Hamiltonian complexity

Hamiltonian complexity or Quantum hamiltonian complexity is a field which deals with problems in quantum complexity theory and condensed matter physics. It mostly studies Constraint satisfaction problems related to ground states of local hamiltonians; that is, hermitian matrices that act locally on a system of interest . . The constraint satisfaction problems in quantum hamiltonian complexity have lead to the quantum version of the Cook–Levin theorem. Quantum hamiltonian complexity has helped physicists understand the difficulty of simulating physical systems.

Local Hamiltonian problem

Main article: QMA § The_local_Hamiltonian_problem

Given a hermitian matrix $H$ and non-negative reals $a$, $b$ with $b\geq a+1$, If $\lambda _{0}(H)\leq a$ output Yes. If $\lambda _{0}(H)>=b$ output No. The k-Local Hamiltonian problem is stated similarly except the hamiltonians have $k$ local interactions. This problem has been shown to be QMA-complete for <math>k \geq 2

Area law

Main article: Entropy_of_entanglement § Area_law_of_bipartite_entanglement_entropy

The area law explains the structure of entanglement present in ground states of physically relevant systems.

Quantum analog of the PCP theorem

Main article: PCP_theorem § Quantum_analog_of_the_PCP_Theorem

The Quantum analog of the PCP theorem is an open question in complexity theory.

See also

Hamiltonian simulation

Density Matrix Renormalization Group

Matrix product state