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Parametrize a reproducible pulse by a finite vector $a$ and evaluate its objective experimentally on freshly prepared systems. Repeatedly update the pulse from measured objective values or finite-difference derivatives, including penalties for resources. This is closed-loop control across experimental runs, even when each pulse is open-loop control within a run. It reduces reliance on an inaccurate dynamical model but still requires preparation reproducibility, sufficient measurement statistics and stability of the experimental conditions.

Feedback control selects a control input using observations of the actual controlled plant. In a tracking loop, compare the reference signal $r$ with the measured output $y_m$ to form the tracking error $e=r-y_m$, then let the controller change $u$. The control input changes the plant output, which is measured again: that return path closes the loop. A controller can use earlier observations as well as the latest one. ^[ref]

For the ideal controlled plant, proportional control $u=2(1-x)$ initially commands $u(0)=2$. As $x$ approaches $1$, the command shrinks. When a positive control disturbance pushes $x$ above $1$, the tracking error becomes negative and the control input pushes back. The result is a bounded offset rather than the continuing drift of open-loop control. Integral action can remove that offset by remembering past tracking errors.

A control selected from the observed current state, rather than specified only as a function of the initial state and time. For $x_{t+1}=x_t+u_t$ and stage cost $x_t^2+2x_t{u}_t+7u_t^2$, the infinite-horizon feedback $u_t=-x_t/3$ gives $x_t=(2/3{)}^t{x}_0$ and total cost $2x_0^2$. The finite-horizon Bellman coefficients satisfy ${\Pi }_0=0$, ${\Pi }_s=6(1+{\Pi }_{s-1})/(7+{\Pi }_{s-1})$ and increase to two, proving optimality.

A controller is the rule or mechanism that chooses a control signal. An open-loop control can depend on a precomputed schedule; a feedback control also uses a measured output and can have its own memory, as in integral action.

Control theory studies how to choose control inputs for a dynamical system so that its plant output follows a reference signal, despite control disturbances and imperfect knowledge of the controlled plant. A controller turns the available information into a control signal. The design must consider both the resulting response and stability theory: a rule that corrects a small mismatch can still produce growing deviations if the interconnection is badly chosen. ^[ref]

The central distinction is informational. Open-loop control executes planned commands; feedback control uses the measured output of the actual controlled plant to update those commands. A reference signal can describe a destination, while control disturbances describe influences that the controller did not command. A model can predict where the controlled plant should be; measurement tells the controller what actually happened. The authors provide a free full-book PDF through ^[ref].

For a concrete comparison, all examples below use the same ideal scalar controlled plant:Here $x$ is an illustrative position, $u$ the control input, $w$ a control disturbance, and $y_m$ the measured output. Both $x$ and $t$ are dimensionless quantities; the numbers are chosen to expose the mechanism, with no empirical interpretation. The ordinary differential equation says that the present control input and control disturbance accumulate into $x$. The measured output equals the entire state only in this ideal example; a general plant output need not reveal the entire state.

Hamiltonian engineering designs accessible interactions or their effective time average to implement desired quantum states or unitary operators. The available control inputs, Quantum Hamiltonian generators and laboratory constraints determine which transformations can actually be achieved. Model-based open-loop control computes a waveform before running it; adaptive experiments instead evaluate its performance on the physical system.

For $\dot{\rho }=-i[H_0+f{H}_1,\rho ]$ and ${\dot{\rho }}_d=-i[H_0,{\rho }_d]$, the Hilbert-Schmidt distance gives $V=\Vert \rho -{\rho }_d{\Vert }^2/2$. The choice $f=\mathrm{Tr}({\rho }_d[-i{H}_1,\rho ])$ yields $\dot{V}=-f^2$. Decrease alone does not prove convergence: orthogonal pure eigenstates of a diagonal $H_0$ give zero input for a transverse Pauli control even though their distance remains nonzero. Such a law can be simulated to produce an open-loop control.

An open-loop control is prescribed as a function of time, possibly after calculation from an initial state and a dynamical model. It is executed without using observations of the actual current state to change the input. A simulated feedback control law can generate an open-loop waveform, but the resulting laboratory control does not automatically correct errors in the actual state or model.

For the ideal controlled plant with $w=0$, plan the control inputIntegration gives $x(t)=\min (t,1)$, so the planned command reaches the reference signal $r=1$ at $t=1$. If an unplanned constant control disturbance $w=0.2$ begins at $t=2$, the same open-loop control gives $x(t)=1+0.2(t-2)$ for $t\ge 2$: $x(4)=1.4$ and $x(8)=2.2$. Knowing the nominal response did not provide a way to correct the actual plant output.

