Universal MBQC with generalised parity-phase interactions and Pauli measurements

We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form $\exp(-i\frac{\pi}{2^{n}} Z\otimes Z)$. When $n = 2$, these are equivalent, up to local Clifford unitaries, to graph states. However, when $n > 2$, their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an important role in these generalised graph states. Second, such a state can be used to realise deterministic, approximately universal computation using only Pauli $Z$ and $X$ measurements and feed-forward. Even though, for $n > 2$, the relevant resource states are no longer stabiliser states, they admit a straightforward, graphical representation using the ZX-calculus. Using this representation, we are able to provide a simple, graphical proof of universality. We furthermore show that for every $n > 2$ this family is capable of producing all Clifford gates and all diagonal gates in the $n$-th level of the Clifford hierarchy.