Constant Depth Exact Time Evolution of Spin Systems based on Cartan Decomposition*

Spin models are ubiquitous in physics, arising directly in materials physics, or as representatives of electronic systems. Simulating spin systems via classical computers is not effective for large systems, which makes Hamiltonian simulation by quantum computers a promising option due to its success on executing certain algorithms exponentially faster than classical computers. However current generation quantum computers have excessive noise due to gate implementation and loss of coherence; thus avoiding additional error from approximations such as Trotter is ideal. Here we present an algorithm that enables exact time evolution for ordered and disordered spin models for a given time t with a fixed depth circuit. The algorithm is based on Cartan Decomposition of the algebra generated by the Hamiltonian. Although in general the circuit depth scales exponentially with the number of qubits, for a certain class models of interest to the community — e.g. XY model, transverse field Ising model, and the Kitaev spin chain — the circuit depth scales polynomially.