For a phase-insensitive thermal Lindblad oscillator, the Heisenberg generator restrictedto polynomials in one rotating-frame quadrature is exactly an Ornstein–Uhlenbeck generator. We use this restriction to determine the integrated autocorrelation time of Ap = Xpfor every degree p. The result is a parity-unified finite rational sum, independent of temperature. We prove the controlled large-degree expansion γτp = 4/p + 2/p2 + 8/p3 + 36/p4 +O(p−5) and show that replacing τp by the inverse spectral gap overestimates it by a factorp/2 − 1/4 − 7/(8p) + O(p−2). For even powers, the same Hermite spectral average equalsthe ratio of the Wasserstein-2 and Fisher pullback metrics along Gaussian Gibbs deformations: γτ2n = 2gWnn/gFnn. The normalized shortfall from the gap bound is therefore an exactFisher–Wasserstein defect. We express its normalization-cleared sequence as a terminatinghypergeometric sum, prove factorial leading growth, and derive an order-three recurrencewith polynomial coefficients and a creative-telescoping certificate. The result provides anexactly solvable hierarchy in which a fixed generator has observable-dependent relaxationtimes parametrically smaller than its inverse gap, while the geometric and dynamical descriptions remain identical term by term.
For a phase-insensitive thermal Lindblad oscillator, the Heisenberg generator restrictedto polynomials in one rotating-frame quadrature is exactly an Ornstein–Uhlenbeck generator. We use this restriction to determine the integrated autocorrelation time of Ap = Xpfor every degree p. The result is a parity-unified finite rational sum, independent of temperature. We prove the controlled large-degree expansion γτp = 4/p + 2/p2 + 8/p3 + 36/p4 +O(p−5) and show that replacing τp by the inverse spectral gap overestimates it by a factorp/2 − 1/4 − 7/(8p) + O(p−2). For even powers, the same Hermite spectral average equalsthe ratio of the Wasserstein-2 and Fisher pullback metrics along Gaussian Gibbs deformations: γτ2n = 2gWnn/gFnn. The normalized shortfall from the gap bound is therefore an exactFisher–Wasserstein defect. We express its normalization-cleared sequence as a terminatinghypergeometric sum, prove factorial leading growth, and derive an order-three recurrencewith polynomial coefficients and a creative-telescoping certificate. The result provides anexactly solvable hierarchy in which a fixed generator has observable-dependent relaxationtimes parametrically smaller than its inverse gap, while the geometric and dynamical descriptions remain identical term by term.
For the damped quantum harmonic oscillator at finite temperature, we study the ratio Δ_q⋆/g_F between a Lindblad-resolvent-induced quadratic form W_q⋆(A) = ⟨A · K_β⋆(A)⟩{ρ_β} (with K_β⋆ = (−ℒ⋆)^{−1} − I, ℒ⋆ the Heisenberg-picture Lindblad generator that admits the thermal state ρ_β as a stationary state) and the exponential-family Fisher variance g_F(A) = Var{ρ_β}(A), evaluated on polynomial perturbations A_p = (a + a†)^p. We establish five exact results. Theorem A (hypergeometric decomposition): the projector weight R_l onto the l-th Hermite eigenmode of −ℒ⋆ admits a closed hypergeometric form. Theorem B (closed-form factorial moments): all factorial moments admit closed product formulas. Theorem 1 (DC ratio): the resolvent-to-Fisher ratio equals −2 for every polynomial order p. Theorem 2 (off-diagonal closed form): gives the off-diagonal deviation matrix. Corollary 3: establishes negative semidefiniteness of the full deviation matrix. Proofs use only Hermite orthogonality, OU spectral decomposition, and the Bény–Osborne Wasserstein gradient flow formulation.
Many physical phenomena, including thermalization in open quantum systems and quantum Gibbs sampling, are modeled by Lindbladians approximating a system weakly coupled to a bath. Understanding the convergence speed of these Lindbladians to their steady states is crucial for bounding algorithmic runtimes and thermalization timescales. We study two such families of processes: one characterizing a repeated-interaction Gibbs sampling algorithm, and another modeling open many-body quantum thermalization. We prove that both converge in polynomial time for several non-commuting systems, including high-temperature local lattices, weakly interacting fermions, and 1D spin chains. These results demonstrate that simple dissipative quantum algorithms can prepare complex Gibbs states and that Lindblad dynamics accurately capture thermal relaxation. Our proofs rely on a novel technical result that extrapolates spectral gap lower bounds from quasi-local Lindbladians to the non-local generators governing these dynamics.
