  We study implications of unitarity for pseudo-orbit expansions of the
spectral determinants of quantum maps and quantum graphs. In particular, we
advocate to group pseudo-orbits into sub-determinants. We show explicitly that
the cancellation of long orbits is elegantly described on this level and that
unitarity can be built in using a simple sub-determinant identity which has a
non-trivial interpretation in terms of pseudo-orbits. This identity yields much
more detailed relations between pseudo orbits of different length than known
previously. We reformulate Newton identities and the spectral density in terms
of sub-determinant expansions and point out the implications of the
sub-determinant identity for these expressions. We analyse furthermore the
effect of the identity on spectral correlation functions such as the
auto-correlation and parametric cross correlation functions of the spectral
determinant and the spectral form factor.
