# Ising spin vs Pauli spin matrices

Are Ising spins scalar or operators? I am not a condensed matter physicist hence having some confusion. I have learnt about Ising models from adiabatic quantum algorithm papers. For example this presentation or this paper encodes their quantum adiabatic algorithm for an Ising model. In an Ising model the spins are $\pm 1$. In his original adiabatic quantum computation paper, Farhi didn't mention about Ising models. Moreover he explicitly used Pauli's spin matrices and not the $\pm 1$ spins.

Can we use $\pm$ spins and Pauli's matrices interchangeability at least in the context of adiabatic quantum algorithms?

• – Tyson Williams May 29 '13 at 16:39
• @TysonWilliams, I wanted see if any extra insight is available on that forum. – Omar Shehab May 29 '13 at 18:30

As you noted, the Ising model has spins that are $\pm 1$ whereas in a full quantum model such as the Heisenberg model, the spins are represented by Pauli matrices.
The Ising Hamiltonian can be written as $$H = J\sum_{\langle i,j\rangle} \sigma_i \sigma_j$$ and all the $\sigma_i \in \{-1,1\}$.
whereas for quantum spins, we'd have $$H = J\sum_{\langle i,j\rangle} \vec{S_i} \cdot \vec{S_j}$$ and the $\vec{S_i}$ are vectors with elements determined by the Pauli matrices. This product can then be expanded as $$\vec{S_i} \cdot \vec{S_j} = S_i^z S_j^z + \frac{1}{2}\left(S_i^+ S_j^- + S_i^- S_j^+\right)$$ where $S_i^+$ and $S_i^-$ are the spin raising and lowering operators. Thus, we get one term that looks just like the Ising term, because $S_i^z$ can be either $-1$ or $1$, but we also get terms that describe how the two spins can flip: They start with opposite spin and then both of them flip.