Magic Class and the Convolution Group

TL;DR

This paper introduces the concept of magic class and convolution group to classify many-body quantum states, providing a new framework that links symmetries, entropy, and renormalization group ideas for understanding quantum phases.

Contribution

It proposes the magic class and convolution group as novel tools for classifying quantum states, including an efficient coarse-graining method and connections to symmetries and entropy.

Findings

Quantum states are classified via fixed points of the convolution group.

Magic classes relate to symmetries and quantum entropy.

The convolution group connects to renormalization group concepts.

Abstract

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.