Software Package for Qubit Simulation and Quantum Optimal Control Phys. Rev. Applied 17, 034036 (2022) doi

  • Piccolo.jl Julia

    Quantum optimal control by direct collocation, solved with an interior-point method.

    Aaron Trowbridge, Aditya Bhardwaj, Kevin He, David I. Schuster and Zachary Manchester Direct Collocation for Quantum Optimal Control preprint (2023) arXiv:2305.03261

  • Juqbox.jl Julia

    Optimal control of closed systems with B-spline envelopes and carrier waves.

    N. Anders Petersson and Fortino Garcia Optimal Control of Closed Quantum Systems via B-Splines with Carrier Waves SIAM J. Sci. Comput. 44, A3592-A3616 (2022) doi

  • Quandary C++

    High-performance optimal control of open quantum systems (Lindblad), parallelized with MPI.

    Stefanie Gunther, N. Anders Petersson and Jonathan L. DuBois Quandary: An open-source C++ package for high-performance optimal control of open quantum systems 2021 IEEE/ACM Second International Workshop on Quantum Computing Software (QCS), 88-98 (2021) doi

  • Spinach MATLAB

    Spin-dynamics simulation for magnetic resonance, with GRAPE optimal control built in.

    H.J. Hogben, M. Krzystyniak, G.T.P. Charnock, P.J. Hore and Ilya Kuprov Spinach – A software library for simulation of spin dynamics in large spin systems Journal of Magnetic Resonance 208, 179-194 (2011) doi

  • QControlPolyOpt Julia

    Global optimization of quantum control problems as polynomial optimization (QCPOP).

    Denys I. Bondar, Llorenç Balada Gaggioli, Georgios Korpas, Jakub Marecek, Jiri Vala and Kurt Jacobs Globally optimal control of quantum dynamics Phys. Rev. Research 7, 043202 (2025) doi · arXiv:2209.05790

  • ECD_control Python

    Optimization of echoed conditional-displacement sequences for an oscillator coupled to a qubit.

    Alec Eickbusch, Volodymyr Sivak, Andy Z. Ding, Salvatore S. Elder, Shantanu R. Jha, et al. Fast universal control of an oscillator with weak dispersive coupling to a qubit Nat. Phys. 18, 1464-1469 (2022) doi

  • Boulder Opal Python · commercial

    Commercial software from Q-CTRL for autonomous calibration and control of quantum hardware (cloud service with a Python client).

Simulation and differentiable dynamics

  • QuTiP Python

    The Quantum Toolbox in Python — closed and open dynamics, the basis of many control packages.

    J.R. Johansson, P.D. Nation and Franco Nori QuTiP: An open-source Python framework for the dynamics of open quantum systems Computer Physics Communications 183, 1760-1772 (2012) doi · arXiv:1110.0573Neill Lambert, Eric Giguère, Paul Menczel, Boxi Li, Patrick Hopf, et al. QuTiP 5: The Quantum Toolbox in Python Physics Reports 1153, 1-62 (2026) doi · arXiv:2412.04705

  • QuantumToolbox.jl Julia

    QuTiP-style toolbox for quantum dynamics in Julia.

  • dynamiqs Python

    GPU-accelerated, differentiable solvers for closed and open systems, built on JAX.

  • quantum-optimal-control Python

    GRAPE with automatic differentiation on GPUs (TensorFlow); the code of the paper below, not actively maintained.

    Nelson Leung, Mohamed Abdelhafez, Jens Koch and David Schuster Speedup for quantum optimal control from automatic differentiation based on graphics processing units Phys. Rev. A 95, 042318 (2017) doi

  • DYNAMO MATLAB

    Unified framework for comparing and benchmarking gradient algorithms (GRAPE, Krotov and hybrids); last updated 2011.

    S. Machnes, U. Sander, S. J. Glaser, P. de Fouquières, A. Gruslys, et al. Comparing, optimizing, and benchmarking quantum-control algorithms in a unifying programming framework Phys. Rev. A 84, 022305 (2011) doi

Pulse sequences and noise

  • Q-CTRL Open Controls Python

    Open library of dynamical-decoupling sequences and composite pulses.

