Simulating quantum dynamics on classical computers is challenging for large systems due to the significant memory requirements. Simulation on quantum computers is a promising alternative, but fully optimizing quantum circuits to minimize limited quantum resources remains an open problem. We tackle this problem presenting a constructive algorithm, based on Cartan decomposition of the Lie algebra generated by the Hamiltonian, that generates quantum circuits with time-independent depth. We highlight our algorithm for special classes of models, including Anderson localization in one dimensional transverse field XY model, where a O(n^2)-gate circuits naturally emerge. Compared to product formulas with significantly larger gate counts, our algorithm drastically improves simulation precision. In addition to providing exact circuits for a broad set of spin and fermionic models, our algorithm provides broad analytic and numerical insight into optimal Hamiltonian simulations.

Data provided here consist of two different Hamiltonians, RMS position of the excitation for each timestep calculated via Trotterization, RMS position of the excitation for each timestep calculated via Cartan Decomposition, and errors of each approach for each Hamiltonian. Initial state is an excitation on the first site (rightmost site with the convention used in the dataset). All the data is obtained via simulating the state vector evolution by using a python code that can be found in https://github.com/kemperlab/cartan-quantum-synthesizer.