Quantum Computing

 

 

 

 

Bloch Sphere

 

In most of simple operations or representations, I used the qbit state circle as shown here. It is very simple and easy to understand, but the Qbit state circle has restrictions. Since it is only in 2 dimension, it cannot represent the result of operation that requires 3D representation. On top of it, it cannot represent the state function that has imaginary probability compoment.

There is another way of visualizing a qbit that can overcome all the restrictions mentioned above. It is called Bloch Sphere which is named after the person (Felix Bloch) who invented the concept.

 

 

 

Definition of Bloch Sphere

 

Simply put, Bloch Sphere is a way to respresent a qbit states in a 3D spherical coordinates. As you know, Spherical Coordinates is a system defined by R (radius), theta and phi. In Bloch sphere, R is fixed to 1.  Only Phi and Theta are variable. The qbit state function in the Bloch Sphere is defined as follows.

 

 

In case not familiar with Euler form of the equation and for easy calculation, we can rewrite the above formula as the equation [2] shown below.

 

 

Even though the qbit state function on Bloch sphere looks a little bit complicated, it is still in the form of probability for two qbit state |0> and |1>.  The probability of |0> and |1> (alpha and beta) is transformed into a little bit complicated form in Bloch sphere as shown below. Here you would notice that the probability of |1> can be expressed in complex number.

 

 

 

 

Bloch Vectors on each Axis

 

Now for practice, let's think of some of the important points in the coordinate and how it is expressed in state function.

 

Let's take the point on the circumference that crosses the +z axis. In spherical coordinate, this point is where theta = 0 and phi = 0. If you plug these values into Bloch sphere state function you will get |0> as shown below.

 

 

 

Now Let's take the point on the circumference that crosses the -z axis. In spherical coordinate, this point is where theta = pi and phi = 0. If you plug these values into Bloch sphere state function you will get |1> as shown below.

 

 

 

Now Let's take the point on the circumference that crosses the -z axis. In spherical coordinate, this point is where theta = pi/2 and phi = 0. If you plug these values into Bloch sphere state function you will get the function with the same proability of |0> and |1> as shown below. In many documents, you would see this function is represented to be |+> for simplicity.

 

 

 

Now Let's take the point on the circumference that crosses the -x axis. In spherical coordinate, this point is where theta = pi/2 and phi = pi. If you plug these values into Bloch sphere state function you will get the function with the same proability of |0> and |1> as shown below. In many documents, you would see this function is represented to be |-> for simplicity.

 

 

 

 

Now Let's take the point on the circumference that crosses the +y axis. In spherical coordinate, this point is where theta = pi/2 and phi = pi. If you plug these values into Bloch sphere state function you will get the function with the same proability of |0> and |1> , but the probability of |1> is imaginary number as shown below. In many documents, you would see this function is represented to be |+i> for simplicity.

 

 

 

 

Now Let's take the point on the circumference that crosses the -y axis. In spherical coordinate, this point is where theta = pi/2 and phi = pi. If you plug these values into Bloch sphere state function you will get the function with the same proability of |0> and |1> , but the probability of |1> is imaginary number as shown below. In many documents, you would see this function is represented to be |-i> for simplicity.

 

 

 

 

Putting All Together

 

Putting together all the examples shown in previous section, the labels on the coordinate left side can be renamed as shown on right ? In many technical document about Bloch Sphere uses the labels shown on right. So try to get familiar to this labeling.

  • +z indicates the state |0> and -z indicates the state |1>

  • +x indicates the state |+> and -z indicates the state |->

  • +y indicates the state |+ i> and -y indicates the state |- i>

 

 

 

 

Gate on Bloch Sphere

 

 

Hadamard Gate

 

 

 

 

X Gate

 

 

 

 

Y Gate

 

 

 

 

Z Gate

 

 

 

 

 

Reference :

 

[1] The Bloch Sphere (simply explained)

[2] Lecture 15 2 BLOCH SPHERE

[3] Quantum Optics || 06 The Bloch Sphere 7 24