Field map

Foundations

The mathematical framework — what can be controlled, how optimality is defined, and what the optimization problem looks like.

Quantum controllability Article Which states or operations are reachable — answered by the dynamical Lie algebra generated by the drift and control Hamiltonians. Problem formulation & figures of merit Article How a control task is written as an optimization problem — dynamics, fidelity measures, penalties and constraints. Pontryagin's maximum principle Article Necessary conditions for optimality of a control, the starting point for both analytic solutions and gradient methods. Geometric & time-optimal control Article Exact optimal solutions for low-dimensional systems using differential geometry, including minimum-time and singular controls. Control landscapes Article The topology of the fidelity as a function of the control — why local optimizers so often find global optima, and when traps appear.

Speed limits & complexity

Fundamental bounds on how fast and with how many resources a quantum system can be steered.

Quantum speed limits Article Lower bounds on the time needed to evolve between states, and their role as the minimal time reachable by optimal control. Control complexity & resources Article How much information, bandwidth or how many control parameters are needed to reach a target — and how this scales with system size.

Numerical algorithms

Optimization methods that compute control pulses — the computational workhorses of the field.

GRAPE Article Gradient Ascent Pulse Engineering — piecewise-constant controls updated with exact gradients of the fidelity. Krotov's method Article A sequential, monotonically convergent update scheme with continuous-time controls and built-in update penalties. CRAB & dCRAB Article Chopped RAndom Basis — expand the control in a small, randomized basis and optimize the few coefficients, often gradient-free. GOAT Article Gradient Optimization of Analytic conTrols — exact gradients for arbitrary smooth, parametrized pulse shapes. QCPOP & polynomial optimization Article Quantum Control via POlynomial oPtimization — recast the control problem as a polynomial optimization and solve it globally, with certificates of optimality where the relaxation is tight. Quasi-Newton & second-order methods Article Using curvature information (BFGS, Newton) to accelerate convergence of gradient-based pulse optimization. Automatic differentiation & differentiable programming Article Computing control gradients with machine-learning frameworks, enabling complex cost functions and models with little derivation effort. Gradient-free & global optimizers Article Nelder–Mead, CMA-ES, evolutionary and swarm methods — useful when gradients are unavailable, noisy or misleading. Pulse parametrization & constraints Article Choosing how controls are represented, and enforcing experimental limits on amplitude, bandwidth, smoothness and transfer functions. Numerical propagation & simulation Article Computing the time evolution — and its derivatives — fast and accurately enough for optimization, from matrix exponentials to Magnus, Chebyshev and Krylov propagators.

Analytic & adiabatic design

Pulse design from closed-form or physically motivated constructions rather than numerical search.

Shortcuts to adiabaticity Article Reach the result of a slow adiabatic process in finite time — counterdiabatic driving, invariant-based engineering, fast-forward. Adiabatic passage Article Robust population transfer by slowly varying the Hamiltonian — rapid adiabatic passage, STIRAP and their variants. DRAG & leakage suppression Article Derivative Removal by Adiabatic Gate — analytic corrections that suppress leakage and phase errors in weakly anharmonic qubits. Floquet engineering & periodic driving Article Designing effective Hamiltonians with periodic drives — high-frequency expansions, average Hamiltonian theory and parametric driving. Lyapunov, tracking & model-predictive control Article Constructive control laws from classical control theory — Lyapunov functions, trajectory tracking and model-predictive control — including feedback-based quantum algorithms.

Robust control

Making controls insensitive to noise, drifts, miscalibration and inhomogeneity.

Robust & ensemble optimal control Article Optimizing a pulse to perform well over a range of uncertain parameters — sampling, averaging and sensitivity-based approaches. Composite pulses Article Sequences of simple rotations whose systematic errors cancel to high order — BB1, CORPSE, SCROFULOUS and relatives. Dynamical decoupling Article Pulse sequences that average away unwanted couplings to the environment — spin echo, CPMG, XY-n, UDD. Filter functions & noise-tailored control Article Frequency-domain description of how a control sequence filters noise, used to design noise-robust pulses and do noise spectroscopy.

Learning & closed-loop control

Optimizing directly on the experiment, and using machine learning to find or adapt controls.

Closed-loop & in-situ optimization Article Optimizing controls with fidelity estimates measured on the real device, sidestepping model errors. Reinforcement & machine learning for control Article Agents and neural networks that learn control policies or pulses, from simulation or directly from experimental data. System identification for control Article Learning the Hamiltonian, noise and hardware distortions well enough that model-based optimal pulses work on the device.

Open systems & feedback

Control in the presence of an environment, of dissipation, and of measurement.

Control of open quantum systems Article Optimal control of dissipative dynamics (Lindblad and non-Markovian), and fighting or exploiting decoherence. Reservoir engineering & dissipative preparation Article Designing the coupling to an environment so that dissipation itself drives the system to the desired state. Measurement-based feedback Article Steering a system using real-time measurement records — quantum filtering, Markovian and Bayesian feedback. Linear quantum systems & coherent feedback Article Systems-engineering theory for quantum networks of oscillators and fields — input–output models, coherent feedback, LQG and H-infinity control.

Many-body & large systems

Controlling systems whose state space grows exponentially, and the bridge to quantum algorithms.

Many-body optimal control Article Optimal control of interacting many-particle systems, often combined with tensor-network simulations. Variational algorithms & QAOA as control Article Viewing variational quantum algorithms and QAOA as optimal control problems — and what each field can learn from the other. Bosonic & continuous-variable control Article Controlling infinite-dimensional systems — oscillators, cavities and their nonlinear couplings — to prepare cat, GKP and other non-Gaussian states.

Control tasks

What the controls are for — gates, states, sensing, thermodynamic cycles, error correction.

Quantum gate synthesis Article Designing pulses that implement target unitaries with high fidelity — single- and multi-qubit gates and leakage control. State preparation & transfer Article Driving a system into a target state — from single-qubit transfers to entangled and many-body states. Control for sensing & metrology Article Using optimized controls to maximize sensitivity — e.g. quantum Fisher information — in the presence of noise. Control in quantum thermodynamics Article Optimal driving of quantum heat engines, refrigerators and batteries — power, efficiency and entropy production trade-offs. Control for quantum error correction Article Pulse design for QEC primitives — fast syndrome measurement, fault-tolerant operations and bosonic-code control. Pulse-level compilation Article Turning quantum circuits into hardware pulses — pulse libraries, optimal-control compilation of gate blocks, and pulse-level programming interfaces.

Physical platforms

How the ideas are realized in specific hardware, each with its own Hamiltonians and constraints.

Superconducting circuits Article Microwave control of transmons and other circuits — leakage, crosstalk, flux pulses and distortion are central concerns. Trapped ions Article Laser and microwave control of ions in electromagnetic traps — entangling gates via motional modes and shuttling. Neutral atoms & Rydberg arrays Article Optical tweezer arrays with Rydberg interactions — optimal gates and many-body state preparation. Ultracold atoms & BECs Article Controlling Bose–Einstein condensates and atoms in optical lattices — transport, splitting and many-body state preparation. Spins — NMR, ESR & colour centres Article Magnetic resonance and solid-state spins such as NV centres — the original home of GRAPE and robust pulse design. Molecules & chemical control Article Laser control of molecular dynamics and reactions — one of the historical roots of quantum optimal control. Photonic & optomechanical systems Article Control of light, cavities and mechanical resonators — photon generation, routing and bosonic modes. Control electronics & software stacks Article The classical machinery that delivers pulses — signal generation, FPGA and real-time controllers, calibration software and control stacks.

Outline