State-Consuming Experimental Design: Greedy Failure and Amortized Planning

An experimental design action normally reveals information and leaves the system available for the next action. A quantum measurement does not: it consumes the state that later measurements needed. We show this is not a quantitative nuisance but a change of problem class. Under back-action the design utility stops being a set function and becomes a function of the measurement sequence, and we prove that greedy selection admits no constant-factor shot-cost guarantee over the stated family with no duration or resource restriction: on an explicit two-parameter family, for every fixed regularizer $\lambda >0$ and every finite $C$ some member forces shot overhead above $C$, with the overhead tending to $1/\sin ϵ$ as the frequency scale $\omega \to 0$. The $(1-1/e)$ argument loses its premise—the utility is not a set function—rather than a degraded ratio; unbounded cost alone would not falsify the bound, and we locate the regularizer-coupled regime where the normalized ratio itself drops below $1-1/e$. Numerically the failure is severe and specifically attributable: greedy pays a $3.3\times$ shot penalty on an instance whose fresh-copy control shows no gap at all ($5.9\times$ on a second, where fresh copies reverse the ranking), and search closes the gap only at $(2d+1)(2|\mathcal{A}|{)}^k$-leaf exhaustive cost. The answer the structure dictates is amortization. A policy trained once per instance distribution, spending zero utility evaluations at deployment, beats greedy on all four conditions of the tested random-Hermitian family ($29$–$40$ of $30$–$40$ paired instances, Holm $p\le 1.3\times 10^{-8}$), and adding a budget-parity repair sweep at greedy's own evaluation count reaches the exact optimum in $17$–$80\%$ of instances against greedy's $0$–$7\%$; the policy trains without the exhaustive-search teacher ("teacher-free" below) at $12.6\times$ lower cost. We state the boundary as plainly: on device-physics instances the improvement requires the simulator-based repair sweep and is therefore not yet hardware-deployable, and "teacher-free" means free of the exhaustive-search teacher, not free of the simulator utility oracle during repair. We preregister a deferred-measurement compilation of the experiment for an ion-trap processor; the experimental artifact package is not included with the current submission.