Hamiltonian complexity

Hamiltonian complexity or quantum Hamiltonian complexity is a topic 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 led 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

Given a Hermitian matrix

H

{\displaystyle H}

, let

λ

0

{\displaystyle \lambda _{0}}

denote the ground state energy of the Hamiltonian

H

{\displaystyle H}

, and let

a

{\displaystyle a}

and

b

{\displaystyle b}

be non-negative real numbers with

b

≥

a

+

1

{\displaystyle b\geq a+1}

. If

λ

0

≤

a

{\displaystyle \lambda _{0}\leq a}

, output Yes. If

λ

0

≥

b

{\displaystyle \lambda _{0}\geq b}

, output No. The k-local Hamiltonian problem is similar except the Hamiltonians have

k

{\displaystyle k}

-local interactions. This problem has been shown to be QMA-complete for

k

≥

2

{\displaystyle k\geq 2}

.

Area law

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