From Domain Walls to Magnons – Phase Transition in Temporal Entanglement of Dual-Unitary Circuits

Generic quantum circuits near the dual-unitary limit exhibit a rich array of equilibrium phases and dynamical phenomena. Their simulation often relies on approximation schemes such as Matrix Product States (MPS), which fail in regimes where the entanglement entropy grows linearly with system size, as in non-integrable or strongly chaotic dynamics. Using the mapping between dual-unitary circuits and statistical-mechanical models, we bound the error of the temporal Matrix Product State (tMPS) method—a computational scheme that evolves the system in space rather than time—and identify a transition, tunable by the entangling power, between two regimes where the upper bounds on the error are determined by the energetics of a magnon and a domain wall, respectively. We solve for the temporal entanglement at the analytically tractable critical point. These results clarify the connection between temporal correlations and the complexity of classical simulation algorithms, and argue rigorously for the efficacy of tMPS in scalable quantum many-body simulations.