Fast quantum algorithms for computing the unit group and class group of a number field

@inproceedings{Hallgren2005FastQA,
  title={Fast quantum algorithms for computing the unit group and class group of a number field},
  author={Sean Hallgren},
  booktitle={STOC},
  year={2005}
}
Computing the unit group and class group of a number field are two of the main tasks in computational algebraic number theory. Factoring integers reduces to solving Pell's equation, which is a special case of computing the unit group, but a reduction in the other direction is not known and appears more difficult. We give polynomial-time quantum algorithms for computing the unit group and class group when the number field has constant degree. 

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