Some properties of a Hamiltonian are unique to its spectrum and are basis-independent. For example, I think whether the Hamiltonian's gap remains constant as $n$ goes to infinity, or whether the Hamiltonian obeys an area law, are independent of the basis in which the Hamiltonian is written.
But it seems to me that some other computationally interesting properties of the $k$-local Hamiltonian problem may be dependent on the basis in which the Hamiltonian is written. Am I correct in saying that the answer to each of the following may depend on the basis in which the Hamiltonian is written - either the standard (computational) basis, or else the Hadamard basis?
- Whether or not the $k$-local Hamiltonian is frustration-free, meaning that the global ground state of the full Hamiltonian corresponds to the lowest energy state of each local term?
- Whether or not each term commutes with one another?
- Whether the Hamiltonian is stoquastic - that is, whether, as a Hermitian matrix, it only includes real, non-positive off-diagonal elements?
- Whether it is $k$-local to begin with, or what its $k$-value is?
Or does it even make sense to ask the above?
Here, I mean locality in the computational sense (and not geometric sense) - i.e., $H$ can be written as the sum of a polynomial number of local terms each acting solely on up to $k$ qudits (tensored with identities elsewhere) - but these qudits need not be geometrically near each other.
This is born out of me rowing various confusion boats - entanglement is clearly basis independent, and generally highly entangled states are computationally harder to work with than otherwise, but I've also believed that being non-stoquastic, or being non-frustration free, or non-commuting, or being a highly non-local Hamiltonian, corresponds to being among the most difficult to find ground states thereof.
I'm also learning slowly about IQP circuits in view of the recent experiments from Quera - hence the question about non-commuting Hamiltonians.
