Exact solution of long-range stabilizer R\'enyi entropy in the dual-unitary XXZ model

TL;DR

This paper derives exact solutions for the long-range stabilizer Re9nyi entropy, a measure of quantum magic, in the dual-unitary XXZ model using ZX-calculus, revealing insights into magic generation in quantum circuits.

Contribution

It provides the first exact analytical results for long-range stabilizer Re9nyi entropy in a dual-unitary quantum model, including for all times and Re9nyi parameters.

Findings

Exact solutions for short-time SRE in the thermodynamic limit.

Exact long-range SRE results for all times and Re9nyi parameters in a specific partition.

Numerical verification of results for low Re9nyi parameters and system sizes.

Abstract

Quantum systems can not be efficiently simulated classically due to the presence of entanglement and nonstabilizerness, also known as quantum magic. Here we study the generation of magic under evolution by a quantum circuit. To be able to provide exact solutions, we focus on the dual-unitary XXZ model and a measure of magic called stabilizer R\'enyi entropy (SRE). Moreover, we focus also on long-range SRE, which cannot be removed by short-depth quantum circuits. To obtain exact solutions we use a ZX-calculus representation and graphical rules for the evaluation of the required expressions. We obtain exact results for SRE after short-time evolution in the thermodynamic limit and for long-range SRE for all times and all R\'enyi parameters for a particular partition of the state. Since the numerical evaluation of these quantities is exponentially costly in the R\'enyi parameter, we verify this numerically for low R\'enyi parameters and accessible system sizes and provide numerical results for the long-range SRE in other bipartitions.