  Quantum algorithms are well-suited to calculate estimates of the energy
spectra for spin lattice systems. These algorithms are based on the efficient
calculation of the discrete Fourier components of the density of states. The
efficiency of these algorithms in calculating the free energy per spin of
general spin lattices to bounded error is examined. We find that the number of
Fourier components required to bound the error in the free energy due to the
broadening of the density of states scales polynomially with the number of
spins in the lattice. However, the precision with which the Fourier components
must be calculated is found to be an exponential function of the system size.
