
Quantum Computing is a hype growing more and more these days, but what is it? Why should we use it and how can we get started with it? This is what I would like to tackle in this post.

## Quantum Computing?

As written in the excellent publication by [Richard P. Feynman - Simulating Physics with Computers](/pdf/Richard_P_Feynman-Simulating_Physics_With_Computers.pdf), we would like to have a kind of computer that does exactly the same as nature. A computer that is able to simulate the real world better, and next to that allows us to tackle larger and more complex problems. Think for example about problems such as:

* Quantum Chemistry
* Quantum Dynamics
* Material Science
* Optimization Problems
* Sampling
* Secure Computing
* Cryptography
* Machine Learning
* Searching

To solve these kind of problems, we need to fundamentally change computers as they work now, transforming from classical `bits` towards `qubits`. However before we get started, I recommend you to take a refresher of [Linear Algebra](/post/2019/1/16/quantum-quantum-linear-algebra) to have the fundamentals needed to continue :) 

## Qubits?

### Introduction

Now what are Qubits? When we look at nature, it does not represent everything binary as we do now, take a look at these examples:
* An electron can be in spin up or spin down
* A photon can have a vertical or horizontal polarization

All these kind of examples can be represented as a qubit, representing a 2-state quantum mechanical system.

*How does an electron look with its spins?*

![./spin-quantum-number.png](./spin-quantum-number.png)

There are 2 main aspects in Quantum Physics:
* **Entanglement:** Particles can become connected so that we can predict the state of the other (e.g. one elctron is spin-up, then we know the other is spin-down if they are entangled)
* **Superposition:** A quantum system can be in multiple states at the same time until it is measured

### Mathmatical Representation

Of course to be useful in Quantum Computing, we need to represent these qubits in a mathmatical way. This we can by using the "ket" notation of before, representing a qubit as:

$$
\vert \psi \rangle = \alpha \vert 0 \rangle + \beta \vert 1 \rangle =
\alpha *
\begin{bmatrix}
1 \\
0
\end{bmatrix}
+ \beta *
\begin{bmatrix}
0 \\
1
\end{bmatrix}
$$

With $\alpha$ and $\beta$ being probability amplitudes (describes the behaviour of the system), with extra constraint that $\vert \alpha \vert^2 + \vert \beta \vert^2 = 1$

Seeing that we are working with a 2-level system that uses complex numbers, we can thus represent this on a sphere called the **Bloch Sphere**:

![./bloch-sphere.png](./bloch-sphere.png)

Where the angles are described by: $\alpha = cos(\frac{\theta}{2})$ and $\beta = e^{i * \theta} * sin(\frac{\theta}{2})$

## Using Quantum Circuits to create Quantum Algorithms

### Introduction

To start working with Quantum Computing, we create circuits by using quantum gates. A good overview of the different gates can be found on [Wikipedia](https://en.wikipedia.org/wiki/Quantum_logic_gate), so I will not go too much on depth on these. These Quantum Gates are the building blocks of circuits and allow us to write our different programs and algorithms.

### A mathmatical example

Let's actually calculate an example. When we for example combine a hadamard gate with a CNOT gate:

> The Hadamard Gate is called the "Fair Coin Flip", inducing an equal probability of the qubit being in either of the states

![./H_CNOTGate.png](./H_CNOTGate.png)

Then we know the following:

(Define Psi, Theta, Hadamard, Identity and CNOT)

$$
\vert \psi \rangle =
\begin{bmatrix}
\alpha \\
\beta
\end{bmatrix}
;
\vert \theta \rangle =
\begin{bmatrix}
\gamma \\
\delta
\end{bmatrix}
;
H
=
\frac{1}{\sqrt(2)}
*
\begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}
;
I
=
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
;
CNOT
=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0
\end{bmatrix}
$$

Which we can work out for 2 cross products already (Psi and Gamma; Hadamard and Identity):

(Work out Cross Product of Psi and Gamma, Hadamard and Identity)

$$
\vert \psi \rangle
\otimes
\vert \theta \rangle
=
\begin{bmatrix}
\alpha \gamma \\
\alpha \delta \\
\beta \gamma \\
\beta \delta
\end{bmatrix}
;
H \otimes I
=
\frac{1}{\sqrt(2)}
*
\begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}
*
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
=
\frac{1}{\sqrt(2)}
*
\begin{bmatrix}
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
1 & 0 & -1 & 0 \\
0 & 1 & 0 & -1
\end{bmatrix}
$$

