The particular case of cyclotomic fields whencomputing unit groups by quantum algorithms

The computation of unit and class groups in arbitrary degree number field is done in polynomial time in a similar fashion to the Shor's factoring algorithm. Contrary to the fixed degree case which was solved in 2001 by Hallgren and a follow-up paper of Schmidt and Vollmer (2005), the arbitrary degree case requires errors estimations and is solved by the conjunction of two papers, Eisenträger et al. (2014) and de Boer et al. (2020). In the particular case of cyclotomic fields we propose a version of the algorithm which makes use of cyclotomic units. Indeed, the Shor-like procedure of Eisenträger et al.'s algorithm produces random approximations of vectors in the dual of the lattice of units. In order to guarantee the correction of the algorithm, they have to do the computations in high precision and hence require a large amount of qubits. Thanks to the lattice of cyclotomic units, one can do the computations in smaller precision and reduce the number of qubits.