Understanding quantum unitary operators
Unitary operators are defined as a unitary transformation of a rigid body rotation of the Hilbert space. When these unitary operators are applied to the basis states of the Hilbert space, for example, the 
and 
state, the results transform the state vector position but it does not change its length. Let’s see what this means for a qubit. The basis states of a qubit are mapped on the Hilbert space 
as described in Chapter 5, Understanding a Qubit, 
and 
, where 
, and 
are linear transformations that preserve orthogonality over unitary transformations. We’ll wrap our heads around this definition a bit by looking at this mathematically first.
A linear transformation on a complex vector space can be described by a 2x2 matrix, U:
Furthermore, if we obtain the complex transpose of the matrix U as 
, by transposing the matrix U and applying the complex conjugate, as illustrated:
Then we can say that the matrix U is...
