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For a real quantum optimal control objective $J=⟨⟨A|{\rho }_v(T)⟩⟩-\mathrm{Re}\int ⟨⟨A_v|{\dot{\rho }}_v-K[f]{\rho }_v⟩⟩dt-{\sum }_m{c}_m\int f_m^{2}dt/2$, first variation gives the state equation, the adjoint equation, $A_v(T)=A$, and $c_m{f}_m=\mathrm{Re}⟨⟨A_v|{\partial }_{f_m}K|{\rho }_v⟩⟩$. The adjoint is taken for the Hilbert-Schmidt inner product. With dissipation it is essential to use $K^{†}$, not silently the same generator as the forward state. These are necessary stationarity equations; they do not establish global optimality.

A nonlinear programming problem minimizes an objective over finitely many real variables subject to constraints, with at least one objective or constraint nonlinear. Parametrizing the pulse amplitudes in quantum optimal control produces such a problem. Gradient descent and related local methods exploit differentiability, but a local stationary point need not be a global optimum without additional convexity or other structure.

An always-on Ising coupling produces evolution during every nominally local pulse, and generally does not commute with the applied $x,y$ Quantum Hamiltonians. A naive product of single-qubit rotations therefore accumulates unwanted conditional phases and can entangle the qubits. Idling is also no longer an identity operation.

For a pulse of duration $\tau$, neglecting the coupling requires $|J_{12}|\tau \ll 1$ in units $\hslash =1$, or $|J_{12}|\tau /\hslash \ll 1$ in ordinary units. For order-one rotation angles, this is the strong-local-drive regime $|B|\gg |J_{12}|$. Large drive amplitude alone does not justify neglect over a long total sequence: the accumulated interaction time or an appropriate error bound must also be small.

If it cannot be neglected, include it in the quantum optimal control model, or refocus it. A $\pi$ pulse $P=R_x^{(1)}(\pi )$ obeys $P(Z\otimes Z)P^{†}=-Z\otimes Z$, giving the ideal Ising spin echoThis cancellation assumes the refocusing pulses are instantaneous relative to $1/|J_{12}|$, or that the coupling during finite pulses is compensated. It explains how fast local controls can suppress the fixed coupling without mistaking it for a controllable zero interaction.

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.

Three standard objectives are state preparation, observable optimization and process implementation. All optimizations are over admissible control inputs and a specified final time $T$.

For state preparation, steer an initial density operator to a target ${\rho }_d$. An exact goal is ${\rho }_f(T)={\rho }_d$, or one may minimize the squared Hilbert-Schmidt distanceFor a pure target ${\rho }_d=|{\psi }_d⟩⟨{\psi }_d|$, maximizing $⟨{\psi }_d|{\rho }_f(T)|{\psi }_d⟩$ is a useful alternative. The target must be physically attainable: Hamiltonian engineering alone preserves the initial spectrum of the density operator.

For an observable objective, choose a Hermitian operator $O$ and maximize its expected final value:Population transfer is a special case with $O$ the projector onto the desired level or subspace; suppressing an unwanted population reverses the sign or minimizes this quantity.

For coherent process engineering, implement a target unitary operator $V$ on every input, rather than merely arranging one state transfer. If the propagator is $U_f(T)$, a phase-insensitive error isThe error vanishes precisely when $U_f(T)=e^{i\varphi }V$: equality in the Hilbert-Schmidt inner product Cauchy-Schwarz inequality bound makes the two operators proportional. For an open-system process, the analogous objective compares the implemented quantum channel with the target channel. These objectives can be supplemented by pulse-energy, amplitude, bandwidth and duration constraints in quantum optimal control.

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.

Quantum control chooses externally adjustable interactions to steer a quantum state or implement a desired unitary operator. Its achievable transformations depend on the Hamiltonian or Lindblad equation, the available control inputs, and physical constraints such as bandwidth and amplitude. Quantum optimal control optimizes performance within those constraints.

Quantum optimal control minimizes a state-transfer, observable, or gate error together with resource costs, subject to a controlled Schrödinger equation or Lindblad equation. Forward propagation of the state and backward propagation of a costate compute the gradient efficiently. The optimum depends on the admissible time-dependent fields and laboratory constraints.