The energy is the last thing to fail: properties, spin and controls for sample-based quantum diagonalization, from N₂ to an iron–sulfur cluster

The energy is the last thing to fail. Sample-based quantum diagonalization (SQD), the method behind the largest chemistry calculations on quantum processors, grades its results by the energy, the quantity least sensitive to errors in the state. This work asks what else the state gets right, on archived hardware data from two sources: the author's published N₂ run on IBM hardware, and IBM's archive for the 77-qubit iron–sulfur demonstration of 2025, the largest SQD calculation whose raw samples are public. Three density properties are followed alongside the energy against converged references: two stop improving while the energy still does; the third, set by the diffuse tail, oscillates while the energy descends smoothly. Four controls show whose work the energy is: random bitstrings seeded with five valid strings lag the hardware data by one iteration and match them at equal subspace size, a coupled-cluster prior matches or beats them there, and the device beats only the published control, uniform in-sector strings, whose loop has nothing to repair. The ordering repeats on the iron–sulfur cluster, where the spin is read alongside the energy and a penalty confirms the classical seed's lead as a singlet energy. Exact samples of the one-layer LUCJ circuit stall on a few hundred configurations: the noise feeds the repair. The recommendation is cheap: report a density property and the spin alongside the energy, at fixed subspace size, against a converged reference. Looking forward, exact samples of a time-evolved state broaden the support ten- to twentyfold over the LUCJ circuit, take the static loop to half a millihartree where LUCJ stalls, and, repaired from a noise channel, lead the hardware shots at equal subspace size at equilibrium and at the stretched bond; generative circuit compression would bring such a sampler inside the Nighthawk gate budget. v2 (2026-09-11): Table 10 gains the one-layer LUCJ circuit with optimized parameters, all-to-all and hardware pattern, as the rung between the coupled-cluster-parametrized circuit and the time-evolved state; §3.6.3 and §3.7.6 state that the configuration prior is coupled cluster's output, not the circuit's input. Changes in v3 (2026-09-13): the optimized-circuit rows of Table 10 restated from four converged optimizations (hardware pattern reproducible to chemical accuracy; three all-to-all optima, a perturbed start returning to one of them); a new §3.6.4 measuring the classical solver's envelope to fifty million determinants on a laptop (Table 12). Changes in v4: Table 10 gains the full-CI weight missing from each sample's string product space, 1 − W(U×U), and two full-CI yardstick rows; §3.6.3 reads it: the missing weight orders the static-loop error where configuration and string counts do not, and a millihartree needs less than about 3 × 10⁻⁴ missing. Changes in v5: a new §3.6.5, "Which root is the state", summarizes in one paragraph a measurement reported in full in a companion paper in preparation: at the stretched geometries the lowest root of a subspace closed under spin inversion is a spin mixture, which a DMRG-X-type overlap rule cannot repair; the abstract and §4 follow it. Table 10's hardware pattern read from the CCSD amplitudes is given as a range over ten builds, because round-off-level differences between coupled-cluster runs change that circuit; three optimizations "from the same starting state".