A Two Unitary Operator Model for Quantum Computation Simulation

We show that knowledge about the eigenvalues of the product of two unitary operators on r-qubits, exp(iH_C) exp(iH_S) = exp (iH_eff), can enhance one's ability to factor the product of two prime numbers.  The effective hamiltonian H_S can be mapped onto a free fermion model, and the effective hamiltonian H_C can be mapped onto a model of spins rotating in a spatially varying magnetic field. However, the product of the two operators leads to an effective hamiltonian, H_eff, that is strongly interacting with up to r-bit interactions.  Thus, it is challenging to solve even approximately.  Numerical results on small systems are presented to evaluate several different approximate techniques: perturbation in a number of different parameters, calculation of the entanglement entropy, and evaluation of the trace of H_eff, as well as a numerical test of the BCH expansion using H_C and H_S. Using the trace of H_eff is the most promising approach, although it is hampered by H_eff only being determined by H_C and H_S up to factors of 2π.