Universality verification for a set of quantum gates

We establish a relationship between the notion of universal quantum gates and the notion of unitary t -designs. We show that a set of qudit gates S ⊂U (d ) is universal if and only if S forms a δ -approximate t (d ) -design, where δ <1 , t (2 )=6 , and t (d )=4 for d ≥3 . Moreover, we argue that from the application point of view sets S with the δ close to 1 should be regarded as nonuniversal. We also provide a second, more algebraic, criterion for the universality verification. It says that S ⊂U (d ) is universal if and only if the matrices that commute with {U^{⊗t (d )}⊗U^{¯⊗t (d )}|U ∈S } commute also with {U^{⊗t (d )}⊗U^{¯⊗t (d )}|U ∈U (d ) } , where t (2 )=3 , and t (d )=2 for d ≥3 . Finally, we show that the complexity of checking this algebraic criterion scales polynomially with the dimension d .