Quantum Computing Fundamentals
Understanding qubits, basis, Hadamard transform, and quantum probability
1. Qubit (quantum bit)
A qubit is a unit vector in a 2-dimensional complex Hilbert space:
\[ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle \]
where
\[ \alpha, \beta \in \mathbb{C}, \quad |\alpha|^2 + |\beta|^2 = 1 \]
- \(|0\rangle, |1\rangle\) are basis states
- \(|\alpha|^2\) = probability of measuring 0
- \(|\beta|^2\) = probability of measuring 1
2. Basis (measurement basis)
A basis is a set of orthonormal vectors used to describe or measure a qubit.
(a) Computational (Z) basis
\[ |0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \]
Measurement probabilities:
\[ P(0) = |\langle 0|\psi\rangle|^2 = |\alpha|^2 \]
\[ P(1) = |\langle 1|\psi\rangle|^2 = |\beta|^2 \]
(b) Hadamard (X) basis
\[ |+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}, \quad |-\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}} \]
Same qubit, different description:
\[ |\psi\rangle = c_+ |+\rangle + c_- |-\rangle \]
3. Basis change (Hadamard transform)
The Hadamard gate:
\[ H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \]
Example:
\[ H|0\rangle = |+\rangle, \quad H|1\rangle = |-\rangle \]
4. Why basis matters (quantum communication)
- Measuring in the wrong basis gives random outcomes
- Non-commuting bases: \[ [Z, X] \neq 0 \]
- Used in protocols like BB84 for eavesdropping detection
A qubit is a vector; a basis is the coordinate system you choose to measure it.
5. Why Hadamard are used?
- Hadmard is used to keep the total probality value 1 because practically probabity cannot be greter than 1. If we do not apply Hadamard then probability of P(0) + P(1) could be 2 (max). Hadamard limts it to P(0) + P(1) = 1/2 + 1/2 = 1 (max)
6. Amplitude and Probability in Quantum Mechanics
In the above you observe the probabity of qbits are directly proportional to their amplitude. Surprised? You should know that the probabily of dection of a wave increases as the legth of wavelength increases. It is a natural phenomena
Amplitude becomes probability by squaring its size.
That rule is called the Born rule.
What is an amplitude? (In general)
- a signed size (can be + or −)
- or a length of an arrow
It is not a probability.
Why amplitude cannot be probability
Amplitudes can be:
- negative
- complex (in advanced cases)
Probabilities must:
- be positive
- be between 0 and 1
So we need a way to turn:
+ or − → always positive
The rule that does this
\[ \text{probability} = |\text{amplitude}|^2 \]
This is not chosen by us — experiments confirm it. This rule is the Born rule.
Simple number example
Amplitude for 0:
\[ a = \frac{1}{\sqrt{2}} \]
Probability:
\[ P(0) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
Same for 1.
Why square and not something else?
- Removes minus signs \[ (-a)^2 = a^2 \]
- Makes probability always positive
- Preserves total probability: \[ a^2 + b^2 = 1 \]
No other simple rule does all three and matches experiments.
Everyday analogy
- Wave height = amplitude
- Energy ∝ (height)²
Double the wave height → 4× energy
Quantum probability behaves the same way.
Example: qubit
Suppose a qubit is:
\[ |\psi\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}} \]
- Amplitude for 0: \(\psi_0 = \frac{1}{\sqrt{2}}\)
- Amplitude for 1: \(\psi_1 = \frac{1}{\sqrt{2}}\)
Probability of measuring 0:
\[ P(0) = |\psi_0|^2 = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
Probability of measuring 1:
\[ P(1) = |\psi_1|^2 = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]