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

How can we construct parent Hamiltonians for purified Gibbs states (thermofield double states) that are both frustration-free and computationally tractable? The authors seek a sum-of-squares (SoS) representation that avoids the need for continuous-time integrals or complex Bohr-frequency decompositions, which are typical in existing thermal state constructions.

The authors utilize modular transformations to define modular annihilators—operators that annihilate the purified Gibbs state. By squaring these annihilators, they construct a frustration-free parent Hamiltonian. This framework allows for the optimization of the Hamiltonian's spectral properties (such as the spectral gap) by choosing different generator bases. For free-fermion systems, the modular transformation acts linearly on Majorana operators, allowing for an exact, analytically solvable construction. For interacting many-body systems, where closed-form modular transformations are unavailable, the authors introduce a Krylov-Lanczos approximation scheme to evaluate the modularly dressed generators and provide rigorous bounds on the resulting ground-state error.

1. Uniqueness: The purified Gibbs state is the unique ground state of the constructed parent Hamiltonian whenever the chosen generators form a unital irreducible algebra.
2. Free-Fermion Optimization: For free-fermion models, the authors identify an optimal coefficient matrix S that maximizes the spectral gap at a fixed operator norm, leading to a mixing time bound that is independent of the inverse temperature.
3. Controlled Approximation: For interacting systems, the Krylov-Lanczos method provides a stable approximation of the modularly dressed operators. The authors prove that for one-dimensional local Hamiltonians, this approximation can be restricted to a finite spatial neighborhood, with the required Krylov dimension scaling polynomially with temperature and interaction range.