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Use geometric quantum state synthesis by adjacent rotations, rather than three full level population swaps. Each resonant pulse implements a complex Givens rotation on one allowed two-state plane. Start with the target amplitude vector, use such rotations to eliminate entries back to the first basis vector, and reverse that sequence. Its angles are fixed by the ratios of target amplitudes, and optical phases supply their relative complex phases.

Here is an explicit real-coordinate construction in the interaction picture. Define a phase-calibrated rotation byA resonant Quantum Hamiltonian proportional to $-i|j⟩⟨k|+i|k⟩⟨j|$ realizes this rotation; the optical phase chooses that quadrature. Let $\beta =\arccos (1/\sqrt{3})$. ThenThe pulse areas equal these rotation parameters in the direct Hamiltonian-coefficient convention of quantum pulse area. A given sign or phase choice for a physical transition depends on its dipole and basis conventions; calibration realizes the specified rotation without assuming every adjacent transition has the same energy ordering.

If the requested amplitudes refer to the laboratory frame at $t_F$, the target for this construction is $U_0(t_F,0{)}^{†}|\Psi ⟩$, not simply $|\Psi ⟩$. Set the rotation phases accordingly, or add the equivalent accessible diagonal phase corrections. This compensates the known drift phases and gives the stated laboratory superposition, not merely the correct level populations. Strong regularity and connected allowed transitions guarantee the required controllability; bandwidth restrictions set how slowly the selective approximation must be implemented.

With $B_y=0$, a pulse of the $x$ field realizes $R_x(\theta )=e^{-i\theta {\sigma }_x/2}$ when $\int B_x(t)dt=\theta /2$. Similarly the $y$ field realizes $R_y(\theta )$ with area $\theta /2$. These are instances of the quantum pulse area rule. A rotation about the z-axis can be synthesized from the two available axes:because $R_x(\pi /2){\sigma }_y{R}_x(-\pi /2)={\sigma }_z$.

For a target $V=\left(\begin{array}{cc}u& v\\ -v^{\ast }& u^{\ast }\end{array}\right)$, one convenient construction isTake $\beta =2atan2(|v|,|u|)$ and choose $\alpha +\gamma =-2\arg u$, $\alpha -\gamma =-2\arg (-v)$ when both entries are nonzero; if an entry vanishes its phase constraint can be omitted. Substitute the three-pulse construction for each $z$ rotation. Apply the factors from right to left in time. Pulses along the available $x,y$ axes therefore implement every single-qubit gate up to global phase. Negative angles use negative field area, or an equivalent full-period rotation.

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.

The Pauli matrices obey ${\sigma }_x^2={\sigma }_y^2=I$ and ${\sigma }_x{\sigma }_y+{\sigma }_y{\sigma }_x=0$, so $A=\cos \varphi {\sigma }_x+\sin \varphi {\sigma }_y$ has $A^2=I$. Its entries are $A_{12}=\cos \varphi -i\sin \varphi =e^{-i\varphi }$ and $A_{21}=e^{i\varphi }$, with zero diagonal. Substitute this matrix into the involution exponential formula:The drive phase selects the transverse rotation axis, while the quantum pulse area selects its rotation angle.