A Partially Random Trotter Algorithm for Quantum Hamiltonian Simulations

Given the Hamiltonian, the evaluation of unitary operators has been at the heart of many quantum algorithms. Motivated by existing deterministic and random methods, we present a hybrid approach, where Hamiltonians with large amplitude are evaluated at each time step, while the remaining terms are evaluated at random. The bound for the mean square error is obtained, together with a concentration bound. The mean square error consists of a variance term and a bias term, arising, respectively, from the random sampling of the Hamiltonian terms and the operator-splitting error. Leveraging on the bias/variance trade-off, the error can be minimized by balancing the two. The concentration bound provides an estimate of the number of gates. The estimates are verified using numerical experiments on classical computers.