Generators and Relations for Un(Z[1/2,i])

Xiaoning Bian
(Dalhousie University)
Peter Selinger
(Dalhousie University)

Consider the universal gate set for quantum computing consisting of the gates X, CX, CCX, omega^dagger H, and S. All of these gates have matrix entries in the ring Z[1/2,i], the smallest subring of the complex numbers containing 1/2 and i. Amy, Glaudell, and Ross proved the converse, i.e., any unitary matrix with entries in Z[1/2,i] can be realized by a quantum circuit over the above gate set using at most one ancilla. In this paper, we give a finite presentation by generators and relations of U_n(Z[1/2,i]), the group of unitary nxn-matrices with entries in Z[1/2,i].

In Chris Heunen and Miriam Backens: Proceedings 18th International Conference on Quantum Physics and Logic (QPL 2021), Gdansk, Poland, and online, 7-11 June 2021, Electronic Proceedings in Theoretical Computer Science 343, pp. 145–164.
Published: 18th September 2021.

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