On the Universality and Membership problems for quantum gates

What is it about?

We study the Universality and Membership Problems for gate sets consisting of a finite number of quantum gates. Our approach relies on the techniques from compact Lie groups theory. We also introduce an auxiliary problem called Subgroup Universality Problem, which helps in solving some instances of the Membership Problem, and can be of interest on its own. The resulting theorems are mainly formulated in terms of centralizers and the adjoint representations of a given set of quantum gates.

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Why is it important?

Our work exploits the connection between some natural problems for quantum gates and Lie theory.

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