# Vectors & Vector Spaces 101, for Quantum Computing

If your a classical developer you will have a basis in maths of some form in any case. To become a quantum developer you have to either refresh your knowledge or get back to learning some linear algebra. We start with some vectors and vector spaces.

## To Vectors We Go

The vector is one of the most important mathematical quantities in quantum computing. It has both direction and magnitude. Consider below we have a vector V with x and y components,

\begin{equation*} V = \begin{pmatrix} 3 \\ 5 \end{pmatrix} \end{equation*}

Visually this is 3 units along the x-axis and 5 units along the y-axis pointing in a direction no matter where its tail starts.

In Quantum Computing , vectors are used to show a particular quantum state. These are called state vectors and are visualised using a Bloch Sphere.

## The Bloch Sphere ( a.k.a The Unity Sphere )

The Block Sphere contains all the possible points, the state space, that represent a quantum state to which our state vectors can point. The red vector below for example |y> corresponds to a superposition between |0> and \1> . Imagine the red vector below being able to move and point to any point inside the sphere, representing a different quantum state essentially.

https://en.wikipedia.org/wiki/Bloch_sphere

A 3 minute video on the Block Sphere …

## The Vector Space

A vector is an element of a vector space. A vector space is a set of objects where 2 conditions hold.

The first condition is that a Vector addition of 2 vectors with real numbers in the vector space V produces a 3rd vector with real numbers in the vector space V

\begin{equation*} \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix} \end{equation*}

The second condition says that the scalar multiplication of a real number vector and some real number value n, is also in the vector space, for all n contained in the real number set.

\begin{equation*} \left\lvert \begin{matrix} n|v \\ \end{matrix} \right\rangle = \begin{pmatrix} nx \\ ny \end{pmatrix} \in V \\ \forall n \in \mathbb{R} \end{equation*}

If you have come this far, your doing well. That’s all for now. Next up, Matrices and Matrix Operations !

## Author: Andrew Penrose

IBM STSM, AI Applications Member, IBM Academy of Technology. IBM Master Inventor.