## 2.10 The Bloch sphere

Unitary operations on a single qubit form a group.
More precisely, the set of all ^{39}
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; in Chapter 3 we will actually relate unitary operations on state vectors to rotations in this Euclidean space, demonstrating this isomorphism.^{40}

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**.^{41}
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 in Chapter 3, but we might as well give some specific results here first, since some are easy enough to calculate “by hand”.
Note that *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

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

**!!!TO-DO!!! points on the intersection of the axes with the Bloch sphere are exactly the eigenstates of the corresponding Pauli operator**

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

One can show^{42} that the effect of the rotation represented by unitary

### 2.10.1 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 and the point
p . 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 unit 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 by the group) to the other; in Chapter 3 we will discuss how to go from one*group*(i.e. the thing acting on the space) to the other.↩︎We will revisit this construction again in more detail, and from a slightly different point of view, in Chapter 3.↩︎