Non-stabilizerness Entanglement Entropy: a measure of hardness in the classical simulation of quantum many-body systems

Classical and quantum states can be distinguished by entanglement entropy, which can be viewed as a measure of quantum resources. Entanglement entropy also plays a pivotal role in understanding computational complexity in simulating quantum systems. However, stabilizer states formed solely by Clifford gates can be efficiently simulated with the tableau algorithm according to the Gottesman-Knill theorem, although they can host large entanglement entropy. In this work, we introduce the concept of non-stabilizerness entanglement entropy which is basically the minimum residual entanglement entropy for a quantum state by excluding the contribution from Clifford circuits. It can serve as a new practical and better measure of difficulty in the classical simulation of quantum many-body systems. We discuss why it is a better criterion than previously proposed metrics such as Stabilizer R\'enyi Entropy. We also show numerical results of non-stabilizerness entanglement entropy with concrete quantum many-body models. The concept of non-stabilizerness entanglement entropy expands our understanding of the ``hardness`` in the classical simulation of quantum many-body systems.