 Solution

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.