Quantum optimal control ($J=\Phi (\rho (T))+{\int }_0^T\ell (f(t))dt$)

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.

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.

For $J=⟨\psi (T)|A|\psi (T)⟩-(\lambda /2){\int }_0^T{f}^{2}dt$ and $i\hslash \dot{\psi }=(H_0+f{H}_1)\psi$, enforce the state equation with $-2\mathrm{Re}\int ⟨\chi |\dot{\psi }+iH\psi /\hslash ⟩dt$. Variation gives the same Schrödinger equation for the costate, propagated backward from $\chi (T)=A\psi (T)$. Control variation gives the displayed real gradient. Forward state propagation, backward costate propagation, and a constrained gradient update form direct-adjoint looping. These are necessary stationarity equations, not a guarantee of a global optimum.

The Cauchy-Schwarz inequality for the Hilbert-Schmidt inner product gives $0\le J\le 1$ for $n$-dimensional unitary operators. Equality $J=0$ holds exactly when $U=e^{i\varphi }{U}_T$. When absolute gate phase matters, use the phase-sensitive Hilbert-Schmidt distance $\Vert U-U_T{\Vert }^2/(2n)=1-\mathrm{Re}\mathrm{Tr}(U_T^{†}U)/n$ instead.

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.

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.