The Von Neumann equation integrates to $\rho (t)=U(t)\rho (0)U(t{)}^{†}$, so the spectrum and purity of a density operator are invariant under any allowed open-loop control. These are the kinematic restrictions.

There is no additional orbit restriction here: the generators $i{\sigma }_z$ and $i{\sigma }_x$, together with their matrix commutator, span the special unitary Lie algebra:For freely chosen durations and piecewise continuous real control values, this gives density operator controllability. The connected reachable group is $SU(2)$; its action by conjugation is the same as that of $U(2)$, because a global phase cancels. Thus exactly the pairs ${\rho }_0\leftrightarrow {\rho }_2$ and ${\rho }_1\leftrightarrow {\rho }_3$ can be interconverted. No coherent control can turn either mixed state into either pure state, even though every orientation within each spectral orbit is reachable.

An open-loop control is computed from a dynamical model and the initial preparation, then applied without changing it in response to measurements during that run. Controllability determines which transformations are possible in principle; field design must also account for duration, available control generators, amplitude and bandwidth limits, and calibration errors.

Three useful model-based strategies are the following. Resonant quantum control chooses carrier frequencies and phases to select transitions, and uses quantum pulse area or elementary rotation gates to compile the desired transformation. Adiabatic quantum control chooses a slowly varying path of Quantum Hamiltonians whose relevant eigenstate connects the initial and target states, maintaining a sufficiently large spectral gap. Quantum optimal control parametrizes or varies the fields and maximizes a fidelity or observable objective, while penalizing resources and enforcing the dynamical equations. These can all produce predetermined laboratory waveforms; experimental feedback is a separate design choice.

For a detailed variational construction, letChoose a positive target observable $O$, for example $O=|{\psi }_d⟩⟨{\psi }_d|$ for pure-state transfer. MaximizeThe terminal term is the target probability or observable expectation, and the quadratic integral penalizes field energy. A costate $\chi (t)$ enforces the Schrödinger equation through the real augmented functionalVariation in $\chi$ recovers the state equation. Integrating the variation in $\dot{\psi }$ by parts gives the terminal boundary term $2\mathrm{Re}⟨O\psi (T)-\chi (T)|\delta \psi (T)⟩$ and the interior adjoint equation. Thus the Euler-Lagrange conditions areThe costate obeys the same Hamiltonian equation but is propagated backward from its terminal value. The field variation isHence the variational costate gradient for Hamiltonian quantum control isAt an unconstrained stationary field,These equations couple forward and backward boundary data, so this is not a single initial-value integration, nor a direct feedback law for an unknown experimental state.

A practical direct-adjoint looping algorithm starts from an admissible trial field. Propagate $\psi$ forward and store its trajectory; set $\chi (T)=O\psi (T)$ and propagate $\chi$ backward; compute $g_k$ along the two trajectories; then update $u_k^{\mathrm{new}}=u_k+\eta g_k$ for a positive step size $\eta$. A line search evaluates the actual objective and decreases $\eta$ until the ascent condition is met. Project into the admissible amplitude range or optimize a finite Fourier or pulse basis to enforce bandwidth constraints. Repeat until the projected gradient and change in objective are small. Repeated initializations help explore different local optima; the stationarity equations alone do not guarantee a global optimum.

For a piecewise constant discretization, define $H_j=H_0+{\sum }_k{u}_{k,j}{H}_k$ and $U_j=e^{-i\Delta t{H}_j/\hslash }$. Forward multiplication gives the state trajectory; backward multiplication gives the costate. The exact matrix-exponential derivative isThis supplies accurate discrete gradients even when $H_j$ and $H_k$ do not commute. The small-step expression $\partial U_j/\partial u_{k,j}\simeq -i\Delta t{H}_k{U}_j/\hslash$ is only an approximation. Forward/backward factorization is the basis of gradient ascent pulse engineering. For a full-gate objective, propagate all basis columns or $U(t)$ itself and use a terminal objective such as the phase-insensitive unitary gate error; optimizing one state alone does not implement a prescribed operation on every input.

