Temporal Entanglement in Chaotic Quantum Circuits

The concept of space evolution (or space-time duality) has emerged as a promising approach for studying quantum dynamics. The basic idea involves exchanging the roles of space and time, evolving the system using a space transfer matrix rather than the time evolution operator. The infinite-volume limit is then described by the fixed points of the latter transfer matrix, also known as influence matrices. To establish the potential of this method as a bona fide computational scheme, it is important to understand whether the influence matrices can be efficiently encoded in a classical computer. Here we begin this quest by presenting a systematic characterization of their entanglement-dubbed temporal entanglement-in chaotic quantum systems. We consider the most general form of space evolution, i.e., evolution in a generic spacelike direction, and present two fundamental results. First, we show that temporal entanglement always follows a volume law in time. Second, we identify two marginal cases-(i) pure space evolution in generic chaotic systems and (ii) any spacelike evolution in dual-unitary circuits-where Renyi entropies with index larger than one are sublinear in time while the von Neumann entanglement entropy grows linearly. We attribute this behavior to the existence of a product state with large overlap with the influence matrices. This unexpected structure in the temporal entanglement spectrum might be the key to an efficient computational implementation of the space evolution.