Parity Quantum Computing As Yz-Plane Measurement-Based Quantum Computing

We show that universal parity quantum computing employing a recently introduced constant depth decoding procedure is equivalent to measurement-based quantum computation (MBQC) on a bipartite graph using only yz-plane measurements. We further show that any unitary MBQC using only yz-plane measurements must occur on a bipartite graph. These results have a number of consequences and open new research avenues for both frameworks.