To implement the result, convert the optimized envelope and quadratures into the calibrated electric or magnetic drive fields. An arbitrary-waveform source can set time-domain quadratures; optical spectral pulse shaping applies a complex mask to the available pulse spectrum,using, for example, a 4f pulse shaper and spatial light modulator. Finite spectral support, mask resolution and actuator limits must be included in the optimization: smoothing an unconstrained optimum afterward may spoil it. Simulating the calibrated final pulse, optionally over an ensemble of detunings or calibration errors, checks its predicted fidelity and robustness. The computed fields are then executed as open-loop control, with all measurements used for later calibration rather than an assumed instantaneous state readout.

In adaptive experimental quantum control, parametrize a reproducible pulse by a finite real vector $c$, for example $f_c(t)={\sum }_{j=1}^p{c}_j{B}_j(t)$ in a calibrated waveform basis. Prepare the same initial state for each trial, apply that pulse without feedback during the run, and estimate the objective $J(c)$ from repeated measurements. No accurate dynamical model is required if the laboratory can evaluate the objective.

One concrete algorithm is experimental finite-difference gradient ascent. For every parameter, run trials at $c\pm \delta e_j$ and estimateUpdate $c$ to an admissible projection of $c+\eta g$, test the new pulse, and reduce the step size if performance deteriorates. Averaging many shots controls statistical noise; choose $\delta$ large enough for the difference to exceed the measurement uncertainty but small enough to approximate a derivative. Repeat until no statistically significant improvement remains. Multiple starting pulses help explore distinct basins, but this does not guarantee a global optimum. Resource penalties and hardware restrictions belong in the objective or the admissible parameter set.

The feedback is across experimental runs: a measured objective chooses the next pulse. Each individual trial is open-loop control. This distinction matters because the algorithm does not require an instantaneous nondestructive state measurement on the system being controlled.

The deterministic law cannot be transplanted unchanged into measurement-based quantum feedback. Evaluating its state-dependent scalar requires knowledge of the current quantum state. In fact $C=\mathrm{Tr}(i[H_1,{\rho }_d]\rho )$ is the expectation value of a Hermitian observable, but that expectation cannot be obtained exactly from a single system without measurement backaction. Tomography uses many independently prepared copies, not a nondisturbing readout of one evolving state.

Continuous weak measurements can supply a noisy record and a conditional state estimate, but then the state follows stochastic measurement dynamics with additional disturbance and decoherence, rather than the closed unitary equation used in part (a). A new stochastic stability analysis is required. One may instead simulate the corrected Lyapunov quantum control from known initial data and apply its waveform as open-loop control, or learn a pulse across repeated experiments as in Question 3. Those procedures do not constitute real-time state feedback on a single quantum trajectory. The separate source error in part (a) remains: even ideal knowledge of the state would not make its printed overlap law monotone.

The target specifies the desired output; the controller selects a control input; an actuator realizes it on the system; and the sensor returns a measured output. The environment supplies additional interactions and disturbances. A simple loop is:

For a quantum sensor, a measurement in quantum mechanics has statistical outcomes and generally changes the state through measurement backaction. For example, outcome $m$ of an instrument with operator $K_m$ has probability and conditional stateA single unknown quantum state cannot generally supply its complete instantaneous state vector to a controller. Noncommuting observables cannot be read simultaneously with arbitrary precision, and reliable quantum state tomography normally needs many reproducible preparations. A conditional state estimate can instead be updated from a continuous record, with its disturbance included in the model.

Quantum actuators change a Quantum Hamiltonian through applied electromagnetic fields, tunable interactions or couplings to auxiliary systems. A closed-system actuator implements unitary operators, so it preserves the spectrum and purity of a density operator; it cannot arbitrarily overwrite a state as a classical assignment operation could. Dissipative actuators or measurement in quantum measurements can change these invariants, but their dynamics must be accounted for.

The environment can entangle with the system, causing decoherence and dissipation when unobserved degrees of freedom are traced out. It is consequently part of the dynamical model, rather than just an additive classical disturbance. Conversely, Markovian reservoir engineering uses such coupling constructively to prepare and stabilize states.

In open-loop control the applied waveform is prescribed without using measurements from the current evolution. In closed-loop control it depends on the available measurement record or inferred state. Measurement-based quantum feedback is one realization of the latter; experimental adaptation across repeated preparations is another. The measurement record is classical information, but its production has quantum backaction.

Hamiltonian engineering chooses accessible interactions so that their actual or effective Quantum Hamiltonian produces the desired evolution. A typical model is $H_f(t)=H_0+{\sum }_k{f}_k(t)H_k$; the controls may change frequencies, amplitudes, phases or coupling strengths. Three standard model-based open-loop control approaches illustrate different mechanisms.

