Locality in Dissipative Ground-State Preparation

Abstract

Can the ground state of a local Hamiltonian be prepared by local dissipation? We address this question for free-fermion Hamiltonians, using a quadratic Lindbladian with linear jump operators for the dissipative ground-state preparation. For translation-invariant targets, Fourier correspondence relates spatial locality to momentum-space analyticity, allowing us to derive a complete spectral criterion for the existence of exponentially local linear jump operators that prepare the ground state. This criterion leads to a no-go theorem that, sign-changing Fermi-level crossings exclude exponentially local linear jump operators for preparing the ground state. While these crossings are the complete obstruction in one dimension, additional obstructions generally arise in higher dimensions. We illustrate these additional obstructions using two quadratic-band-touching models with algebraically decaying ground-state correlations. We further show that short-range ground-state correlations, although not necessary, are sufficient for local dissipative preparation even without translation invariance. Our results establish how the spectral structure of the target Hamiltonian governs the locality of dissipative ground-state preparation.

Can a geometrically local free-fermion Hamiltonian have its ground space prepared by exponentially local dissipation? This paper studies quadratic Lindblad dynamics with linear fermionic jump operators, requiring the entire ground space to be dark and globally attractive. Locality includes exponentially decaying tails, with uniform bounds across system sizes. For translation-invariant targets, Fourier analyticity yields a necessary-and-sufficient spectral criterion: the characteristic polynomial of the Bloch Hamiltonian must factor analytically into positive- and non-positive-energy factors. Necessity follows by extracting a meromorphic representation of the positive-energy sector from analytic jump coefficients; sufficiency follows from the explicit construction $G(q)=\sqrt{\kappa}\,h(q)\chi_-(h(q),q)$. Analytic continuation then excludes sign-changing Fermi-level crossings. These crossings are the complete obstruction in one dimension, but not in higher dimensions. Two two-dimensional quadratic-band-touching examples, both with algebraic ground-state correlations, distinguish the possibilities: one admits strictly finite-range jumps, while the other violates the analytic-factorization criterion. Without translation invariance, exponentially decaying ground-state correlations suffice: $G\propto hP_+=h(I-C)$ gives exponentially local jumps. For this energy-weighted construction, positive-energy modes decay at rates $\kappa\varepsilon_\nu^2$, giving $\tau_{\rm mix}(\epsilon)\le(\kappa\Delta_h^2)^{-1}\log(N/\epsilon^2)$. A spectral-filter implementation avoids detailed eigenstate knowledge, with time-integral truncation error $O(\sqrt{\kappa}/T)$. The classification assumes translation invariance and linear dark-state jumps; interacting systems remain open. Local preparability does not imply rapid mixing for gapless targets. Preparation may retain initial-state dependence within a degenerate ground space.