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Gradient ascent pulse engineering parametrizes each control by time-slice amplitudes and optimizes a differentiable quantum-control objective. Forward products of slice unitary operators and backward adjoint propagation give derivatives with respect to every amplitude. For $U_j=e^{-i{H}_j\Delta t}$, the exact derivative is ${\partial }_{f_m}{U}_j=-i{\int }_0^{\Delta t}{e}^{-i{H}_j(\Delta t-\tau )}{H}_m{e}^{-i{H}_j\tau }d\tau$. Updating all slice amplitudes along these derivatives is gradient ascent for a fidelity, or gradient descent for its negative.

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.

For a concrete variational method, maximize the terminal-observable objective with a fluence penalty,where $\lambda >0$ weights the control cost. Enforce the dynamics with a complex costate $|\chi (t)⟩$ and the real augmented functionalVariation of $\chi$ restores the state equation. Integrate the $\dot{\psi }$ variation by parts; because $\delta \psi (0)=0$, the terminal coefficient gives $\chi (T)=A\psi (T)$, and the interior coefficient givesThis equation has final data, so it is integrated backward. Variation of the real control gives the variational costate gradient for Hamiltonian quantum controlIn particular an unconstrained stationary pulse satisfies $f(t)=2\mathrm{Im}⟨\chi |H_1|\psi ⟩/(\lambda \hslash )$. The state, adjoint and control equations form a coupled two-boundary-value problem, so the stationarity formula is not an independent explicit solution for $f$.

Use direct-adjoint looping: start from an admissible pulse, propagate the state forward, set the terminal costate, propagate it backward, compute the gradient on the time grid, and update the pulse with an ascent step and line search. Project amplitude bounds and impose bandwidth constraints within the parametrization or update. Repeat until the projected gradient is small. For gate objectives the forward variable is the propagator and the costate is matrix-valued; for mixed-state objectives it is the density operator. Gradient ascent pulse engineering applies the same efficient forward/backward differentiation to time-slice amplitudes. The variational conditions identify local stationary controls and require numerical convergence checks; they do not prove global optimality of a nonconvex control landscape.

A quantum optimal control problem specifies a terminal task, admissible fields and resource costs. Pure-state transfer can minimize $1-|⟨{\psi }_d|\psi (T)⟩{|}^2$; mixed-state transfer on an accessible spectral orbit can minimize $\Vert \rho (T)-{\rho }_d{\Vert }_{\mathrm{HS}}^2/2$; preparing an observable value can minimize $-\mathrm{Tr}(O\rho (T))$. A gate task can minimize $1-|\mathrm{Tr}(U_d^{†}U(T)){|}^2/n^2$ for an $n$-dimensional Hilbert space, which is insensitive to an overall phase. These different terminal functions encode different tasks even when the dynamics are the same. Fluence penalties, amplitude bounds, bandwidth limits, or the total duration can be added to express experimental requirements.

For example, minimizesubject to $\dot{\rho }={\mathcal{L}}_f(\rho )$ and a specified $\rho (0)$. For a closed system, ${\mathcal{L}}_f(\rho )=-i[H_0+{\sum }_m{f}_m{H}_m,\rho ]$; for an open system add the known Lindblad equation dissipator. Introducing a Hermitian costate $\Lambda$ and taking the Hilbert-Schmidt inner product gives the forward and backward equations and functional gradientFor Hamiltonian controls the second term is $\mathrm{Tr}(\Lambda [-i{H}_m,\rho ])$. The adjoint is defined by the Hilbert-Schmidt inner product; in particular, the closed-system costate equation is $\dot{\Lambda }=-i[H,\Lambda ]$, integrated backward from its terminal condition. An unconstrained interior optimum has zero gradient; amplitude constraints replace this by the corresponding constrained first-order conditions.

Numerically, represent the fields by piecewise constant amplitudes $f_{m,j}$ on slices, or by a finite basis $f_m(t)={\sum }_j{a}_{mj}{b}_j(t)$ such as splines or Fourier modes. For piecewise constant closed-system dynamics, $U_j=e^{-i{H}_j\Delta t}$ and $U(T)=U_K\cdots U_1$. Forward products and backward costates provide all slice derivatives in one pair of propagations. In gradient ascent pulse engineering, update these amplitudes to improve the chosen objective, with projection onto amplitude bounds if needed. The exact slice derivative is the derivative of the matrix exponentialReplacing it by $-i\Delta t{H}_m{U}_j$ is only a short-slice approximation when $H_j$ and $H_m$ do not commute. Other gradient descent or nonlinear optimization methods can optimize the same finite parameter set; direct-adjoint looping avoids a separate dynamical simulation for each control coefficient.

A pulse is accepted after propagation with the actual constrained waveform confirms its terminal error and resource use. Sampling model uncertainties in the objective can favor robust pulses. Nonconvexity means a numerical stationary point need not be a global optimum, so initial guesses, convergence tolerances and a sufficiently resolved time grid matter.