Quantum T-design - Unitary Designs

Unitary Designs

Elements of the unitary design are elements of the unitary group, U(d), the group of unitary matrices. A t-design of unitary operators will generate a t-design of states.

Suppose is your unitary design (i.e. a set of unitary operators). Then for any pure state let . Then will always be a t-design for states.

Formally define a unitary t-design, X, if

Observe that the space linearly spanned by the matrices over all choices of U is identical to the restriction and This observation leads to a conclusion about the duality between unitary designs and unitary codes.

Using the permutation maps it is possible to verify directly that a set of unitary matrices forms a t-design.

One direct result of this is that for any finite

With equality if and only if X is a t-design.

1 and 2-designs have been examined in some detail and absolute bounds for the dimension of X, |X|, have been derived.