Spectral Core-Tail Architecture for Locally Certified Gibbs-State Preparation

Abstract

Preparing a quantum Gibbs state requires reproducing both its thermal distribution and the associated many-body eigenspaces. In this work, we formalize the spectral core-tail architecture (SCTA) as a framework comprising a structured thermal core, a geometrically characterized unitary tail, and an exact residual measuring the remaining core-frame Hamiltonian mismatch. We derive a local Gibbs-state error bound separating core-preparation, tail-implementation, and modeling errors. The modeling contribution is volume uniform when its Kubo-Mori response meets the shell summability condition and the relevant local data remain uniform. We then highlight three anchor Hamiltonian classes which admit exact core-tail constructions with zero residuals. For small deformations of an exact anchor, we employ a Schrieffer-Wolff reduction procedure to construct a corrected core-tail pair that formally removes the deformation order by order. Under uniform locality and solvability assumptions, we show that the first-order reduction yields a residual that remains bounded and is quadratic in the deformation strength. Furthermore, we numerically test the first-order reduction on deformed graph-stabilizer Hamiltonians. The numerical results show approximately quadratic suppression of both the Hamiltonian-level mismatch and the local Gibbs-state error in the perturbative regime. We also find that the constructed correction circuit structure remains useful as a variational ansatz beyond perturbative control, often improving on both the bare and prescribed first-order states.