Resulting in:

$$
CNOT * (H \otimes I) * (\vert \psi \rangle \otimes \vert \theta \rangle)
=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0
\end{bmatrix}
\frac{1}{\sqrt(2)}
*
\begin{bmatrix}
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
1 & 0 & -1 & 0 \\
0 & 1 & 0 & -1
\end{bmatrix}
\begin{bmatrix}
\alpha \gamma \\
\alpha \delta \\
\beta \gamma \\
\beta \delta
\end{bmatrix}
\\
=
\frac{1}{\sqrt(2)}
*
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0
\end{bmatrix}
\begin{bmatrix}
\alpha \gamma + \beta \gamma \\
\alpha \delta + \beta \delta \\
\beta \gamma - \beta \gamma \\
\beta \delta - \beta \delta
\end{bmatrix}
=
\frac{1}{\sqrt(2)}
*
\begin{bmatrix}
\alpha \gamma + \beta \gamma \\
\alpha \delta + \beta \delta \\
\alpha \delta - \beta \delta \\
\alpha \gamma - \beta \gamma
\end{bmatrix}
$$

### Quantum Algorithms

Interesting Quantum Algorithms are:

* **Grover Search:** Find an element in an unordered array. Worse case using Quantum here is $\sqrt(n)$ which is normally N using convential algorithms.
* **Short's Factoring:** Factor an integer in its prime factors. With Quantum this is possible in Polynomial time (RSA encryption is based on Prime Factors, showing the impact quantum can have here)

For an implementation of these algorithms, feel free to check this awesome [article](https://people.cs.umass.edu/~strubell/doc/quantum_tutorial.pdf)

Quantum Computing - An Introduction


Quantum requires a lot of Linear Algebra, so much even that it's very interesting to have a quick reminder about it. In here I will explain the basics of Linear Algebra that are needed to work with Quantum Computing.

## Basic Concepts

Most often used in Quantum Computing are the so called "Ket" vectors, which are a representation of the "Dirac Notation".

These vectors can look like: $<\phi|\psi>$ where we have several elements:
* `< x` = Our "bra", mostly representing a row vector
* `|`   = Action of a linear functional on a vector or the scalar product of vectors
* `y >` = our "ket", mostly representing a column vector

## Matrices 

A matrix is a rectangular scheme of numbers with `m rows` and `x columns`. It is an easy way to calculate formulas that are represented in some sort of structure. Most commonly used in Quantum Computing is the need for being able to Multiply these matrices. However to be able to multiple these matrices, we first need to look if the *base requirement of the columns of matrix a being equal to the rows of matrix b* is met. Once that is done, we can start multiplying them.

To multiple those, we then take every row and multiply it with every column, taking the sum of the results.

**Example:**

$$
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}

\begin{bmatrix}
x \\
y
\end{bmatrix}

=

\begin{bmatrix}
ax + by \\
cx + dy
\end{bmatrix}
$$

> Notice that our first matrix is a 2 x 2 matrix, and our second one is a 2 x 1 --> 2 x **2** & **2** x 1, the insides 2's have to be equal.

Putting this into practice:

$$
\begin{bmatrix}
1 & 2 & 3 \\
3 & 4 & 5 \\
6 & 7 & 8
\end{bmatrix}

\begin{bmatrix}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{bmatrix}

=

\begin{bmatrix}
1*0+2*0+3*1 & 1*0+2*1+3*0 & 1*1+2*0+3*0 \\
4*0+5*0+6*1 & 4*0+5*1+6*0 & 4*1+5*0+6*0 \\
7*0+8*0+9*1 & 7*0+8*1+9*0 & 7*1+8*0+9*0
\end{bmatrix}

=

\begin{bmatrix}
3 & 2 & 1 \\
6 & 5 & 4 \\
9 & 8 & 7
\end{bmatrix}
$$

## Complex Numbers

Complex numbers are written by $i$, they solve the fundamental problem of $\sqrt{-1}$ which is impossible to solve, stating it as a new symbol $i$. 

We can visualize complex numbers by using a circle of unit length 1, generating an Imaginary axis and a Real axis.

![./complex-numbers2.png](./complex-numbers2.png)

Where we can find the following statements:

* $z = x + y*i$ 
* $z= e^{j\theta}$ 
* $z = \mid z \mid * (cos(\theta) + i*sin(\theta))$
* $\mid z \mid = \sqrt{x^2 + y^2}$
* $\theta = tg^{-1}(\frac{y}{x})$
