Sub-Riemannian geometry of measurement-based quantum computation

The computational power of quantum phases of matter with symmetry can be accessed through local measurements, but what is the most efficient way of doing so? In this work we show that minimizing operational resources in measurement-based quantum computation (MBQC) on subsystem symmetric resource states amounts to solving a sub-Riemannian geodesic problem between the identity and the target logical unitary. This reveals a geometric structure underlying MBQC and offers a principled route to optimize quantum processing in computational phases.