Unveiling Connections between Tensor Network and Stabilizer Formalism by Cutting in Time (2505.09512v1)

Abstract: Tensor network and stabilizer formalism are two main approaches for classical simulations of quantum systems. In this work, we explore their connections from a quantum resource perspective on simulating $UOU^\dag$ with $O$ an arbitrary operator ranging from pure state density matrices to Pauli operators. For the tensor network approach, we show that its complexity, quantified by entanglement, is governed by the interplay of two other types of quantum resources, coherence and magic. Crucially, which one has the predominant influence on entanglement is elucidated by what we term the time entanglement of $O$. As time entanglement increases, the dominant resource shifts from coherence to magic. Numerical experiments are conducted to support this claim. For the stabilizer formalism approach, we propose an operator stabilizer formalism to enable its application to arbitrary $O$, whose complexity always relies on magic. The operator stabilizer formalism is also powerful than the standard one for simulating some quantum circuits with special structures. Therefore, as the time entanglement increases, the governing resources between the two approaches change from being uncorrelated to highly correlated.