Long-range nonstabilizerness of topologically encoded states from mutual information

TL;DR

This paper investigates how mutual information can diagnose long-range nonstabilizerness in topologically encoded states, providing a classification method and implications for fault-tolerant quantum gates.

Contribution

It extends the use of mutual information as a diagnostic tool for LRN to 2D topologically ordered systems, with full classification in the toric code.

Findings

Mutual information detects LRN in all non-stabilizer states of the toric code.

The approach partially classifies LRN in non-abelian string-net models with Fibonacci order.

Results constrain the types of logical gates feasible in topological quantum computing.

Abstract

We study long-range nonstabilizerness (LRN), namely the obstruction to remove nonstabilizerness with shallow-depth local quantum circuits. In one-dimensional settings, the mutual information between disconnected spatial regions has proven to be a powerful tool to diagnose LRN. In this work, we focus on encoded states of two-dimensional topologically-ordered systems, and explore the ability of the mutual information to serve as a diagnostic of LRN. Focusing on the concrete setting of lattice models defined on a torus, we show that information about LRN can be gained from the analysis of the mutual information between non-overlapping regions containing non-contractible loops, and of the change of such mutual information under modular real-space transformations. We exemplify this idea in the toric code and the non-abelian string-net model with doubled Fibonacci topological order. In the former case, we show that the mutual information provides a full classification, certifying LRN for all encoded non-stabilizer states. In the latter case, instead, our approach does not lead to a full classification, as it detects LRN for all states except from a finite subset with special transformation properties under the modular group. Finally, we discuss how our results on LRN constrain the logical gates that can be implemented fault-tolerantly on the torus.