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.
Read more about this topic: Quantum T-design
Famous quotes containing the word designs:
“It is a mighty error to suppose that none but violent and strong passions, such as love and ambition, are able to vanquish the rest. Even idleness, as feeble and languishing as it is, sometimes reigns over them; it usurps the throne and sits paramount over all the designs and actions of our lives, and imperceptibly wastes and destroys all our passions and all our virtues.”
—François, Duc De La Rochefoucauld (1613–1680)