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

A Partially Random Trotter Algorithm for Quantum Hamiltonian Simulations