  • filter_functions Python

    Generalized filter functions and quantum processes of gate sequences under classical noise.

    Tobias Hangleiter, Pascal Cerfontaine and Hendrik Bluhm Filter-function formalism and software package to compute quantum processes of gate sequences for classical non-Markovian noise Phys. Rev. Research 3, 043047 (2021) doi

Platforms and hardware

  • Pulser Python

    Pulse-level and analog programming of neutral-atom arrays, with emulation.

    Henrique Silvério, Sebastián Grijalva, Constantin Dalyac, Lucas Leclerc, Peter J. Karalekas, et al. Pulser: An open-source package for the design of pulse sequences in programmable neutral-atom arrays Quantum 6, 629 (2022) doi

  • qutip-qip Python

    Compiles circuits to pulses on simulated processors and simulates them with noise.

    Boxi Li, Shahnawaz Ahmed, Sidhant Saraogi, Neill Lambert, Franco Nori, et al. Pulse-level noisy quantum circuits with QuTiP Quantum 6, 630 (2022) doi

  • PennyLane Python

    Differentiable quantum programming, including pulse programs with parametrized Hamiltonians.

  • QICK Python

    Firmware and software for RFSoC-based qubit control and readout.

  • Qibolab Python

    Drivers and pulse execution for several control-electronics platforms.

General numerical tools

  • JAX Python

    Automatic differentiation and GPU/TPU compilation; the engine of several packages above.

  • SciPy optimize Python

    L-BFGS-B, trust-region and other optimizers used by many Python control codes.

  • NLopt C

    Library of local and global, gradient-based and derivative-free optimizers.

  • pycma Python

    Reference implementation of the CMA-ES evolution strategy.

    Nikolaus Hansen and Andreas Ostermeier Completely Derandomized Self-Adaptation in Evolution Strategies Evolutionary Computation 9, 159-195 (2001) doi

  • TSSOS Julia

    Sparse moment–SOS hierarchy for global polynomial optimization.

Reading

Start here

Reviews and tutorials that introduce the whole field.

  • Steffen J. Glaser, Ugo Boscain, Tommaso Calarco, Christiane P. Koch, Walter Köckenberger, et al. Training Schrödinger’s cat: quantum optimal control Eur. Phys. J. D 69, 279 (2015) doi

    The 2015 European roadmap — state of the art, goals and open challenges, across platforms.

  • Christiane P. Koch, Ugo Boscain, Tommaso Calarco, Gunther Dirr, Stefan Filipp, et al. Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe EPJ Quantum Technol. 9, 19 (2022) doi

    The 2022 update of the roadmap, with a long list of open questions in controllability and control design.

  • Q Ansel, E Dionis, F Arrouas, B Peaudecerf, S Guérin, et al. Introduction to theoretical and experimental aspects of quantum optimal control J. Phys. B: At. Mol. Opt. Phys. 57, 133001 (2024) doi · arXiv:2403.00532

    A recent introduction to the theory and the experimental practice of optimal control.

  • Constantin Brif, Raj Chakrabarti and Herschel Rabitz Control of quantum phenomena: past, present and future New J. Phys. 12, 075008 (2010) doi

    Past, present and future of controlling quantum phenomena, from chemistry to quantum information.

  • J Werschnik and E K U Gross Quantum optimal control theory J. Phys. B: At. Mol. Opt. Phys. 40, R175-R211 (2007) doi

    A compact tutorial on quantum optimal control theory and its numerical methods.

  • D. Dong and I.R. Petersen Quantum control theory and applications: a survey IET Control Theory Appl. 4, 2651-2671 (2010) doi · arXiv:0910.2350

    A survey from the control-engineering side — controllability, feedback, robust and learning control.

  • Luigi Giannelli, Sofia Sgroi, Jonathon Brown, Gheorghe Sorin Paraoanu, Mauro Paternostro, et al. A tutorial on optimal control and reinforcement learning methods for quantum technologies Physics Letters A 434, 128054 (2022) doi

    Tutorial on optimal control and reinforcement learning methods for quantum technologies.

