  The classification of many-body quantum states plays a fundamental role in
the study of quantum phases of matter. In this work, we propose an approach to
classify quantum states by introducing the concept of magic class. In addition,
we introduce an efficient coarse-graining procedure to extract the magic
feature of states, which we call the ``convolution group (CG).'' We classify
quantum states into different magic classes using the fixed points of the CG
and circuit equivalence. We also show that magic classes can be characterized
by symmetries and the quantum entropy of the CG fixed points. Finally, we
discuss the connection between the CG and the renormalization group. These
results may provide new insight into the study of the state classification and
quantum phases of matter.
