Dynamical spectral functions from bitstring-sampled quantum subspaces: entanglement, not one-body magic, tracks the sampling cost

Sample-based quantum diagonalization (SQD) and quantum-selected configuration interaction (QSCI) are the electronic-structure methods with most hardware traction, yet their canonical target – the ground-state energy – is where classical methods have caught up. We move the target to dynamics and the resource question. From one bitstring-sampling primitive – computational-basis measurements of a shallow real-time circuit, with no Hadamard or controlled unitaries – we reconstruct, from sampled subspaces, the single-particle spectral functions $A(\omega )$ and $A(k,\omega )$ and the neutral-sector dynamical structure factors $S(q,\omega )$ and $S^{zz}(q,\omega )$, each built classically in the Lehmann representation from its own subspace. The reconstruction matches exact diagonalization on Hubbard chains and, for $A(\omega )$, across nineteen molecules (FCI-verified to $<{10}^{-5}$ Ha), and runs on the IBM Heron processor. Second, we ask which resource controls the cost – the determinant support $\mathrm{\mid }\mathcal{S}\mathrm{\mid }$ the sampler must populate. On number-conserving states the fermionic AntiFlatness collapses to one 1-RDM invariant, ${\mathcal{F}}_1=4\mathrm{t}\mathrm{r}[\gamma (1-\gamma )]=2N_u$. An orbital-rotation (Gaussian) invariant while $\mathrm{\mid }\mathcal{S}\mathrm{\mid }$ is basis dependent, ${\mathcal{F}}_1$ is provably decoupled from the cost; the cost is instead lower-bounded and tracked by the entanglement – the minimal bond dimension $\chi$ (Spearman $\rho =0.90$). One-body magic is thus a faithful multireference diagnostic but an unreliable cost predictor; any genuine advantage lives in the non-Gaussianity of the higher-body cumulants. We prove moment exactness and a sampling bound polynomial in $\mathrm{\mid }\mathcal{S}\mathrm{\mid }$, independent of Hilbert-space dimension. Self-consistent configuration recovery improves the subspace under device noise, while a learned generative model does not beat that classical baseline.