# Cliffords in constant depth

Daniel Grier and Luke Schaeffer released a wonderful paper on arXiv recently. In it, they highlight some surprisingly stronger separations
between QNC^{0}, constant depth quantum circuits, and other classes of constant and logarithmic depth classical circuits. For background,
Bravyi, et al.â€™s major result two years ago was a strict separation between QNC^{0} and NC^{0} by a problem called the â€śHidden
Linear Function problem.â€ť But Grier and Schaeffer prove a stronger and surprising separation between QNC^{0} and NC^{1},
logarithmic depth bounded fan-in classical circuits. And it was my first time seeing the use of Barringtonâ€™s Theorem, and itâ€™s amazing just how
general but powerful it is!

In it they point to a fairly old theorem in measurement-based quantum computation which I was even more surprised by. In short, itâ€™s that any Clifford circuit can be implemented in constant depth by preparation of a constant degree grid state and constant depth MBQC. Clifford circuits are just so rich.