# Sean Hallgren: A quantum algorithm for computing the unit group of an arbitrary degree number field

Monday, October 20, 2014 — 2:30 PM to 3:30 PM EDT

## Sean Hallgren, Pennsylvania State University

Computing the group of units in a field of algebraic numbers is one of the central tasks of computational algebraic number theory. It is believed to be hard classically, which is of interest for cryptography. In the quantum setting, efficient algorithms were previously known for fields of constant degree. We give a quantum algorithm that is polynomial in the degree of the field and the logarithm of its discriminant. This is achieved by combining three new results. The first is a classical algorithm for computing a basis for certain ideal lattices with doubly exponentially large generators. The second shows that a Gaussian weighted superposition of lattice points, with an appropriate encoding, can be used to provide a unique representation of a real-valued lattice. The third is an extension of the hidden subgroup problem to continuous groups and a quantum algorithm for solving the hidden subgroup problem (HSP) over $\mathbb R^n$. Joint work with Kirsten Eisentraeger, Alexei Kitaev, and Fang Song.

Location
QNC - Quantum Nano Centre
0101
200 University Avenue West

Waterloo, ON N2L 3G1

### May 2021

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