Measurement-based quantum computation (MBQC) is a paradigm for quantum computation where computation is driven by local measurements on a suitably entangled resource state. In this work we show that MBQC is related to a model of quantum computation based on Clifford quantum cellular automata (CQCA). Specifically, we show that certain MBQCs can be directly constructed from CQCAs which yields a simple and intuitive circuit model representation of MBQC in terms of quantum computation based on CQCA. We apply this description to construct various MBQC-based Ansätze for parameterized quantum circuits, demonstrating that the different Ansätze may lead to significantly different performances on different learning tasks. In this way, MBQC yields a family of Hardware-efficient Ansätze that may be adapted to specific problem settings and is particularly well suited for architectures with translationally invariant gates such as neutral atoms.