Magic Resources of the Heisenberg Picture

TL;DR

This paper introduces a resource theory for operators' non-stabilizerness, providing a computable measure that quantifies operator magic, with applications in many-body systems and quantum circuit complexity.

Contribution

It develops an operator stabilizer Rènyi entropy as a new magic monotone, linking it to circuit complexity and local dynamical properties, with analytical results in random and integrable systems.

Findings

Operator magic reaches near-maximal value under random evolution.

Operator stabilizer entropy saturates quickly in integrable XXZ circuits.

The measure provides bounds on non-Clifford gate counts and insights into many-body magic generation.

Abstract

We study a non-stabilizerness resource theory for operators, which is dual to that describing states. We identify that the stabilizer R\'enyi entropy analog in operator space is a good magic monotone satisfying the usual conditions while inheriting efficient computability properties and providing a tight lower bound to the minimum number of non-Clifford gates in a circuit. Operationally, this measure quantifies how well an operator can be approximated by one with only a few Pauli strings -- analogous to how entanglement entropy relates to tensor-network truncation. A notable advantage of operator stabilizer entropies is their inherent locality, as captured by a Lieb-Robinson bound. This feature makes them particularly suited for studying local dynamical magic resource generation in many-body systems. We compute this quantity analytically in two distinct regimes. First, we show that under random evolution, operator magic typically reaches near-maximal value for all R\'enyi indices, and we evaluate the Page correction. Second, harnessing both dual unitarity and ZX graphical calculus, we solve the operator stabilizer entropy for interacting integrable XXZ circuit, finding that it quickly saturates to a constant value. Overall, this measure sheds light on the structural properties of many-body non-stabilizerness generation and can inspire Clifford-assisted tensor network methods.