Magic spreading under unitary Clifford dynamics

TL;DR

This paper investigates how quantum magic, a resource in quantum computing, spreads in Clifford circuits, revealing ballistic growth and delocalization of magic, with implications for quantum error correction and complexity.

Contribution

It introduces a method to infer magic distribution from stabilizer codes and identifies two operational magic length scales with ballistic growth behavior.

Findings

Magic length scales grow ballistically at early times.

Magic delocalizes after initial ballistic growth.

The study links magic dynamics to entanglement velocity.

Abstract

Nonstabilizerness, or quantum magic, presents a valuable resource in quantum error correction and computation. We study the dynamics of locally injected magic in unitary Clifford circuits, where the total magic is conserved. However, the absence of physical observables quantifying magic precludes a direct microscopic or hydrodynamic description of its local distribution and dynamics. Using insights from stabilizer quantum error correcting codes, we rigorously show that the spatial distribution of magic can be inferred from a canonical representation of low-magic states, dubbed the bipartite magic gauge. Moreover, we propose two operationally relevant magic length scales. We numerically establish that, at early times, both length scales grow ballistically at distinct velocities set by the entanglement velocity, after which magic delocalizes. Our work sheds light on the spatiotemporal structure of quantum resources and complexity in many-body dynamics, opening up avenues for investigating their transport properties and further connections with quantum error correction.