Higher-order methods for simulations on quantum computers

To implement many-qubit gates for use in quantum simulations on quantum computers efficiently, we develop and present methods reexpressing exp[-i(H₁+H₂+···)Δt] as a product of factors exp[-iH₁Δt], exp[-iH₂Δt],, which is accurate to third or fourth order in Δt. The methods we derive are an extended form of the symplectic method, and can also be used for an integration of classical Hamiltonians on classical computers. We derive both integral and irrational methods, and find the most efficient methods in both cases.