Si-Yuan Chen

Classical simulation of quantum systems is fundamental to understanding the boundary between classical and quantum computing. The two leading approaches, tensor networks (TN) and the stabilizer formalism (SF), have traditionally been viewed as distinct, with seemingly disconnected sources of computational hardness. The complexity of TN methods is dictated by entanglement, while SF complexity is governed by the amount of "magic". This leads to a disconnect: states that are simple for one formalism can be maximally complex for the other. For instance, highly entangled stabilizer states are trivial for SF but can be intractable for TN methods. This raises a crucial question: Is there a unified framework or a single indicator that can diagnose the relationship between these two simulation paradigms? In this work, we provide such an indicator, which we term bra-ket entanglement (BKE). We investigate the classical simulation of the general process U OU†, where O can be any operator, from a quantum resource perspective. We show that BKE serves as a crucial diagnostic tool that reveals a deep connection between the resources governing TN and SF. Our central finding is that as the BKE of the initial operator O increases, the simulation resources required by the two approaches transition from being uncorrelated to being highly correlated.