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Magic Class and the Convolution Group

Kaifeng Bu, Arthur Jaffe, Zixia Wei

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.

Magic Class and the Convolution Group

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.
Paper Structure (6 sections, 7 theorems, 14 equations, 1 figure)

This paper contains 6 sections, 7 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Conclusion and Discussion
  5. Quantum convolutions on DV systems
  6. The proofs of the Theorems

Key Result

Theorem 1

Any two pure, $n$-qudit states $\ket{\psi}$ and $\ket{\phi}$ in the same circuit magic class are in the same CG magic class.

Figures (1)

  • Figure 1: A diagram of the CG as an iterative coarse-graining quantum circuit. Pure $n$-qudit states can be classified into $n+1$ CG magic classes according to the fixed point under the CG flow.

Theorems & Definitions (13)

  • Theorem 1: Relating the magic classes
  • Example 2
  • Theorem 3: Symmetry characterization of CG magic class
  • Theorem 4: Entropy characterization of CG magic class
  • Lemma 5: BGJ23aBGJ23b
  • Definition 6: Key Unitary for qubit systems
  • Definition 7: Convolution of three states for qubit systems
  • Theorem 8: Restatement of Theorem \ref{['thm:equiv']}
  • proof
  • Theorem 9: Restatement of Theorem \ref{['thm:sym']}
  • ...and 3 more