  • U. Boscain, M. Sigalotti and D. Sugny Introduction to the Pontryagin Maximum Principle for Quantum Optimal Control PRX Quantum 2, 030203 (2021) doi

    Pontryagin’s maximum principle explained for quantum control, with worked examples.

Books

  • Domenico D'Alessandro Introduction to Quantum Control and Dynamics Chapman and Hall/CRC (2021) doi

    Mathematical introduction to quantum control — Lie-algebraic controllability, optimal control, dynamics.

  • Howard M. Wiseman and Gerard J. Milburn Quantum Measurement and Control Cambridge University Press (2009) doi

    The standard text on quantum measurement and feedback control.

  • Moshe Shapiro and Paul Brumer Quantum Control of Molecular Processes Wiley (2011) doi

    Coherent control of molecular processes, from the founders of the interference approach.

  • Andrei A. Agrachev and Yuri L. Sachkov Control Theory from the Geometric Viewpoint Springer Berlin Heidelberg (2004) doi

    Geometric control theory — reachable sets, the maximum principle, sub-Riemannian problems.

  • Hendra I Nurdin and Naoki Yamamoto Linear Dynamical Quantum Systems Springer International Publishing (2017) doi

    Linear quantum systems and coherent feedback.

Reviews by topic

  • Christiane P Koch Controlling open quantum systems: tools, achievements, and limitations J. Phys.: Condens. Matter 28, 213001 (2016) doi

    Control of open quantum systems — tools, achievements and limitations.

  • Raj Chakrabarti and Herschel Rabitz Quantum control landscapes International Reviews in Physical Chemistry 26, 671-735 (2007) doi

    Quantum control landscapes — critical points and what they mean for optimization.

  • Matthias M Müller, Ressa S Said, Fedor Jelezko, Tommaso Calarco and Simone Montangero One decade of quantum optimal control in the chopped random basis Rep. Prog. Phys. 85, 076001 (2022) doi

    Ten years of the chopped random basis (CRAB) method.

  • D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot and J. G. Muga Shortcuts to adiabaticity: Concepts, methods, and applications Rev. Mod. Phys. 91, 045001 (2019) doi

    Shortcuts to adiabaticity — concepts, methods and applications.

  • Nikolay V. Vitanov, Andon A. Rangelov, Bruce W. Shore and Klaas Bergmann Stimulated Raman adiabatic passage in physics, chemistry, and beyond Rev. Mod. Phys. 89, 015006 (2017) doi

    Stimulated Raman adiabatic passage and its many uses.

  • Sebastian Deffner and Steve Campbell Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control J. Phys. A: Math. Theor. 50, 453001 (2017) doi

    Quantum speed limits, from the uncertainty principle to optimal control.

  • Daniel A. Lidar Review of Decoherence‐Free Subspaces, Noiseless Subsystems, and Dynamical Decoupling Quantum Information and Computation for Chemistry, 295-354 (2014) doi

    Decoherence-free subspaces, noiseless subsystems and dynamical decoupling.

  • L. M. K. Vandersypen and I. L. Chuang NMR techniques for quantum control and computation Rev. Mod. Phys. 76, 1037-1069 (2005) doi

    NMR techniques for quantum control and computation — where many control ideas were first tested.

  • Christiane P. Koch, Mikhail Lemeshko and Dominique Sugny Quantum control of molecular rotation Rev. Mod. Phys. 91, 035005 (2019) doi

    Quantum control of molecular rotation.

  • Alicia B. Magann, Christian Arenz, Matthew D. Grace, Tak-San Ho, Robert L. Kosut, et al. From Pulses to Circuits and Back Again: A Quantum Optimal Control Perspective on Variational Quantum Algorithms PRX Quantum 2, 010101 (2021) doi

    Variational quantum algorithms seen as optimal control problems.

