# How to show mathematically the equivalency between Ising Model and QUBO?

It is said that the Ising Model using spin variables $$s ∈ \{−1, 1\}$$ $$H(s)=\sum_{i}h_is_i+\sum_{i and a Quadratic Unconstrained Binary Optimization (QUBO) problem with binary variables $$x ∈ \{0, 1\}$$ $$Obj(x,Q)=x^{T} ·Q·x,$$ are equivalent under a simple change of basis. How one could mathematically show this?

The relation between "Ising" and binary variables is following $$x_i = \frac{1 + s_i}{2},$$ where $$s_i$$ is a spin and $$x_i$$ is a binary variable. Clearly setting $$s_i = -1$$ leads to $$x_i = 0$$ and if $$s_i = 1$$ we get $$x_i$$ = 1. So, this simple linear transform changes spins to binary variables and conversely.
Quadratic terms in QUBO objective functions are equivalent to two-body interactions in Ising Hamiltonian ($$Z_i \otimes Z_j$$ terms in the Hamiltonian) and linear terms are equivalent to independent $$Z_i$$ terms in the Hamiltonian.