  We compute the Reidemeister-Turaev torsion of homology cylinders which takes
values in the $K_1$-group of the $I$-adic completion of the group ring
$\mathbb{Q}\pi_1\Sigma_{g,1}$, and prove that its reduction to
$\widehat{\mathbb{Q}\pi_1\Sigma_{g,1}}/\hat{I}^{d+1}$ is a finite-type
invariant of degree $d$. We also show that the $1$-loop part of the LMO
homomorphism and the Enomoto-Satoh trace can be recovered from the leading term
of our torsion.
