# What are XX, YY, YZ etc. couplings?

The D-wave quantum computer allows us to be able to minimize Ising models. In reading other questions and responses, particularly What would be the simplest addition that would make the D-Wave architecture universal?, XX couplings (and others have been mentioned. They (XX, YY etc. couplings) look like names for connections between qubits based on an axis but beyond that I get lost (especially since the couplers on D-waves architecture seem to be identical from pair to pair). I addition I fail to understand how they fit in with the Ising model math; what are these couplers/what purpose do they serve?

The other question that mentioned XX, YY etc. couplings: In D-Wave's universal quantum computer, why does the YY term have to be driven along with the linear X term?

In the mentioned context, what is meant is that, between a pair of qubits that are coupled, an XX coupling means something of the form $$X\otimes X\equiv\left(\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right),$$ tensored with identity between all other qubits, where $$X$$ is the standard Pauli matrix. You then sum these terms over every possible pair of coupled qubits, possibly with different strengths. For example, in a spin chain, qubits labelled 1 to $$n$$, with a nearest-neighbour coupling (i.e. between all pairs $$i$$ and $$i+1$$), the XX coupling means $$\sum_{i=1}^{N-1}J_i\mathbb{I}^{\otimes(i-1)}\otimes X\otimes X\otimes\mathbb{I}^{\otimes(n-i-1)}$$ for real parameters $$J_i$$. Similarly, $$YY$$ means replacing the $$X\otimes X$$ with $$Y\otimes Y$$.
In other contexts, the terminology can be used slightly differently. For example, in the condensed matter community, they usually talk about an "XX Hamiltonian". This does not mean a Hamiltonian with XX couplings. Instead, it means a Hamiltonian with terms of the form $$XX+YY$$ between coupled pairs of qubits. This is also called the exchange interation. To make this notation clearer, let me give further examples. An $$XXX$$ Hamiltonian, normally called the Heisenberg Hamiltonian, would mean couplings of the for $$XX+YY+ZZ$$, while $$XXZ$$ means $$XX+YY+\Delta ZZ$$ for some parameter $$\Delta$$. In other words, there are generally 3 terms that you have to worry about in a two-qubit coupling: $$aXX+bYY+cZZ$$, and a notation like "XX" or "XXX" tells you the number of non-zero terms $$(a,b,c)$$ which are the same. While "XXZ" tells you two values are the same ($$a=b$$), but that one value is different. The further complication is that this notation is not consistently used. Sometimes people use "XY" to mean $$XX+YY$$ and "XYZ" to mean $$XX+YY+ZZ$$.
Let me try to give some insight as to why that might be (I don't pretend that this applies directly). To that end, consider the simple geometry of a one-dimensional chain. You'd have some sort of Hamiltonian $$H=\sum_{i=1}^{n-1}\Delta_iZ_iZ_{i+1}+\sum_{i=1}^nB_iX_i.$$ The great thing about this Hamiltonian, the transverse Ising model, is that it's exactly solvable via Bogoliubov and Jordan-Wigner transformations. But that means that we can classically simulate its effects and so it's not interesting from a computation perspective. However, if we add extra terms to make the Hamiltonian $$H=\sum_{i=1}^{n-1}\Delta_iZ_iZ_{i+1}+\sum_{i=1}^{n-1}\tilde\Delta_iX_iX_{i+1}+\sum_{i=1}^nB_iX_i.$$ then we don't know how to simulate it, and it has the potential to perform interesting computations (this is a long way from proving that simulation of this Hamiltonian is BQP complete).