A Reproducible Pipeline for Symmetry-Respecting Excited States on Near-Term Quantum Computers: The H2O/STO-3G Case

Variational excited-state quantum algorithms fail for reasons usually studied in isolation: barren plateaus, symmetry contamination, finite-sampling instability, and hardware cost. Using one small but complete system – H${}_2$O in the STO-3G basis (12 qubits, Jordan–Wigner) – we assemble these into a single reproducible pipeline, checking every claim against exact diagonalization. The bare qubit Hamiltonian interleaves cation ($N=7$) states below the neutral manifold; hardware-efficient and number-conserving ansätze stall at Hartree–Fock, an exact stationary point by Brillouin’s theorem, while ADAPT-VQE escapes; variational deflation inherits the contamination and inverts the spectrum, whereas the quantum equation-of-motion (qEOM) subspace method restores the ladder to sub-milli-Hartree accuracy. Particle number is protected \emph{structurally} under shot noise, and a realistic measurement model collapses the thousands of subspace matrix elements to $\sim \text{\hspace{0.17em}⁣}{10}^5$ commuting groups; a matrix-aware shot allocation then reaches chemical accuracy at $\sim \text{\hspace{0.17em}⁣}3\times {10}^9$ total shots – a thousandfold below the naive per-element estimate and reachable in days – leaving single-circuit gate fidelity, not measurement, as the binding constraint. This work is a teaching and benchmarking reference, not a new method; all code, parameters, and figures are released.