The LHZ architecture is quantum annealing architecture which guarantee all-to-all connectivity. By introducing the parity qubit, all-to-all connected spin system is implemented.
There is interacting $N$-spin system,
$$ \hat{H}{\rm logical} = \sum{(j,k)}^N J_{jk}\hat{\sigma}^z_j\hat{\sigma}^z_k $$
The interaction $J$ is quasi-local. To address all-to-all connectivity, the parity qubit is introduced.
$$ \hat{\sigma}^z_j\hat{\sigma}^z_k \rightarrow \hat{\tilde{\sigma}}^z_{(j,k)} $$
Therefore, the logical qubit system is mapped to
$$ \hat{H}{\rm logical} = \sum{(j,k)}^N J_{jk}\hat{\sigma}^z_j\hat{\sigma}^z_k \quad \rightarrow \quad \hat{H}{\rm physical} = \sum{l=(j,k)}^{N_P}J_{l}\hat{\tilde{\sigma}}^z_{l} + \sum_{n}^{N_C} C_n $$
where $C$ is constraint for guaranteeing logic. There is spin redundancy of cloosed-loop, $\tilde{\sigma}^{z}{(j,k)}\tilde{\sigma}^{z}{(k,l)}\tilde{\sigma}^{z}{(l,m)}\tilde{\sigma}^{z}{(m,j)}=(\sigma^{z}_j\sigma^{z}_k)(\sigma^{z}_k\sigma^{z}_l)(\sigma^{z}_l\sigma^{z}_m)(\sigma^{z}_m\sigma^{z}_j)= (\sigma^{z}_j\sigma^{z}_k\sigma^{z}_l\sigma^{z}_m)^2=+1$, where $j\rightarrow k\rightarrow l \rightarrow m \:(\rightarrow j)$ generate closed plaquette.
$(j,k)\rightarrow (k,l) \rightarrow (l,m) \rightarrow (m,j)\: (\rightarrow (j,k))$
Therefore, there is even-parity constraint,
$$ \sigma^z_{(j,k)}\sigma^z_{(k,l)}\sigma^z_{(l,m)}\sigma^z_{(m,j)} = +1 $$
It means, when the spin arrangement of the physical qubits is following even-parity constraint, the energy is much lower than otherwise.
One example of the even-parity implementation is using four-body interaction.
$$ C^{\rm even}{n=(j,k,l,m)} = -C \hat{\sigma}^z{(j,k)}\hat{\sigma}^z_{(k,l)}\hat{\sigma}^z_{(l,m)}\hat{\sigma}^z_{(m,j)} $$
The system is under the even-parity state, $C^{\rm even} = -C$, however the system is under the odd-parity state, $C^{\rm even} = C$. Therefore, the physical system energetically favor even-parity state.