Georgia Institute of Technology Finds Limits to Ground State Preparation

Space analyticity derives a complete spectral criterion for the existence of exponentially local linear jump operators that prepare the ground state. The resulting criterion leads to a no-go theorem stating that sign-changing Fermi-level crossings exclude exponentially local linear jump operators for preparing the ground state. These crossings represent the complete obstruction in one dimension; however, additional obstructions generally arise in higher dimensions. Researchers illustrate these additional obstructions using two quadratic-band-touching models with algebraically decaying ground-state correlations. Furthermore, short-range ground state correlations, while not necessary, are sufficient.

## Spectral Properties Dictate Locality During Dissipative Ground State Preparation or even local dissipative preparation without translation invariance. Our results establish how the spectral structure of a target Hamiltonian governs locality during dissipative ground-state preparation. Preparing ground states of quantum many-body systems is a fundamental, yet challenging goal for quantum simulation. Engineered dissipation has recently gained considerable attention among methods like adiabatic state preparation, imaginary-time evolution, and coherent Hamiltonian deformations.

By designing dissipation to remove excitations while preserving the ground-state subspace, it offers a natural route from arbitrary initial states. Rapid convergence of such protocols has been established for free and weakly interacting models; recent experimental demonstration occurred in a one-dimensional transverse-field Ising chain using trapped ions. Researchers are also exploring engineered dissipation for topologically ordered states like toric code and fractional quantum Hall states.

Most literature focuses on time complexity, but comparatively little study addresses the spatial structure of dissipations. Removing excitations requires spectral construction specifying jump operators through energy-lowering transitions between eigenstates of the target Hamiltonian. Physical realizations prefer geometrically local jump operators in real space. This raises an important question: does locality of the target Hamiltonian guarantee local dissipative ground state preparation, or can its spectrum necessitate nonlocal jumps?

We address this question for geometrically local free-fermion Hamiltonians in arbitrary dimensions with quadratic Lindbladian dynamics. Geometric locality includes quasi-locality exhibiting exponentially decaying tails. For translation-invariant systems, momentum space connects spectral structure to spatial locality; we establish a criterion for local preparation within a dark subspace. We prove that no local linear jumps can prepare the ground state of a translation-invariant free-fermion Hamiltonian displaying sign-changing Fermi level crossings, a complete obstruction in one dimension but not guaranteeing success even without such crossings in higher dimensions.

Two quadratic band touching models illustrate this distinction: both have algebraically decaying correlations, yet only one admits local linear jumps. Specifically, any geometrically local free-fermion Hamiltonian with exponentially decaying ground-state correlations allows construction of local jump operators to prepare its ground state.

Here ρ is the system’s density matrix under evolution; μ labels dissipative channels and Kμ denotes associated jump operators. The goal is to relax the system towards the target ground-state subspace ΩH of H, PGS represents projection onto it. Together with darkness this ensures convergence to a steady state on ΩH. Degenerate cases may have multiple steady states dependent upon the initial conditions.

Here h is a 2N × 2N kernel. We assume couplings are geometrically local with possible exponential tails having uniform bounds across spatial positions and system sizes leading to spectral bound ||h|| ≤ Emax where Emax >0 is size-independent.

Global attraction requires each positive mode coupled to a channel giving Σ μ |Gμψν|2 > 0 for εν >0. These requirements impose spectral selection, untouched components with nonpositive energy while those with positive energy couple to dissipation. The coefficients Gμ;(j,a) determine the spatial structure of linear jumps. For channels centred at xμ exponential locality demands decaying coefficients across channels and system sizes.

Eigenvalue decomposition gives h(q)un(q)=εn(q)un(q). The Fourier transform relates spatial locality to analyticity: continuous functions have exponentially decaying coefficients if analytic extension exists beyond real domain. Here we use same function across system sizes L sampled at momenta qk=2πk/L for finite size.

A criterion for target spectra can be seen starting with a single orbital per cell (single band model). Assuming analytic Gμ, then E(q) is quotient of two analytic functions and therefore meromorphic.

This requires positive energy branch not crossing Fermi level restricting the Hamiltonian’s spectrum for local preparation extending to multi-band cases where a matrix M consists powers Gμ h with r independent rows equal number of positive modes at momentum q. The team at Georgia Institute of Technology demonstrated that short-range ground state correlations are sufficient for localized dissipation even without translation invariance, improving upon previous methods focused on time complexity which required it; previously such preparations were impossible without this symmetry. This work establishes criteria determining when dissipation can reliably guide a quantum system towards its lowest energy state using localised interactions with the environment, revealing how spectral qualities dictate feasibility.

By connecting spatial properties to momentum characteristics they proved sign changing Fermi level crossings prevent exponentially local jump operators from preparing free fermion Hamiltonians’ ground states representing complete obstructions in one dimension but additional challenges in higher dimensions as illustrated by two quadratic band touching models.

While demonstrating sufficiency represents progress beyond earlier work requiring translational symmetry quantifying precisely *how* “short-range” these correlations need to be remains open and simply establishing sufficiency doesn’t address practical limitations scaling up complexity.

This research determined that the structure of a Hamiltonian’s spectrum governs whether its ground state can be prepared through localized interactions with an environment, specifically, using linear jump operators for dissipation. The authors found sign-changing Fermi level crossings prevent this preparation in one dimension, while more complex obstacles arise in higher dimensions, demonstrated using models with algebraically decaying ground-state correlations. This work clarifies how spectral features dictate the feasibility of guiding quantum systems to their lowest energy states via dissipation.

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