## Introduction

The blog post provides an introduction to quantum mechanics and will describe the math necessary to understand quantum mechanics. This shall set the stage to understand quantum computation and quantum algorithms in future posts.

To start with, what is quantum mechanics?. Quantum mechanics is a theory to explain the mechanics/relationships of particles which are sub-atomic in size. Particles at sub-atomic levels exhibits behavior that can't be described by classical mechanics. Let's jump on and perform an experiment akin to what Thomas Young did in the year 1801. Let's observe the behavior of this experiment and let's formulate the necessary math required to explain the behavior.

## Double-slit Experiment and Wave Particle Duality? of light

Let's take a cheap light source such as laser pointer and pass it through a plate that which has two slits in it. Let there be a bare wall after the plate with two slits. Newton theorized light to be made of particles. If light were particles, what would one observe on the wall were two dark bands corresponding to the light that was shone on the plate. This dark bands is similar to what you would get if you fire bullets across the slits. They would hit the wall at the points straight behind them. In contrast, the examination showed a alternating patterns of dark light streaks across the wall. This behavior is typical of a wave which has the phenomena of interference. The waves from the two slits oscillate (the ripples that you see when you drop a pebble on to the water in a pond) and travels towards the wall. At some points they reinforce each other and at some points they cancel out, there by producing the dark light streaks across the wall. This clearly establish the fact that light is a wave.

Now if we position a detector and decide to see which path light took through the slit , we observe two illuminated spots on the wall. This is surprising! and this indeed could happen if only light were to be made up of particles!!. This gives rise to what people call as

**wave particle duality**- that of some time light exhibits behavior of particle and rest of time as that of wave.
The very act of observing which slit the light particles came from disturbs the system state. Before observation the system is in a

**superposition**of states.**Our observing**the system determines it's fate. It is like as if the very act of observation creates a**illusion of existence**. Quantum mechanics is somewhat counter intuitive and expands the realms of science into meta science and human consciousness which is somewhat controversial. However the aim of this article shall be to take a mathematical approach to understand the above phenomena. It sufficiently brings out the beauty that exists in quantum mechanics and enables one to tap into unforeseen computation that nature offers mankind to solve certain problems with super exponential speedup.## A Probabilistic Interpretation

Let's try and calculate the probability of a point p being illuminated. This is given by:

$$ P({point\ p\ being\ illuminated}) = P({photons\ via\ slit\ A}) + P({photons\ via\ slit\ B}) $$

But the above equation clearly can't explain interference. The simple fact from above equation is that the probability by it's fundamental definition can never go down, that is, it can only be a positive value greater than the value of it's constituent probabilities that is being summed. The above equation accounts for the particle nature of light but not it's wave nature.
Let's introduce a new formalism of probability that would explain the above thereby providing a new area of exciting mathematical fun. We introduce the notion of what is something called the

**. Let's denote it by the greek letter $\psi$. This probability amplitude is something similar of analogue of a classical wave and it's value be of the representative of the height of the wave. Thus it can be both positive or negative. In addition, it can take complex values!. Well just a small detour to explain this complex number in that it is neither complex nor imaginary as the words relate to it. Generally the first introduction to complex number would have been done by your high school math teacher. It is generally introduced as roots of the equation $x^2=-1$. This made up the***probability amplitude**imaginary*as everything when squared shall yield positive. But there is something more and beautiful way one can interpret complex numbers.
Complex numbers can relate to transformations which include scaling and rotation. A positive real number represents scaling, say a number 7 means scale by a factor of 7. Likewise negative number represents a flip, that is a rotation of $180$° around the origin. $-1$ represents a half rotation i.e rotation of 180°. If you apply twice the half-rotation, you remain where you are i.e $-1 \times -1 = 1$ (Here multiplication means do the operation one after the other). What would a 90° rotation be? It is nothing but the number $\mathbb{i}$. If we twice make a rotation of 90°, in effect we have made a rotation of 180° i.e, $\mathbb{i}\times\mathbb{i}=-1$. In a similar way the 45° rotation is $\frac{1}{\sqrt 2}(1+\mathbb{i})$. (Note Addition is nothing but do operations in parallel and lay end to end). Thus any arbitrary rotation and scaling could be represented by complex numbers.

We could derive from complex numbers the two quantities the magnitude and phase and thus they can relate to the wave amplitudes. If the phases are opposite, constructive interference and phases being opposite result in destructive interference. One things is that the probabilities are positive real number and hence we take the square of the absolute value of the amplitude to be the probability. We can now arrive at the equation for probability of any point being illuminated, which explains the interference behavior as:

$$ P({point\ p\ being\ illuminated}) = \lvert {A(point\ p\ illuminated)} \rvert^2 = \lvert { A({photons\ via\ slit\ A}) + A({photons\ via\ slit\ B}) } \rvert^2 $$

## Math Required for Quantum Mechanics

Below I detail some of the math and definitions that is required for us to understand quantum mechanics in subsequent blog series posts.

