Consider a square lattice where at each site we have a variable $s_i$ that can take two values $s_i=\pm 1$. The sites can be thought to represent atoms that have a spin $s_i$. The spins are interacting and the system has energy

$$ H = -\sum_{i,j=1}^N J_{ij}\, s_i s_j. $$

[Explain what are the possible values of this Hamiltonian. Start we two spins.]

If each spin has an additional energy (for example, due to an external field) then the energy is

$$ H = -\sum_{i,j=1}^N J_{ij\,} s_i s_j - \sum_{i=1}^N h_i s_i. $$

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The problem is to find the minimum of the energy $H$ for given $J_{ij}, h_i$. We are looking for the configuration $s_i$ at the minimum of $H$.

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Exercise. (Two and three Ising spins) Set $h_i=0$ in the Ising energy.

(a) Consider two spins and write the possible values of the energy. What is the state of minimum energy? Consider the cases $J>0$ and $J<0$.

(b) Consider three spins and $J_{ij}=J>0$ for $i,j=1,2,3$. What are the states of minimum energy?

Such optimization problems are usually solved using variants of Monte Carlo methods on large-scale high performance classical computers, .

A simplification is to consider that $J_{ij}=J$ is a constant for neighboring spins and it is 0 otherwise (this is called near neighbour interaction).

Exercise. Write the model (the energy) for a chain of $N$ spins. What is the minimum energy configuration? (consider both cases for $J>0$ and for $J<0$.

Exercise. (QUBO) In quadratic unconstrained binary optimization (QUBO) problems, we use a binary variable $x_i \in \{0,1\}$ defined on each vertex in a set $V$. QUBO is given by

$$ E(\{\mathbb{x}\}) = \sum_{i,j\in V} w_{ij\,} x_i x_j + \sum_{i\in V} c_i x_i. $$

Apply a transformation to the Ising model to obtain the Quadratic Unconstrained Binary Optimization problem.