I'm trying to read this quantum computing explainer. I got to the part where they introduce the Hadamard gate, which is this 2x2 unitary matrix $$ H = \frac{1}{\sqrt 2}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} $$, considered over the complex field.
They explain that $HH=I$, and I know from undergrad that $H$ acts as an isometry since its columns are unitary. I know that for any N, all NxN isometries belong to a group whose operation is function composition (or matrix multiplication). But, I want a more thorough understanding of this matrix and similar ones, preferably in terms of symmetry groups.
- What is a conventional name for the group of 2x2 unitary matrices? For now, I'll call it George.
- Can George be generated by a set of rotations and reflections?
- Is George the symmetry group of a set? Maybe the unit circle?
- Can this particular element of George be represented as a series of rotations and reflections? What are they and how did you find them? How many are required?
Thanks all!
The group of $2 \times 2$ unitary matrices is called $U(2)$. It is not generated by rotations and reflections, because rotations and reflections are real, and not every unitary matrix is real.
$H$ itself is a reflection.