Modular-Annihilator Parent Hamiltonians for Purified Gibbs States: Spectral Design and Controlled Approximation

Abstract:Purified Gibbs states provide a bridge between finite-temperature physics, dissipative dynamics, and ground-state methods. In this work, we study the exact finite sum-of-squares (SoS) construction of their parent Hamiltonians and the associated Lindbladian based on modular annihilators. Given a finite set of Hermitian generators, the corresponding modular annihilators yield a frustration-free SoS representation without continuous time integrals or an explicit decomposition into Bohr-frequency sectors. The purified Gibbs state remains a common zero mode while the freedom to choose and combine the generators can be used to optimize the spectral properties of the parent Hamiltonian. For free-fermion Hamiltonians, modular transformations act linearly on Majorana operators, leading to an analytically solvable family of parent Hamiltonians parameterized by a real symmetric coefficient matrix \(S\). For the scalar-functional subclass $S=f(h)$, we show that, at fixed operator norm, the choice $S_{\mathrm{opt}}\propto 1/\sqrt{\cosh(2\beta h)}$ has mixing time upper bound $2\log(2N/\epsilon)$ for any $\beta$, which exhibits rapid mixing and is irrelevant to the inverse temperature $\beta$. For interacting systems, where the modularly dressed generators are not available in closed form, we introduce a Krylov--Lanczos approximation scheme and bound the resulting ground-state error in terms of the modular-approximation error and the parent-Hamiltonian gap. Numerical results illustrate the free-fermion spectral advantage and show how the accuracy of the interacting construction depends on temperature, interaction strength, and Krylov dimension.

arXiv:2608.30272 [quant-ph]

(or arXiv:2608.30272v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.30272 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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