  • Phila Rembold, Nimba Oshnik, Matthias M. Müller, Simone Montangero, Tommaso Calarco and Elke Neu Introduction to quantum optimal control for quantum sensing with nitrogen-vacancy centers in diamond AVS Quantum Science 2, 024701 (2020) doi

    Optimal control for quantum sensing with NV centres.

Landmark papers

A small selection; the timeline below gives more.

  • Anthony P. Peirce, Mohammed A. Dahleh and Herschel Rabitz Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications Phys. Rev. A 37, 4950-4964 (1988) doi

    Optimal control theory formulated for quantum-mechanical systems.

  • Richard S. Judson and Herschel Rabitz Teaching lasers to control molecules Phys. Rev. Lett. 68, 1500-1503 (1992) doi

    Closed-loop learning control — let the experiment optimize the pulse.

  • Navin Khaneja, Roger Brockett and Steffen J. Glaser Time optimal control in spin systems Phys. Rev. A 63, 032308 (2001) doi

    Time-optimal control of coupled spins by geometric methods.

  • Navin Khaneja, Timo Reiss, Cindie Kehlet, Thomas Schulte-Herbrüggen and Steffen J. Glaser Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms Journal of Magnetic Resonance 172, 296-305 (2005) doi

    GRAPE, the gradient algorithm used everywhere.

  • Herschel A. Rabitz, Michael M. Hsieh and Carey M. Rosenthal Quantum Optimally Controlled Transition Landscapes Science 303, 1998-2001 (2004) doi

    Why optimal control is easy so often — trap-free landscapes.

  • T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Montangero, et al. Optimal Control at the Quantum Speed Limit Phys. Rev. Lett. 103, 240501 (2009) doi

    Optimal control at the quantum speed limit.

  • Patrick Doria, Tommaso Calarco and Simone Montangero Optimal Control Technique for Many-Body Quantum Dynamics Phys. Rev. Lett. 106, 190501 (2011) doi

    CRAB — optimal control for many-body dynamics with few parameters.

Every topic article ends with the key papers on that topic.

