Introduction

A qubit is the basic unit of information in quantum computing, while classical bits can be described as a 1 or 0, a qubit is simultaneously a combination of both.
It is an integral part of the future of computing and cryptography.

Fundamental Laws of Qubits

Born’s Rule

The probability of getting outcome X is given by square of the overlap between the actual state and vector representation of outcome X.

When you measure a qubit $\ket{\psi} = \alpha\ket{0} + \beta\ket{1}$ in the standard basis:

The probability of yielding 0 is $P(0) = \vert\vert\alpha\vert\vert^2$
The probability of yielding 1 is $P(1) = \vert\vert\beta\vert\vert^2$

No-Cloning Theorem

It is mathematically impossible to create an identical copy of an arbitrary unknown quantum state.

Mathematical Notation

Bra-ket Notation

We represent quantum states using ‘kets’ $\ket{\psi}$. The standard computation basis vectors are defined as orthogonal column vectors:

0 state: $\ket{0} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$
1 state: $\ket{1} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$

We represent quantum conjugate states using ‘bra’ $\bra{\psi}$.
Note that $a^*$ means the conjugate of $a$.

If $\ket{\psi} = \begin{pmatrix} a \\ b \end{pmatrix}$, then $\bra{\psi} = \begin{pmatrix} a^* \ b^* \end{pmatrix}$.

Inner Product

The inner product $\braket{\phi|\psi}$ multiplies a Bra and a Ket to yield a scalar. It measure the overlap between 2 quantum states:
-$\braket{\phi|\psi} = \begin{pmatrix} a^* \ b^* \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix} = a^*\alpha + b^*\beta$

Superposition State

A general qubit state, $\ket{\psi}$ is a linear combination of these basis states:
-$\ket{\psi} = \alpha\ket{0} + \beta\ket{1} = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}$

The Normalization Law

When a qubit is measured, it must collapse to either 0 or 1, hence the total probability must equal to 1:
-$\vert\vert\alpha\vert\vert^2 + \vert\vert\beta\vert\vert^2 = 1$

Qubit States, Basis and Operations

The Hadamard (Diagonal) Basis

State (Diagonal): $\ket{+} = \frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$
State (Anti-Diagonal): $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$
Both Hadamard Basis if measured using a standard basis detector, have a 50/50 exact change of collapsing into 0 or 1.

Tensor Product

To combine multiple qubits into a single system, we use the tensor product. For two qubits:
-$\ket{0} \otimes \ket{0} = \ket{00} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \cdot 1 \\ 1 \cdot 0 \\ 0 \cdot 1 \\ 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}$

General Formula for two-qubit tensor product:
-$a \otimes b = \ket{ab} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} \otimes \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 \cdot \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} \\ a_2 \cdot \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} a_1b_1 \\ a_1b_2 \\ a_2b_1 \\ a_2b_2 \end{pmatrix}$

The four basis states for a two-qubit system are $\ket{00}$, $\ket{01}$, $\ket{10}$, $\ket{11}$.

Tensor products represent the combination of 2 individual qubit into 1 qubit. Reminder that all qubits are simply vectors and the tensor product simply converts 2 2-dimensional vectors into 1 4-dimensional vector.

Bell States

Bell States are important states in quantum computing and cryptography, the Bell State is the smallest unit of entanglement (Between 2 qubits) also known as an ebit.

$\ket{\phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11}) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}$, 50% Chance -> 00 state, 50% Chance -> 11 state
$\ket{\phi^-} = \frac{1}{\sqrt{2}}(\ket{00} - \ket{11}) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ -1 \end{pmatrix}$, 50% Chance -> 00 state, 50% Chance -> 11 state
$\ket{\psi^+} = \frac{1}{\sqrt{2}}(\ket{01} + \ket{10}) = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \end{pmatrix}$, 50% Chance -> 01 state, 50% Chance -> 10 state
$\ket{\psi^-} = \frac{1}{\sqrt{2}}(\ket{01} - \ket{10}) = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ 1 \\ -1 \\ 0 \end{pmatrix}$, 50% Chance -> 01 state, 50% Chance -> 10 state