Introduction to Quantum Computing: Notation

Hi All, I figured as I am learning Quantum Computing I could create a curriculum out of the material and teach others. So let’s begin with the notation and a basic mathematical concept, a vector.

A vector is a list of numbers. The dimension of a vector is the number of numbers in the list.

vector = [ 4 5 6 ] 

A vector can be written horizontally or vertically:

horizontal_vector = [ 0 5 3 ]
vertical_vector = [ 4
                    5
                    6 ]
    

Paul Dirac was the scientist that established the foundation of quantum mechanics and computing that we see today. His notation of how we describe quantum states is popular and used widely today because it easy to adopt.

Figure 1: Paul Dirac in some older years

For the word braket, for quantum notation denote bra and ket as horizontal (w) and vertical vector (v).

In the language, we most commonly use|v> and <w| as this will be more apparent. This will take a little time to pickup so be patient with yourself.

ket = v = |v> = vertical vector 
bra = w = <w| = horizontal vector

So lets start off easy with two kets a and b with dimension of 3 for both:

|a> = [ a1
        a2
        a3 ]

|b> = [ b1
        b2
        b3
      ]

If we take the sum of these two vectors then:

|a + b> = [ a1 + b1
            a2 + b2
            a3 + b3 ]

If we think about this problem

What is the sum of kets a and b?

|a> = [ 3
        1 ]

|b> = [ 1
        2 ]

Which is intuitive:

|a + b> = [ 4
            3 ]

If you think about it graphically it would look something like this:

Figure 2: Graphical Representation of a vector addition

If we now take the the bra a and ket b which is with a dimension length of 3 for both.

<a| = [ a1 a2 a3]

|b> = [ b1
        b2
        b3 ]

Then if we multiply these two vectors together:

<a|b> = [ a1 a2 a3][ b1
                     b2
                     b3 ]

What if we do this:

<a | a> = a1^2 + a2^2 and a3^2

If determine some values and what properties it means for the vectors. When two vectors are multiplied and their values are zero it denotes the vectors are 90 degrees from each other.

Figure 3: Orthogonal Vectors

So this means in our quantum notation:

<a | b> = 0

If we have a value where, this is a unit vector, an established property where a vector multiplied by itself is 1:

<a | a> = 1

An orthonormal basis set means that in a set of vectors there will be one or more orthogonal vectors.

Homework #1

For kets a, b, and c with a dimension of 2, determine which vectors are orthogonal from each other and what is the orthonormal basis set.

|a> = [ 3 
        1 ]

|b> = [ 1
        2 ]

|c> = [ 2
        6 ]

Hint:

<a | b> = ?
<a | c> = ?
<b | c> = ?

Happpy cheminformatics and quantum learning!