From Ground States to Local Hamiltonians

Abstract

Traditional quantum physics solves ground states for a given Hamiltonian, while quantum information science asks for the existence and construction of certain Hamiltonians for given ground states. In practical situations, one would be mainly interested in local Hamiltonians with certain interaction patterns, such as nearest neighbour interactions on some type of lattices. A necessary condition for a space $V$ to be the ground-state space of some local Hamiltonian with a given interaction pattern, is that the maximally mixed state supported on $V$ is uniquely determined by its reduced density matrices associated with the given pattern, based on the principle of maximum entropy. However, it is unclear whether this condition is in general also sufficient. We examine the situations for the existence of such a local Hamiltonian to have $V$ satisfying the necessary condition mentioned above as its ground-state space, by linking to faces of the convex body of the local reduced states. We further discuss some methods for constructing the corresponding local Hamiltonians with given interaction patterns, mainly from physical points of view, including constructions related to perturbation methods, local frustration-free Hamiltonians, as well as thermodynamical ensembles.

11 pages, 2 figures, to be published in PRA

References in corpus (5)

• Matrix product states represent ground states faithfully
• Novel schemes for measurement-based quantum computation
• Only n-Qubit Greenberger-Horne-Zeilinger States are Undetermined by their Reduced Density Matrices
• N-Qubit W States are Determined by their Bipartite Marginals
• Correlations in excited states of local Hamiltonians

Cited by in corpus (20)

• Quantum Machine Learning for Chemistry and Physics
• Magic-state resource theory for the ground state of the transverse-field Ising model
• Quantum state tomography via reduced density matrices
• Local-measurement-based quantum state tomography via neural networks
• Uniqueness of Quantum States Compatible with Given Measurement Results
• Local Hamiltonians Whose Ground States are Hard to Approximate
• Hamiltonian-Driven Shadow Tomography of Quantum States
• Symmetric Extension of Two-Qubit States
• Reconstructing quantum states from local data
• Quantum imaginary time evolution steered by reinforcement learning
• Measurement-Based Quantum Computing with Valence-Bond-Solids
• Determining system Hamiltonian from eigenstate measurements without correlation functions
• Unified Quantum State Tomography and Hamiltonian Learning Using Transformer Models: A Language-Translation-Like Approach for Quantum Systems
• Uniquely determined pure quantum states need not be unique ground states of quasi-local Hamiltonians
• When can a local Hamiltonian be recovered from a steady state?
• A variation principle for ground spaces
• Quantum marginals, faces, and coatoms
• Parent Hamiltonian Reconstruction via Inverse Quantum Annealing
• Determining non-Hermitian parent Hamiltonian from a single eigenstate
• Investigating Pure State Uniqueness in Tomography via Optimization