## 3.4 Unitaries as rotations

Now we can finish off our previous discussion (Section 2.10) of the Bloch sphere: we know how single-qubit state vectors correspond to points on the Bloch sphere, but now we can study how *rotations* of this sphere.

Geometrically speaking, the group of **special** (meaning the determinant is equal to

In quantum theory, any two unitary matrices that differ by some global multiplicative phase factor represent the same physical operation, so we are “allowed to” fix the determinant to be

Here we’re going to work with

This last restriction on ^{74}

An alternative way of writing this expression is

Now comes a remarkable connection between two-dimensional unitary matrices and ordinary three-dimensional rotations:

The unitary

The fact that the angle is ^{75} is to parametrise so that

For example,
^{76}

Rotating a state about a **Pauli axis** (the **Pauli rotation**.
We can write these as

Now we can show that the Hadamard gate

In somewhat abstract terms, we make the connection between unitaries and rotations by looking how the unitary group *with zero trace*.
All such matrices

The vector space of **traceless matrices** (i.e. matrices **Lie algebra**.
These arise when studying **Lie groups** — which are a combination of groups and manifolds, i.e. “a geometric space which has an algebraic structure” — via the notion of a **tangent space**.

In particular, the space of **skew-Hermitian** (*Lie* group.

You might be wondering why we have suddenly switched to *skew*-Hermitian instead of Hermitian, but this is really just a mathematician/physicist convention: you can go from one to the other by simply multiplying by

Now,

Next, note that this map is an **isometry**^{77} (a distance preserving operation), since it preserves the scalar product in the Euclidean space: for any two vectors **orthogonal**: orthogonal transformations preserve the length of vectors as well as the angles between them.

Furthermore, we can show^{78} that *only* isometries in three dimensional Euclidean space (which are described by orthogonal matrices with determinant

Thus, in the mathematical lingo, we have established a group homomorphism^{79}
**special orthogonal group** in three dimensions — the group of all rotations about the origin of three-dimensional Euclidean space *real*) matrices.
It follows from Equation (

This mathematical argument is secretly using the language of unit **quaternions**, also known as **versors**, since these provide a very convenient way of describing spatial rotation, and are often used in e.g. 3D computer graphics software.

Physicists, however, usually prefer a more direct demonstration of this rotation interpretation, which might go roughly as follows.
Consider the map

As you can see, we often make progress and gain insights simply by choosing a convenient parametrisation.↩︎

It is a good exercise to show that you can write any

U in this way as well.↩︎Be careful: the precise definition can vary a lot between different texts, with some including a factor of

1/2 , or even a negative sign.↩︎Some mathematicians might say that

\det R_U=1 because “any matrix in\mathrm{U}(2) can be smoothly connected to the identity”.↩︎Recall that a

**homomorphism**is a structure-preserving map between two algebraic structures of the same type; in our case, two groups. An**isomorphism**between algebraic structures of the same type is a homomorphism that has an inverse homomorphism.↩︎