Open Problems
- Bose–Hubbard CGW14 proof on different geometries. The proof of [CGW14] is for symmetric $0$–$1$ graphs with at most one self-loop. What about simple graphs? Lattices? No self–loops?
- What are the complexities of the HaF instances of Hamiltonians?
- What is the complexity of varying Ising Model instances on different geometries?
- What are the complexity classifications of Hamiltonians with interaction strengths of the form $J = O(1)$?
- The general definition of the models admits algebraic locality and not geometrical locality. It is therefore natural to ask what the complexity is for interaction strengths of the form, $J_{ij} = |i - j|^{-\alpha}$.
- The BPP algorithm presented by Bravyi [Bra15] for calculating the partition function of the ferromagnetic TIM is not practical. Is there an improved version that can offer practicality?
- The advent of guided local Hamiltonian instances begs the question as to whether similar results can be found for the likes of the Fermi–Hubbard, Bose–Hubbard, and Electronic Structure Hamiltonian.
- Are hybrid interaction systems of interest and can their complexity also be defined?
- What is the complete impact of guiding states on the complexity of local Hamiltonians?
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Can a robust framework of local Hamiltonians with restricted ground states be defined?
Guiding states have been proposed that have a semi–classical strucutre. Recently, local Hamiltonians with succinct ground states have been proposed. Furthermore, finding extremal product states for local Hamiltonians has been studied. Is there a better framework with which we can discuss these ideas and find possible links?
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What alternate ideas exist for circuit–to–Hamiltonian constructions?
Are there good constructions without the need for a clock? Furthermore, are there constructions that are more physically motivated?
- Is it #P-hard to approximate partition functions at real temperature?