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 $XX+YY+ZZ$.

what purpose do they serve

The purpose is to change the maths. The whole point is that without these extra terms, D-wave's quantum computer is not universal - it is incapable of implementing an arbitrary quantum computation.

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).


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