### Complex Vector Space

A complex vector space is one in which the scalars are complex numbers. Vectors are usually represented by column matrices and hence

$$ v = \begin{pmatrix}

x_1 \\

x_2 \\

\vdots \\

x_n \\

\end {pmatrix}, x_i = a_i + ib_i \in \mathbb{C}$$

### Basis

A

$ \begin{pmatrix}**basis**for a given vector space is a set of vectors which can be used in a linear combination to produce any vector in that space. All the basis form an orthogonal set , that is, their scalar (inner) product is zero. Usually the basis vectors are normalized and hence they form an orthonormal set. There can be infinitely many sets of basis vector. For example, in a 2D, the very obvious basis set are1 \\

0 \\

\end{pmatrix},

\begin{pmatrix}

0 \\

1 \\

\end{pmatrix}

$

### Hilbert Space

Hilbert space is a linear vector space with scalar product in $\mathbb{C}$. This space can be of finite or infinite dimensions. I will discuss here the finite dimension Hilbert space. It is denoted by $\mathcal{H}$. The scalar product in Hilbert space is from the set of complex number $\mathbb{C}$ as opposed to the reals $\mathbb{R}$ in the Euclidean case.

${{ x = \begin{pmatrix}

x_1 \\

x_2 \\

\vdots \\

x_n \\

\end {pmatrix}

, y = \begin{pmatrix}

y_1 \\

y_2 \\

\vdots \\

y_n \\

\end {pmatrix}

, x_i, y_i \in \mathbb{C} }}

$

and the vectors belong to Hilbert space, $ x, y \in \mathcal{H}$. The scalar product of the vectors is defined as the $$ xy = (x_1, ...., x_n) \begin{pmatrix}

y_1 \\

y_2 \\

\vdots \\

y_n \\

\end {pmatrix}

= \sum_{i=1}^n x_i^*y_i \in \mathcal{C}

$$

where $x^*$ denotes the complex conjugate.

The norm of a vector $\|x\|$ is defined as

$$ \|x\| = \sqrt {xx} \in \mathbb{R^+} $$

The operators on the Hilbert space map one vector into another. They are linear transformations on the vector space which can be represented by matrices.

$x = Ay$. This can be expanded as:

$$x = Ay \Leftrightarrow \begin{pmatrix}

x_1 \\

x_2 \\

\vdots \\

x_n \\

\end {pmatrix} = \begin{pmatrix}

A_{11} & A_{12} & \cdots & A_{1n} \\

A_{21} & A_{22} & \cdots & A_{2n} \\

\vdots & \vdots& \ddots & \vdots \\

A_{21} & A_{22} & \cdots & A_{2n} \\

\end{pmatrix} \begin{pmatrix}

y_1 \\

y_2 \\

\vdots \\

y_n \\

\end {pmatrix}$$

### Ket Notation and it's dual Bra

The vectors of Hilbert space is defined by the "ket" which is $$ |\psi\rangle \in \mathcal{H}$$ This is nothing but a convenient shorthand notation introduced by Paul Dirac.

There is the dual vector space $\mathcal{H^*}$ to the Hilber vector space. This space is the space of linear functionals over the vector space. The "bra" vectors are the members of the dual vector space and are denoted by $$ \langle\phi| \in \mathcal{H^*}$$

We can write the scalar product as "bra" acting on "ket" and the "bra-ket" is $$\langle\phi|\psi\rangle = \sum_k \psi_k^*\phi_k$$ where $*$ denotes complex conjugation. The following are the properties of the scalar product: $$ \langle\phi|\psi\rangle^* = \langle\psi|\phi\rangle $$ $$\langle\phi|c_1\psi_1 + c_2\psi_2\rangle = c_1\langle\phi|\psi\rangle + c_2\langle\phi|\psi\rangle {\ linear\ in\ ket}$$ $$\langle c_1\phi_1 + c_2\phi_2 | \psi\langle = c_1^*\langle\phi_1|\psi\rangle + c_2^*\langle\phi_2|\psi\langle {\ antilinear\ in\ bra}$$ $$ \langle\psi|\psi\rangle \gt 0 \forall \psi \neq 0 , \langle\psi|\psi\rangle = 0 \iff \psi = 0 \ positive\ definite$$

One can also associate with bra and ket an outer product which is $|\psi\rangle\langle\phi|.

The covectors in the dual space form a one to one correspondence with the vectors in the Hilbert space. We can turn a vector into a covector by the Hermitian conjugate:$$|\psi\rangle^\dagger = \langle\psi|$$. Also the following holds: $$ A|\psi\rangle \rightarrow \langle\psi|A^\dagger $$ $$AB|\psi\rangle \rightarrow \langle\psi|{A^\dagger}{B^\dagger}$$

Please note $A^\dagger$ is called as adjoint to A and is defined as the complex conjugation of the transposed matrix or vice versa i.e $A^\dagger = (A^*)^T = (A^T)^*$

### Mathematical Definitions

- Unitary: Unitary matrices satisfy the property $UU^\dagger=I$ where I is the identity matrix. This means $U^{-1}=U^\dagger$. As a consequence, the unitary matrices preserve inner product and the norm. The eigen values of unitary matrices have the value 1.
- Normal: Normal matrices are ones which satisfy the property $N^\dagger N=NN^\dagger$. We can prove the normal matrices are unitarily diagonalizable and the converse if the matrix is unitarily diagonalizable, then it is Normal. Note $N=U^\dagger DU$. The eigenvalues of $N^\dagger$ are complex conjugates of eigenvalues of N.
- Hermitian: Hermitian matrices are normal matrices satisfy the property $H=H^\dagger$. A hermitian matrix is called positive semidefinite if all its eigenvalues are positive.

## Conclusion

In this blog post, I have introduced the preliminaries required to understand quantum computation. In the next blog post on quantum computation, I shall explain the qubit and the basic quantum gates, that form the building blocks of the quantum circuits. Further series of blog posts shall explain quantum circuits and finally the Shor Algorithm! (Due to time constraints, I am planning to write the Shor algorithm! first and write the other series. Let's see how it goes.)

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