  In this paper, we calculate the upper bound on quantum space complexity of
the quantum algorithms proposed by Biasse and Song (SODA'16) for solving class
group computation and the principal ideal problem using the reductions to
$S$-unit group computation. We follow the approach of Barbulescu and Poulalion
(AFRICACRYPT'23) and the framework given by de Boer, Ducas, and Fehr
(EUROCRYPT'20) and Eisentr\"{a}ger, Hallgren, Kitaev, and Song (STOC'14).