Glossary

Average gate fidelity
The fidelity between an implemented operation and a target gate, averaged uniformly over all pure input states. For a \(d\)-dimensional system it is an affine function of the entanglement fidelity, which makes it simple to compute (Nielsen’s formula). → Problem formulation & figures of merit, Quantum gate synthesis
Bilinear control system
Dynamics of the form \(\dot x = (A + \sum_k u_k(t) B_k)\,x\) — linear in the state and in the controls separately. The Schrödinger equation \(i\dot\psi = (H_0 + \sum_k u_k(t) H_k)\psi\) is the standard example. → Quantum controllability
Closed-loop (learning) control
Optimization in which every candidate control is tried on the experiment and the measured figure of merit is fed back to the optimizer; no accurate model is needed. → Closed-loop & in-situ optimization
Coherent feedback
Feedback in which the controller is itself a quantum system, connected to the plant without measurement — for example through optical fields. → Linear quantum systems & coherent feedback, Measurement-based feedback
Composite pulse
A sequence of simple rotations whose errors cancel to some order, so that the whole sequence is robust to a systematic error (amplitude, detuning) that would spoil a single pulse. → Composite pulses
Control landscape
The figure of merit viewed as a function of the controls. Its critical points — and whether it has traps — decide how hard an optimization is. → Control landscapes
Controllability
The property that every target (state, or unitary) can be reached with admissible controls. For finite-dimensional closed systems it is decided by the Lie algebra generated by the drift and control Hamiltonians. → Quantum controllability
Counterdiabatic driving
An added Hamiltonian that cancels the non-adiabatic transitions of a time-dependent Hamiltonian, so that the system follows its instantaneous eigenstates at any speed. Also called transitionless driving. → Shortcuts to adiabaticity
CRAB
Chopped RAndom Basis: the control is expanded in a few randomized basis functions, and the coefficients are optimized, often without gradients. dCRAB changes the basis during the optimization. → CRAB & dCRAB
DRAG
Derivative Removal by Adiabatic Gate: adding a quadrature component proportional to the derivative of the pulse envelope suppresses leakage and phase errors in weakly anharmonic qubits such as transmons. → DRAG & leakage suppression
Drift Hamiltonian
The part of the Hamiltonian that cannot be switched off, \(H_0\) in \(H(t) = H_0 + \sum_k u_k(t)H_k\). A drift outside the control algebra sets intrinsic time scales and makes minimal times positive. → Quantum controllability, Quantum speed limits
Dynamical decoupling
Sequences of fast pulses that average the coupling between a system and its environment to zero, protecting the system from slow noise. → Dynamical decoupling
Ensemble (simultaneous) control
One control that steers a whole family of systems, which differ in an unknown or inhomogeneous parameter, to their targets at the same time — the mathematical basis of robust pulses. → Robust & ensemble optimal control
Figure of merit
The quantity an optimal control problem maximizes or minimizes — a fidelity, an expectation value, an error — often combined with costs on the control or the time. → Problem formulation & figures of merit
Filter function
A function of frequency that measures how strongly a given control sequence lets noise at each frequency affect the system; the error is, to leading order, its overlap with the noise spectrum. → Filter functions & noise-tailored control
Floquet engineering
Designing an effective time-independent Hamiltonian by driving a system periodically, using Floquet theory and high-frequency expansions. → Floquet engineering & periodic driving
GKSL (Lindblad) equation
The general form of a Markovian master equation, \(\dot\rho = -i[H,\rho] + \sum_k \big(V_k\rho V_k^\dagger - \tfrac12\{V_k^\dagger V_k,\rho\}\big)\), named after Gorini, Kossakowski, Sudarshan and Lindblad. → Control of open quantum systems
GOAT
Gradient Optimization of Analytic conTrols: exact gradients for controls given by analytic functions, obtained by integrating the equations of motion of the propagator’s derivatives. → GOAT
GRAPE
GRadient Ascent Pulse Engineering: piecewise-constant controls, with the gradient of the figure of merit with respect to every time slice computed from one forward and one backward propagation. → GRAPE
Krotov's method
An iterative method that updates the whole control sequentially in time and is constructed to improve the figure of merit monotonically at every iteration. → Krotov's method
Leakage
Population leaving the computational subspace, for example to the third level of a transmon. → DRAG & leakage suppression, Superconducting circuits
Lie algebra rank condition
A finite-dimensional closed system is controllable on \(SU(N)\) if the Lie algebra generated by \(iH_0\) and the \(iH_k\) is \(\mathfrak{su}(N)\) (or \(\mathfrak u(N)\)). → Quantum controllability
Measurement-based feedback
Control that uses the outcomes of measurements, typically continuous weak measurements processed by a quantum filter, to choose the next action. → Measurement-based feedback
Piecewise-constant control
A control that is constant on each of a finite number of time slices — the standard parametrization of GRAPE and the natural output of arbitrary waveform generators. → Pulse parametrization & constraints, GRAPE
Pontryagin's maximum principle
Necessary conditions for optimality: an optimal control maximizes a pseudo-Hamiltonian built from the state and a costate that evolves backwards in time. It is the basis of time-optimal and geometric control. → Pontryagin's maximum principle
Quantum speed limit
A lower bound on the time needed for a quantum evolution or a control task. In optimal control it is often the minimal time below which a target cannot be reached with the available controls. → Quantum speed limits
Reachable set
All states (or operations) that can be reached from a given initial condition with admissible controls, in a fixed time or in any time. → Quantum controllability
Reservoir engineering
Designing the coupling of a system to an environment so that dissipation itself drives the system to a desired state. → Reservoir engineering & dissipative preparation
Rotating-wave approximation
Neglecting rapidly oscillating counter-rotating terms in a driven system, valid when the drive is weak and close to resonance. → Problem formulation & figures of merit
Shortcuts to adiabaticity
Protocols that reach the result of a slow adiabatic process in a much shorter time — by counterdiabatic driving, invariant-based inverse engineering or related methods. → Shortcuts to adiabaticity
STIRAP
Stimulated Raman adiabatic passage: complete population transfer in a three-level system through a dark state, using a counter-intuitive order of two pulses. → Adiabatic passage
Time-optimal control
Reaching a target in the shortest possible time with bounded controls or under fixed couplings; its solutions are often bang-bang or singular arcs of Pontryagin’s principle. → Geometric & time-optimal control, Quantum speed limits
Trap
A control that is a local but not a global optimum of the figure of merit; a local optimizer started near it stops there. → Control landscapes

Timeline

A selection of milestones, each with the paper it refers to.

