  Unit group computations are a cryptographic primitive for which one has a
fast quantum algorithm, but the required number of qubits is $\tilde O(m^5)$.
In this work we propose a modification of the algorithm for which the number of
qubits is $\tilde O(m^2)$ in the case of cyclotomic fields. Moreover, under a
recent conjecture on the size of the class group of $\mathbb{Q}(\zeta_m +
\zeta_m^{-1})$, the quantum algorithms is much simpler because it is a hidden
subgroup problem (HSP) algorithm rather than its error estimation counterpart:
continuous hidden subgroup problem (CHSP). We also discuss the (minor) speed-up
obtained when exploiting Galois automorphisms thanks to the Buchmann-Pohst
algorithm over $\mathcal{O}_K$-lattices.
