Explicit States with Two-sided Long-Range Magic

TL;DR

This paper constructs explicit quantum states with two-sided long-range magic, demonstrating they cannot be generated by simple Clifford and FDU circuits, thus advancing understanding of quantum circuit complexity.

Contribution

It introduces explicit states with two-sided long-range magic, proving they lie outside the first level of the magic hierarchy and connecting magic to many-body phases and error correction.

Findings

Constructed the 'magical cat' state with two-sided long-range magic.

Identified ground states of nonabelian topological orders with this property.

Provided new proof techniques linking magic, phases, and quantum error correction.

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, $|\psi\rangle \propto |0^n\rangle + |+^n\rangle$, 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.