  1. 1950

    Hahn observes the spin echo — refocusing of dephasing by a pulse, the ancestor of dynamical decoupling.

    E. L. Hahn Spin Echoes Phys. Rev. 80, 580-594 (1950) doi
  2. 1954

    Carr and Purcell introduce trains of refocusing pulses; Meiboom and Gill’s modification (1958) makes them robust.

    H. Y. Carr and E. M. Purcell Effects of Diffusion on Free Precession in Nuclear Magnetic Resonance Experiments Phys. Rev. 94, 630-638 (1954) doi
    S. Meiboom and D. Gill Modified Spin-Echo Method for Measuring Nuclear Relaxation Times Review of Scientific Instruments 29, 688-691 (1958) doi
  3. 1979

    Levitt and Freeman use a composite pulse for robust population inversion in NMR.

    Malcolm H. Levitt and Ray Freeman NMR population inversion using a composite pulse Journal of Magnetic Resonance (1969) 33, 473-476 (1979) doi
  4. 1983

    Huang, Tarn and Clark study the controllability of quantum-mechanical systems with Lie-algebraic methods.

    Garng M. Huang, T. J. Tarn and John W. Clark On the controllability of quantum-mechanical systems Journal of Mathematical Physics 24, 2608-2618 (1983) doi
  5. 1985

    Tannor and Rice propose controlling the selectivity of chemical reactions by timing pulse sequences.

    David J. Tannor and Stuart A. Rice Control of selectivity of chemical reaction via control of wave packet evolution The Journal of Chemical Physics 83, 5013-5018 (1985) doi
    David J. Tannor, Ronnie Kosloff and Stuart A. Rice Coherent pulse sequence induced control of selectivity of reactions: Exact quantum mechanical calculations The Journal of Chemical Physics 85, 5805-5820 (1986) doi
  6. 1986

    Brumer and Shapiro propose coherent control through the interference of excitation pathways.

    Paul Brumer and Moshe Shapiro Control of unimolecular reactions using coherent light Chemical Physics Letters 126, 541-546 (1986) doi
  7. 1988

    Peirce, Dahleh and Rabitz formulate optimal control theory for quantum-mechanical systems.

    Anthony P. Peirce, Mohammed A. Dahleh and Herschel Rabitz Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications Phys. Rev. A 37, 4950-4964 (1988) doi
  8. 1992

    Judson and Rabitz propose closed-loop learning control — “teaching lasers to control molecules”.

    Richard S. Judson and Herschel Rabitz Teaching lasers to control molecules Phys. Rev. Lett. 68, 1500-1503 (1992) doi
  9. 1998

    Feedback-optimized femtosecond pulses control a chemical reaction in the laboratory; in NMR, Glaser et al. determine the largest signal that unitary control of spin ensembles can achieve.

    A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, et al. Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses Science 282, 919-922 (1998) doi
    S. J. Glaser, T. Schulte-Herbrüggen, M. Sieveking, O. Schedletzky, N. C. Nielsen, et al. Unitary Control in Quantum Ensembles: Maximizing Signal Intensity in Coherent Spectroscopy Science 280, 421-424 (1998) doi
  10. 1999

    Viola, Knill and Lloyd formulate dynamical decoupling of open quantum systems.

    Lorenza Viola, Emanuel Knill and Seth Lloyd Dynamical Decoupling of Open Quantum Systems Phys. Rev. Lett. 82, 2417-2421 (1999) doi
  11. 2001

    Khaneja, Brockett and Glaser derive time-optimal controls for coupled spins with geometric methods.