This research establishes an exact, matrix-free framework for spectral analysis of Fibonacci-constrained quantum systems, demonstrating how exponentially expanding Hilbert spaces can admit computational descriptions whose complexity grows only linearly with chain length at any fixed spectral moment order. The investigation focuses on the open PXP quantum chain, whose admissible configurations obey a nearest-neighbor exclusion constraint and form a Hilbert space of dimension dim(Hₙ) = Fₙ₊₂. By combining Fibonacci combinatorics, Zeckendorf indexing, temporal-mask transfer operators, and exact integer arithmetic, the study develops a unified mathematical approach to spectral moments, constrained quantum dynamics, and gauge-sensitive observables without explicitly constructing the exponentially large Hamiltonian matrix. The central result is a universal rational hierarchy of even spectral moments, expressed through the generating function Σₙ₌₀∞ Tr(Aₙ²ᵐ) zⁿ = Pₘ(z) / (1 − z − z²)ᵐ⁺¹, with deg Pₘ ≤ 3m + 2. This theorem establishes that every fixed-order even spectral moment possesses a rational generating function governed by a power of the Fibonacci characteristic polynomial. Consequently, infinitely many spectral moments across increasing chain lengths can be reconstructed from a finite collection of exact initial values. Explicit rational expressions are derived for the fourth, sixth, and eighth spectral moments, accompanied by integer recurrence relations and independently reproducible computational certificates. The underlying temporal-mask transfer identity reduces fixed-order moment evaluation to O(n) arithmetic operations, even with spatially inhomogeneous hopping amplitudes and local diagonal fields. Its computational constants depend exponentially on moment order, but not on the exponentially growing Hilbert-space dimension. A complementary result introduces an exact gauge-sensitive fourth-moment certificate for quantum chains carrying occupation-dependent hopping phases. The spectral response obeys the identity Tr(Hₙ,φ⁴) − Tr(Hₙ,₀⁴) = 8(cos φ − 1) Sₙ, where Sₙ is obtained through a Fibonacci-based weighted recurrence requiring only O(n) arithmetic operations. Remarkably, this identity remains valid under arbitrary static real diagonal disorder, provided the diagonal energies and hopping magnitudes remain unchanged between phase settings. It therefore isolates a precisely defined gauge-dependent contribution to the fourth spectral moment without requiring diagonalization or knowledge of the complete spectrum. The certificate reconstructs a dimensionless flux response rather than an absolute quantum-action scale. The broader significance lies in demonstrating that exponential state-space growth does not necessarily imply exponential computational complexity for every exact spectral observable. The results connect constrained quantum many-body physics, Fibonacci graph theory, rational generating functions, and finite-state transfer methods within a common algebraic framework. All reported coefficients are supported by exact-integer verification rather than numerical fitting, and the established complexity bounds concern arithmetic operations rather than total bit complexity. The work provides a reproducible mathematical foundation for further investigations of spectral invariants, gauge responses, quantum constraints, and scalable symbolic computation, while restricting its formal claims to the specified finite-dimensional Hamiltonian models. Keywords: Fibonacci-constrained quantum chains · PXP Hamiltonian · Rational spectral hierarchy · Matrix-free computation · Temporal-mask transfer · Zeckendorf indexing · Gauge invariants · Exact spectral moments · Quantum many-body systems · Integer verification
For a Schrödinger cat state |α⟩+|−α⟩ evolving under thermal Lindblad dynamics in an open bosonic system, we derive a closed-form expression for the time-evolved density matrix ρ(t) via the Glauber-Sudarshan P-representation, reproducing numerical Lindblad evolution at fidelity above 99% across |α|∈[2,3] and β∈[0.5,8]. On this trajectory we compute and compare two quantitatively independent observables. The first is the Wigner negativity W_neg(t), a coherence-amplitude indicator with smooth exponential decay analytic via the Joos-Zeh rate γ_coh = 2γ|α|²(2n̄+1). The second is the integrated autocorrelation time τ_int(ρ(t)), a dynamical-memory indicator extracted via the quantum regression theorem on non-stationary ρ(t) through Liouvillian eigendecomposition and spectral integration. Both observables decay from cat-specific values toward thermal stationary values along the trajectory, but with qualitatively different functional forms and parameter dependence. Across a 16-case parameter scan in (α,β), the half-life ratio tτ/tW has mean 6.03 and coefficient of variation 41%, confirming that τ_int carries independent dynamical-memory information beyond what coherence amplitude captures. The observable τ_int is the trajectory extension of the Wasserstein-Fisher ratio at the center of the stationary dissipation-curvature theorems, computed here in the non-stationary regime via quantum regression. Supplementary topological, information-geometric, and informational indicators (cubical persistence H1 count, dissipation-curvature ratio, purity) are tracked and discussed but do not carry primary contributions of the paper.
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 Sopt∝1/cosh(2βh) has mixing time upper bound 2log(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.
Preparing Gibbs states through dissipative dynamics requires controlling convergence for the chosen Hamiltonian H 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 β. 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) 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.