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 $\sim\!10^5$ commuting groups; a matrix-aware shot allocation then reaches chemical accuracy at $\sim\!3\times10^9$ 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.