In resonant quantum control, select a transition with a near-resonant field. In a rotating frame, after neglecting rapidly oscillating terms in a justified regime, an effective two-level drive can be $H_{\mathrm{eff}}(t)=\Omega (t){\sigma }_x/2$. When its axis is fixed,The pulse area sets the rotation angle and the drive phase selects the transverse axis. A resonant pulse of area $\pi$ swaps the two level populations. This requires knowledge of transition frequencies, couplings and the validity of the selective-drive approximation.

In adiabatic quantum control, design a slowly varying Quantum Hamiltonian whose desired instantaneous eigenstate connects the initial and final states. For a nondegenerate eigenstate separated by a gap, the quantum adiabatic theorem explains why sufficiently slow evolution follows that state up to a phase. The relative size of matrix elements of $\dot{H}$ and squared spectral gaps controls the approximation; degeneracies or rapid changes invalidate simple following. This engineers a state path rather than a prescribed resonant rotation.

In average Hamiltonian engineering, apply a predetermined sequence of fast pulses $P_j$ and let the system evolve for intervals ${\tau }_j$ in the corresponding frames. For cycle length $T_c={\sum }_j{\tau }_j$, the leading effective generator isChoose the sequence to cancel unwanted terms or retain desired couplings. For instance, equal intervals under $H_0=\omega {\sigma }_z/2$ and its ${\sigma }_x$ conjugate cancel the leading generator because ${\sigma }_x{\sigma }_z{\sigma }_x=-{\sigma }_z$. Noncommuting terms produce higher-order commutator corrections, so fast cycling, finite pulse errors and the accuracy of the model matter. All three methods determine a waveform or sequence before its execution; none automatically corrects an unknown mismatch during that run.

The Lyapunov quantum control law requires an exact instantaneous expectation value. Cyclicity writes it aswhere $A(t)$ is a known Hermitian observable. A measurement on one system produces a random outcome, not its exact ensemble expectation, and changes the system state. Repeated measurements for state estimation or continuous weak measurements introduce noise and backaction absent from the assumed closed-system equation. Thus the law cannot be implemented unchanged as exact measurement-based quantum feedback while retaining the dynamics and guarantee in part (a). A conditional-state feedback scheme must include the measurement evolution and rederive its stability properties.

If the initial density operator and Hamiltonian model are known, we can instead solve the deterministic equations in a computer, calculate $f(t)$ along the simulated trajectory, and apply that precomputed waveform in the laboratory as open-loop control. Ensemble measurements can help calibrate such a waveform, but they do not supply nondisturbing instantaneous feedback from an individual evolving system.

Spectral pulse shaping implements a calculated optical control by modifying its complex frequency spectrum. If the available input pulse has spectrum ${\tilde{E}}_{\mathrm{in}}(\omega )$, program a calibrated mask $M(\omega )=a(\omega )e^{i\phi (\omega )}$ so thatThe temporal waveform follows from Fourier inversion. For a real applied field, its spectrum has conjugate symmetry, so the mask must satisfy $M(-\omega )=M(\omega {)}^{\ast }$. Spectral phase controls interference between frequencies and hence their arrival times, chirp and temporal subpulses; spectral attenuation changes their weights. The optimized field envelope is translated into this complex mask, with compensation for known phase shifts and losses in the optical system.

A typical 4f pulse shaper contains a first diffraction grating, a focusing lens, a mask or spatial light modulator in the common focal plane, a second lens and a second grating. The first grating separates the frequencies by angle. The first lens converts those angles into different transverse positions at the spectral Fourier plane. The pixels of the spatial light modulator modify the amplitude and phase of the corresponding frequency bins. The second lens and grating reverse this spatial dispersion and recombine the components into one shaped output beam. In the ideal symmetric layout the four consecutive propagation distances are $f,f,f,f$, giving the name $4f$.

A phase-only spatial light modulator needs suitable polarization optics or another modulation stage for independent amplitude shaping. A passive mask has $0\le a(\omega )\le 1$ and cannot create frequencies absent from the input; finite bandwidth and finite pixel resolution limit temporal features and the useful shaping window. These constraints must be included when converting a mathematical optimum into a laboratory pulse. The programmed mask is an open-loop control, although repeated measurements of the output pulse or system performance can be used to calibrate or optimize it experimentally.