Classical Ising Model

• The “green barriers” allow us to change the interactions between the magnets; giving the $\exists_{ij}$ coefficient
• The “pink big magnets” represent an external magnetic field, which in theory, can give a larger interaction between the small magnets; giving the $\sum h_i \sigma_i$ term

• For commuting Hamiltonians they correspond to classical systems, so for any quantum mechanical effects, there has to be something more → transverse Ising model

• This is a difficult problem for nature because we might get stuck in a local minima (not the lowest energy level)

• Similar to the problem in CS; QUBO

• For the actual Hamiltonian, we usually write it with the minus signs

  

Transverse Field Ising Model

• $\sigma_x\sigma_z \ne \sigma_z\sigma_x$ (not commutative)
• So we add a transverse field $\sum g_i\sigma_i^x$ to make it non-commutative

  • The $\sigma_x$ tries to push it to a superposition (its eigenvectors)