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Explicit States with Two-sided Long-Range Magic

Zhi Li

Abstract

Nonstabilizerness, or magic, is a necessary resource for quantum advantage beyond the classically simulatable Clifford framework. Recent works have begun to chart the structure of magic in many-body states, introducing the concepts of long-range magic -- nonstabilizerness that cannot be removed by finite-depth local unitary (FDU) circuits -- and the magic hierarchy, which classifies quantum circuits by alternating layers of Clifford and FDUs. In this work, we construct explicit states that provably possess two-sided long-range magic, a stronger form of magic meaning that they cannot be prepared by a Clifford circuit and a FDU in either order, thus placing them provably outside the first level of the magic hierarchy. Our examples include the ``magical cat" state, $|ψ\rangle \propto |0^n\rangle + |+^n\rangle$, and ground states of certain nonabelian topological orders. These results provide new examples and proof techniques for circuit complexity, and in doing so, reveal the connection between long-range magic, the structure of many-body phases, and the principles of quantum error correction.

Explicit States with Two-sided Long-Range Magic

Abstract

Nonstabilizerness, or magic, is a necessary resource for quantum advantage beyond the classically simulatable Clifford framework. Recent works have begun to chart the structure of magic in many-body states, introducing the concepts of long-range magic -- nonstabilizerness that cannot be removed by finite-depth local unitary (FDU) circuits -- and the magic hierarchy, which classifies quantum circuits by alternating layers of Clifford and FDUs. In this work, we construct explicit states that provably possess two-sided long-range magic, a stronger form of magic meaning that they cannot be prepared by a Clifford circuit and a FDU in either order, thus placing them provably outside the first level of the magic hierarchy. Our examples include the ``magical cat" state, , and ground states of certain nonabelian topological orders. These results provide new examples and proof techniques for circuit complexity, and in doing so, reveal the connection between long-range magic, the structure of many-body phases, and the principles of quantum error correction.

Paper Structure

This paper contains 21 sections, 14 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. ZX-cat State
  3. $\mathsf{Clifford}\circ\mathsf{FDU}$
  4. $\mathsf{FDU}\circ\mathsf{Clifford}$
  5. Approximated Case
  6. Technical Comments
  7. Nonabelian Topological Order
  8. Proof of \ref{['thm:nonTO']}
  9. Comments on Transversal Gates
  10. Via Universality of Mapping Class Group
  11. Via Symmetries in TQFT
  12. Discussion and Outlook
  13. ZX-cat State: Alternative Proof
  14. Improved Bound for Approximated "Codes" with Large Distances
  15. ZX-cat States as Approximated "Codes"
  16. ...and 6 more sections

Key Result

Theorem 1

The state $\ket\psi$ cannot be prepared by $\mathsf{Clifford}\circ\mathsf{FDU}$, or $\mathsf{FDU}\circ\mathsf{Clifford}$, within a constant approximation error.

Figures (2)

  • Figure 1: The backward light cone $\mathcal{L}_b$ and the forward light cone $\mathcal{L}_f$.
  • Figure 2: Yellow stripe $S=S_v\cup S_h$. Red dots denote excitations in $U\ket{\psi}$. $T=\partial T\cup \mathring{T}$.

Theorems & Definitions (28)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Definition 1: topologically transversal gate, TTG
  • ...and 18 more