From time evolution to a quantum circuit
Exact evolution is described by U(t)=exp(-iHt). Neighbouring interaction terms generally do not commute, so we cannot replace this operator by a product of individual gates without introducing an approximation. Trotterization divides the evolution into smaller pieces.
Our quantum implementation uses the interaction picture with respect to the local field:
U(t)=e^{-iH_ft}U_I(t),\qquad
H_I(t)=e^{iH_ft}H_{\mathrm{XXZ}}e^{-iH_ft}.
\]
The field has not disappeared from the physics. Its effect reappears as a time-dependent phase in the exchange interactions. The ZZ term remains unchanged. The imbalance commutes with the field, so the final field rotation does not change this observable. This helps us formulate a circuit specifically for the measurement we need.
Within each time interval, the interactions are divided into even and odd bonds and applied in a symmetric half-full-half sequence: half an even-bond layer, a full odd-bond layer, and another half even-bond layer. The source code uses interval-averaged interactions, including a sinc factor for the changing phase. This remains an approximation: time ordering and noncommuting interactions still require checks.
We use 20 Trotter steps for each requested final time. The step size is therefore t/20 and increases with the final time. The classical TEBD calculation instead uses a fixed smaller time step and reuses the state along the entire trajectory. Both methods target the same Hamiltonian dynamics, but they do not execute the same algorithm.