## 2.9 The Bloch sphere

Unitary operations on a single qubit form a group.
More precisely, the set of all *non-abelian*) group under matrix multiplication, denoted by ^{52}
That is,

There are many ways to introduce this isomorphism.
Here we will just show how to represent single-qubit state vectors in terms of Euclidean vectors in three dimensions; later (in Section 3.4) we will actually relate unitary operations on state vectors to rotations in this Euclidean space, demonstrating this isomorphism.^{53}

Any single-qubit state can be written as

The parametrisation in terms of

We call this sphere the **Bloch sphere**, and the unit vector **Bloch vector**.
This is a very useful way to visualise quantum states of a single qubit and unitary operations that we perform on it.
Any unitary action on the state vector will induce a rotation of the corresponding Bloch vector.
But what kind of rotation?

We give a complete answer to this question soon, in Section 3.4, but we might as well give some specific results here first, since some are easy enough to calculate “by hand”.
Here is one fundamental observation: *any two orthogonal state vectors appear on the Bloch sphere as two Bloch vectors pointing in opposite directions*.
Now, the two eigenvectors of a single-qubit unitary *This* is the axis about which the Bloch vector is rotated when

It is instructive to work out few simple cases and get a feel for the rotations corresponding to the most common unitaries.
For example, it is easy to check that a phase gate

As previously mentioned, the Pauli operator

How about the Hadamard gate?
Like the Pauli operators, it squares to the identity (

We will eventually show that the effect of the rotation represented by unitary

### 2.9.1 Drawing points on the Bloch sphere

We know that the state

- Calculate
\lambda=\beta/\alpha (assuming that\alpha\neq0 , since otherwise|\psi\rangle=|1\rangle ). - Write
\lambda=\lambda_x+i\lambda_y and mark the pointp=(\lambda_x,\lambda_y) in thexy -plane (i.e. the plane\{z=0\} ). - Draw the line going through the south-pole (which corresponds to
|1\rangle ) and the pointp . This will intersect the Bloch sphere in exactly one other point, and this is exactly the point corresponding to|\psi\rangle .

Note that this lets you *draw* the point on the sphere, but doesn’t (immediately) give you the *coordinates* for it.
That is, this method is nice for geometric visualisation, but the parametrisation method is much better when it comes to actually doing calculations.

Note that

\mathrm{U}(1)\cong\mathbb{C}^\times , where\mathbb{C}^\times is the multiplicative group of invertible elements of the complex numbers, i.e. the set\mathbb{C}\setminus\{0\} with the group operation given by multiplication.↩︎That is, we have the group

\mathrm{U}(2) acting on the space of single-qubit state vectors, and we have the group\mathrm{SO}(3) acting on the unit sphereS^2\subset\mathbb{R}^3 . In this chapter we will discuss how to go from one*space*(i.e. the thing being acted upon) to the other; in Section @ref(unitaries-as-rotations we will discuss how to go from one*group*(i.e. the thing doing the acting) to the other.↩︎