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arXiv · 1503.08579

Translating between the roots of the identity in quantum computers

Wouter Castryck·Jeroen Demeyer·Alexis De Vos·Oliver Keszocze·Mathias Soeken
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Abstract

The Clifford+$T$ quantum computing gate library for single qubit gates can create all unitary matrices that are generated by the group $\langle H, T\rangle$. The matrix $T$ can be considered the fourth root of Pauli $Z$, since $T^4 = Z$ or also the eighth root of the identity $I$. The Hadamard matrix $H$ can be used to translate between the Pauli matrices, since $(HTH)^4$ gives Pauli $X$. We are generalizing both these roots of the Pauli matrices (or roots of the identity) and translation matrices to investigate the groups they generate: the so-called Pauli root groups. In this work we introduce a formalization of such groups, study finiteness and infiniteness properties, and precisely determine equality and subgroup relations.

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Wouter Castryck, Jeroen Demeyer, Alexis De Vos, Oliver Keszocze, Mathias Soeken. 2015-03-30. Translating between the roots of the identity in quantum computers. https://arxiv.org/abs/1503.08579

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