= Solution
Replace evaluation in a possibly inaccurate dynamical model by an experimental objective. Parametrize an admissible pulse as $f(t;a)$, with $a$ collecting time-slice amplitudes or basis-function coefficients. Prepare the same initial state, apply the pulse, measure performance, and update $a$ before the next run. This is <adaptive experimental quantum control>: the loop closes through experimental results, even though each individual pulse is applied open loop.
For example, minimize
$$
\boxed{J_{\mathrm{exp}}(a)=1-\operatorname{Tr}[\rho_d\rho_a(T)]
+\eta\int_0^T\|f(t;a)\|^2\,dt,\qquad a\in\mathcal A,}
$$
for a pure target $\rho_d$, with a nonempty closed <convex set> $\mathcal A$ enforcing amplitude and bandwidth bounds. The overlap is an experimentally estimable <probability>. An <observable> objective can use its measured expectation instead; a gate objective generally needs more input states and measurement settings. The true dynamics constrain the physically produced $\rho_a(T)$, but no accurate formula for those dynamics is required to evaluate this objective.
A direct search compares measured scores for candidate pulses, accepts improvements and refines the search region. Alternatively estimate a <gradient> experimentally by finite differences,
$$
\widehat{\partial_{a_k}J}=
\frac{\widehat J(a+\varepsilon e_k)-\widehat J(a-\varepsilon e_k)}{2\varepsilon},
\qquad a_{n+1}=\Pi_{\mathcal A}(a_n-\gamma_n\widehat{\nabla J}),
$$
where $\Pi_{\mathcal A}$ is the <Euclidean projection onto a convex set>. Population-based searches can also explore several candidates in parallel. These methods use experimental data rather than simulated propagation or a model-derived adjoint <gradient>.
Each score needs enough repeated preparations to manage measurement noise; uncertainty in score differences should determine replication and stopping decisions. Excessively small finite-difference steps amplify that noise. A model-based pulse can provide a useful starting point, but adaptation corrects its systematic mismatch using the device itself. This mitigates model uncertainty, not arbitrary fluctuations: reproducible preparation, calibrated measurements and sufficiently slow drift remain necessary. Nor does a local numerical search guarantee a global optimum.
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