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, especially in tensor network states related methods. 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 nonstabilizerness 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 measure of difficulty in the classical simulation of quantum many-body systems, which takes both entanglement and nonstabilizerness into consideration. We discuss why it is a better criterion than previously proposed metrics such as stabilizer Rényi entropy when measuring the difficulty in the classical simulation with tensor network states related method. We also show numerical results of nonstabilizerness entanglement entropy with concrete quantum many-body models. The concept of nonstabilizerness entanglement entropy expands our understanding of the “hardness” in the classical simulation of quantum many-body systems.

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• Quantum information theory
• Quantum simulation
• Quantum many-body systems
• Density matrix renormalization group