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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>y</a:mi><a:mi>z</a:mi></a:math>-plane measurements. We further show that unitary MBQC using only <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>y</c:mi><c:mi>z</c:mi></c:math>-plane measurements must occur on a bipartite graph. These results have a number of consequences and open new research avenues for both frameworks.
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Smith et al. (2024) studied this question.
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