Decomposition methods of the Suzuki-Trotter type of various orders have been derived in different fields. Applying them both to classical ordinary differential equations (ODEs) and quantum systems allows to judge their effectiveness and gives new insights for many body quantum mechanics where reference data are scarce. Further, based on data for 6x6 system we conclude that sampling with sign (minus-sign problem) is probably detrimental to the accuracy of fermionic simulations with determinant algorithms.

A theorem by Yoshida which states that the discretization error for the Suzuki-Trotter (ST) decomposition is one order smaller than for the corresponding approximated Hamiltonian has long been overlooked with grave consequences for the accuracy of algorithms which use decomposition schemes. In our error analysis for the ground state energy of the Hubbard Hamiltonian by the projector quantum Monte Carlo method, we have used various orders of ST-decompositions, including for the first time in this field, pseudo-symplectic methods. We show that higher order does not necessarily imply better convergence, and identify the condition to obtain good convergence with higher order ST-decompositions. For the first order ST-decomposition, the ground state energy does not converge to the values for numerical diagonalization, in agreement with Yoshida's theorem, which may have caused confusion in connection with the "fermionic sign problem". We show how the use of the sign of the fermion determinant is in fact a reweighing method which for the ground state energy violates some basic Monte Carlo properties (the Gaussian distribution of observables), in contrast to sampling without sign. Whether sampling with or without the sign deviates less from the exact ground state energy depends on the system parameters (interaction, filling) and on the test wave function, so sampling with sign gives not necessarily better or more physical than sampling without sign, a result which can be supported by arguments on the error compensation. We discuss the implications for related methods.

We reinvestigate the error in the time reversibility for a self-gravitational sys tem of Komatsu et al.'), who showed an increase of the error with decreasing stepsize for a standard Verlet integrator, to identify the reason for the irreversibil ity under the many possible causes. For (reversible) velocity Verlet and implicit Runge-Kutta type of integrators, the dependence on the stepsize vanishes, but there is a lower limit for the exponential time evolution of the error. The exponent of the error in the reversibility corresponds to the Lyapunov exponent, which means it can not be overcome with improved numerical methods. Because reversible integrators are very often tested with few particle systems, our finding is important as a caveat that for many-particle system, even with reversible integrators and all other numerical errors well controlled, the system trajectories become irreversible if the nonlinear character of the system is too pronounced.