To efficiently implement many-qubit gates for use in quantum simulations on
quantum computers we develop and present methods reexpressing exp[-i (H_1 + H_2
+ ...) \Delta t] as a product of factors exp[-i H_1 \Delta t], exp[-i H_2
\Delta t], ... which is accurate to 3rd or 4th order in \Delta t. The methods
we derive are an extended form of symplectic method and can also be used for
the integration of classical Hamiltonians on classical computers. We derive
both integral and irrational methods, and find the most efficient methods in
both cases.Comment: 21 pages, Latex, one figur