Fig. 2: Illustrative diagrams of designing product formulas in different scenarios for short-time simulations.

a A general depiction of the edge-set partition and the corresponding interactive decomposition regarding S. Layers signify edge sets \({\{{E}_{k}^{S}\}}_{k}\), with the line between \({E}_{k-1}^{S}\) and \({E}_{k}^{S}\) representing sub-Hamiltonian interactions \({H}_{k}^{S}\) for integer k ≥ 1. b Visualizing the support expansion of the operator O(t) in the second-order Trotter formula circuit. Here, we adopt the interactive decomposition and even-odd permutation. Each colored block represents the matrix exponential, with blue denoting sub-Hamiltonians with even subscripts and red denoting odd ones. Shaded blocks indicate unitaries outside the expanding support of O(t). We only need to implement the effective (bright) blocks, as outlined in Alg. 1. c Illustration of the regrouping and coloring procedure for a two-dimensional nearest-neighbor lattice Hamiltonian. The regrouping preserves nearest-neighbor interactions, with graph edges corresponding to lattice edges. Edges are colored based on their parities along the vertical and horizontal axes.