  By use of the recently derived $universal$ discrete imaginary-time propagator
of the harmonic oscillator, both thermodynamic and Hamiltonian energies can be
given analytically, and evaluated numerically at each imaginary time step, for
$any$ short-time propagator. This work shows that, using only currently known
short-time propagators, the Hamiltonian energy can be optimized to the twelfth
order, converging to the ground state energy of the harmonic oscillator in as
few as three beads. This study makes it absolutely clear that the widely used
second-order primitive approximation propagator, when used in computing the
thermodynamic energy, converges extremely slowly with increasing number of
beads.
