Captured weight and boundary leakage bound the error of sample-based spectral functions

Abstract

Sample-based quantum diagonalization builds a subspace from a device's measured bitstrings. From one protocol -- computational-basis samples of a real-time evolved probe, no Hadamard test -- we reconstruct the single-particle $A(\omega)$ and $A(k,\omega)$ of the 1D Hubbard model on that subspace. We prove that for Rayleigh-Ritz on any orthogonal projector the relative $L_1$ error lies between $1-w$ and $\min\{1+w,(1+\sqrt{w})(\sqrt{1-w}+\widehat\Lambda(\eta)/\eta)\}$, with $w$ the captured weight and $\widehat\Lambda$ the boundary coupling at resolution $\eta$, assembled from the Ritz pairs without the answer. Weight alone cannot certify: an analytic counterexample forces any bound indexed on it to its trivial value, and on the probe's support, where $w=1$ exactly, the error is still $0.35$--$0.43$ at $\eta=0.18\,t$ against a ceiling of $2$. A subspace containing the order-$K$ Krylov space reproduces the moments through order $2K+1$, and no further. Measured to dimension $10\,306\,296$, the bound is vacuous at every operating fraction of the resource scan, at all five sizes, by $1.66$--$8.61$, and informative only well above them; it crosses the trivial bound at a true error of $1.2$--$7.3\times10^{-3}$, though a two-constant rule in $1-w$ is tighter on 11 of 14 splits. In the determinant basis the probe's Krylov space has coordinate support on every symmetry-allowed determinant, so at full weight only a subspace holding all of them has zero leakage; for a momentum probe a momentum block, $1/L$ of the sector, does. At full weight a second-order certificate follows; below it, where every operating fraction lies, one is open. One-body fermionic magic does not predict the support in advance: its rank correlation with it is $-1$ in each of six strata (exact $p=3.3\times10^{-13}$ against independent orderings) but changes sign between families.