Deciding universality of quantum gates

E. Solano, França M. Santos, P. Milman

Adam Sawicki, Katarzyna Karnas

We consider the problem of deciding if a set of quantum one-qudit gates $$\mathcal {S}=\{g_1,\ldots ,g_n\}\subset G$$ is universal, i.e. if $${<}\mathcal {S}{>}$$ is dense in G, where G is either the special unitary or the special orthogonal group. To every gate g in $$\mathcal {S}$$ we assign the orthogonal matrix $$\mathrm {Ad}_g$$ that is image of g under the adjoint representation $$\mathrm {Ad}:G\rightarrow SO(\mathfrak {g})$$ and $$\mathfrak {g}$$ is the Lie algebra of G. The necessary condition for the universality of $$\mathcal {S}$$ is that the only matrices that commute with all $$\mathrm {Ad}_{g_i}$$ ’s are proportional to the identity. If in addition there is an element in $${<}\mathcal {S}{>}$$ whose Hilbert–Schmidt distance from the centre of G belongs to $$]0,\frac{1}{\sqrt{2}}[$$ , then $$\mathcal {S}$$ is universal. Using these we provide a simple algorithm that allows deciding the universality of any set of d-dimensional gates in a finite number of steps and formulate a general classification theorem.

Adam Sawicki, Katarzyna Karnas

Compact-form criteria are provided for deciding universality of a given set of gates in a finite number of steps. The criteria may also indicate what types of gates can be added to a set of nonuniversal gates to turn it into a universal set.

Emmanuel Jeandel

Long, Gui-Lu, Liu +3 more

Yinuo Xue, Qian Chen, Jing-Song Huang

We present a criterion that serves as the basis for a polynomial-time algorithm to decide whether a finite set of qudit gates exponentiated by some Hamiltonians is universal. Our approach formulates universality in Lie algebraic terms and applies Borel--de Siebenthal theory with a diagonal generator having incommensurate spectrum. In this framework, nonuniversality is detected by invariant subspaces, equivalently by a graph-connectivity obstruction, while universality is repaired by adding generators that couple disconnected components. We further prove that two generators are sufficient for universal control. Our work reveals a profound link between qudit universality and irreducibility of Lie algebra representations.