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arXiv:2608.30272

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

Changhao Yi, Jun Takahashi, Cunlu Zhou

quant-phSubmitted Aug 31, 2026• Announced Sep 1, 2026• 0 citations
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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 $S_{\mathrm{opt}}\propto 1/\sqrt{\cosh(2βh)}$ has mixing time upper bound $2\log(2N/ε)$ for any $β$, which exhibits rapid mixing and is irrelevant to the inverse temperature $β$. 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.

Comments
41 pages, 4 figures

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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)S=f(h)S=f(h), we show that, at fixed operator norm, the choice Sopt∝1/cosh⁡(2βh)S_{\mathrm{opt}}\propto 1/\sqrt{\cosh(2βh)}Sopt​∝1/cosh(2βh)​ has mixing time upper bound 2log⁡(2N/ε)2\log(2N/ε)2log(2N/ε) for any βββ, which exhibits rapid mixing and is irrelevant to the inverse temperature βββ. 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.
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Gibbs state preparation is an important subroutine in quantum computing. In this work we use the detectability lemma to improve Gibbs state preparation. Specifically, we design new Gibbs state preparation methods that do not rely on simulating Lindbladian evolution, thus avoiding the overhead from it. For local Lindbladians consisting of MMM terms, this approach reduces the cost by a factor of O(M)O(M)O(M). We also combine the detectability lemma operator and quantum singular value transformation to implement ground state projection operators of frustration-free Hamiltonians, resulting in a quadratic speedup in the spectral gap dependence. Applying this method to Lindbladians for the Gibbs state of local commuting Hamiltonians, we achieve quadratically better dependence on the Lindbladian spectral gap.
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Preparing Gibbs states through dissipative dynamics requires controlling convergence for the chosen Hamiltonian HHH and coupling operators. However, efficient implementation requires convergence guarantees from the spectral gap of the generator of the dynamics, which remains a crucial challenge. Recent results address the challenge by restricting the dynamics with structural assumptions about the Hamiltonian and its interactions. For the Chen-Kastoryano-Gilyen (CKG) construction and arbitrary finite-dimensional Hamiltonians, we prove an explicit comparison that converts a spectral gap estimate for the infinite-temperature generator into a lower bound on the spectral gap of the CKG quantum Gibbs sampler at every finite positive inverse temperature βββ. It allows a single spectral gap estimate at infinite temperature to certify convergence across Hamiltonians and temperatures. Such an estimate is often known from a classical spectral gap estimate, such as the classical random walk spectral gap. For Hodge Laplacians of simplicial complexes as Hamiltonians, we construct coupling operators whose generator at infinite temperature has the same spectral gap as a classical simplicial down-up walk. A positive spectral gap of the walk and bounded spectral width W=λmax⁡(H)−λmin⁡(H)W=λ_{\max}(H)-λ_{\min}(H)W=λmax​(H)−λmin​(H) give a generator spectral gap independent of the number of simplices at every fixed temperature. We then establish conditions under which the sampler approximately prepares zero-energy (harmonic) states in polynomial evolution time. This provides a conditional convergence guarantee for the state preparation in the thermal approach to quantum topological data analysis.
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