    Navin Khaneja, Roger Brockett and Steffen J. Glaser Time optimal control in spin systems Phys. Rev. A 63, 032308 (2001) doi
  12. 2003

    Palao and Kosloff extend optimal control to unitary transformations — quantum gates.

    José P. Palao and Ronnie Kosloff Optimal control theory for unitary transformations Phys. Rev. A 68, 062308 (2003) doi
  13. 2004

    Rabitz, Hsieh and Rosenthal argue that quantum control landscapes are generically free of traps.

    Herschel A. Rabitz, Michael M. Hsieh and Carey M. Rosenthal Quantum Optimally Controlled Transition Landscapes Science 303, 1998-2001 (2004) doi
  14. 2005

    Khaneja et al. introduce GRAPE for NMR pulse design.

    Navin Khaneja, Timo Reiss, Cindie Kehlet, Thomas Schulte-Herbrüggen and Steffen J. Glaser Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms Journal of Magnetic Resonance 172, 296-305 (2005) doi
  15. 2007

    Uhrig finds the optimal pulse timing for protecting a qubit from dephasing.

    Götz S. Uhrig Keeping a Quantum Bit Alive by Optimized π -Pulse Sequences Phys. Rev. Lett. 98, 100504 (2007) doi
  16. 2009

    DRAG pulses suppress leakage in superconducting qubits; optimal control is pushed to the quantum speed limit; transitionless driving is formulated.

    F. Motzoi, J. M. Gambetta, P. Rebentrost and F. K. Wilhelm Simple Pulses for Elimination of Leakage in Weakly Nonlinear Qubits Phys. Rev. Lett. 103, 110501 (2009) doi
    T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Montangero, et al. Optimal Control at the Quantum Speed Limit Phys. Rev. Lett. 103, 240501 (2009) doi
    M V Berry Transitionless quantum driving J. Phys. A: Math. Theor. 42, 365303 (2009) doi
  17. 2010

    Shortcuts to adiabaticity are introduced for fast frictionless cooling of atoms in traps.

    Xi Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Guéry-Odelin and J. G. Muga Fast Optimal Frictionless Atom Cooling in Harmonic Traps: Shortcut to Adiabaticity Phys. Rev. Lett. 104, 063002 (2010) doi
  18. 2011

    The CRAB method makes optimal control practical for many-body dynamics.

    Patrick Doria, Tommaso Calarco and Simone Montangero Optimal Control Technique for Many-Body Quantum Dynamics Phys. Rev. Lett. 106, 190501 (2011) doi
  19. 2015

    The first European roadmap for quantum optimal control is published.

    Steffen J. Glaser, Ugo Boscain, Tommaso Calarco, Christiane P. Koch, Walter Köckenberger, et al. Training Schrödinger’s cat: quantum optimal control Eur. Phys. J. D 69, 279 (2015) doi
  20. 2018

    GOAT gives exact gradients for analytic controls; reinforcement learning is applied to quantum control.

    Shai Machnes, Elie Assémat, David Tannor and Frank K. Wilhelm Tunable, Flexible, and Efficient Optimization of Control Pulses for Practical Qubits Phys. Rev. Lett. 120, 150401 (2018) doi
    Marin Bukov, Alexandre G. R. Day, Dries Sels, Phillip Weinberg, Anatoli Polkovnikov and Pankaj Mehta Reinforcement Learning in Different Phases of Quantum Control Phys. Rev. X 8, 031086 (2018) doi
  21. 2022

    The updated European roadmap for quantum optimal control in quantum technologies is published.

    Christiane P. Koch, Ugo Boscain, Tommaso Calarco, Gunther Dirr, Stefan Filipp, et al. Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe EPJ Quantum Technol. 9, 19 (2022) doi
  22. 2025

    QCPOP finds globally optimal controls with polynomial optimization.

    Denys I. Bondar, Llorenç Balada Gaggioli, Georgios Korpas, Jakub Marecek, Jiri Vala and Kurt Jacobs Globally optimal control of quantum dynamics Phys. Rev. Research 7, 043202 (2025) doi · arXiv:2209.05790