qunatum gate

Quantum Computing — Study Notes

Single Qubit Gates

Complete Reference Guide for Students

📄 Topic 3.1

⏱️ ~45 min read

📑 Table of Contents

1. What is a Qubit?

8. Quick Reference Table

2. What are Quantum Gates?

9. Bloch Sphere Visualization

3. Pauli-X Gate

10. Important Relationships

4. Pauli-Y Gate

11. Worked Example

5. Pauli-Z Gate

12. Key Takeaways

6. Hadamard Gate

13. Common Exam Questions

7. Phase Gates (S & T)

14. Rotation Gates (Rx, Ry, Rz)

01 What is a Qubit?

A qubit is the fundamental unit of quantum information. Unlike a classical bit (which is either 0 or 1), a qubit can exist in a superposition of both states simultaneously.

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$$

Where:

• $\alpha, \beta \in \mathbb{C}$ are complex numbers called probability amplitudes
• $|\alpha|^2 + |\beta|^2 = 1$ — the normalization condition (total probability = 1)

📐 Column Vector Representation

$$|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$

Any qubit state can be written as: $|\psi\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}$

02 What are Quantum Gates?

Property	Classical Gates	Quantum Gates
Reversibility	Reversible or Irreversible	Always Reversible
Operates on	Classical bits	Qubits
Representation	Truth tables	Unitary Matrices
Determinism	Can be non-deterministic	Always deterministic

KEY PROPERTY — Unitarity

$$U^\dagger U = UU^\dagger = I$$

Where $U^\dagger$ is the conjugate transpose (adjoint) of $U$, and $I$ is the identity matrix.

🔄

Reversible

Every gate has inverse

📏

Norm-Preserving

Output always valid

🎯

Deterministic

Same input → same output

03 Pauli-X Gate NOT Gate Bit Flip

The Pauli-X gate is the quantum equivalent of the classical NOT gate. It flips the state of a qubit: $|0\rangle \leftrightarrow |1\rangle$.

$$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$

Action on |0⟩

$$X|0\rangle = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} = |1\rangle$$

Action on |1⟩

$$X|1\rangle = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} = |0\rangle$$

💡 Bloch Sphere: X gate rotates the state by 180° around the X-axis.

Self-inverse: $X^2 = I$, so $X^{-1} = X$

04 Pauli-Y Gate Bit + Phase Flip

The Pauli-Y gate combines a bit flip with a phase flip, introducing imaginary numbers.

$$Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$$

Action on |0⟩

$$Y|0\rangle = i|1\rangle$$

Action on |1⟩

$$Y|1\rangle = -i|0\rangle$$

🎯 Bloch Sphere: Rotates by 180° around the Y-axis. Self-inverse: $Y^{-1} = Y$

05 Pauli-Z Gate Phase Flip

The Pauli-Z gate leaves $|0\rangle$ unchanged but flips the phase of $|1\rangle$ by multiplying it by $-1$.

$$Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

Action on |0⟩

$$Z|0\rangle = |0\rangle \quad \text{(unchanged)}$$

Action on |1⟩

$$Z|1\rangle = -|1\rangle \quad \text{(phase flipped)}$$

⚠️ Important: Z does NOT change measurement probabilities!

$|\alpha|^2$ and $|\beta|^2$ remain the same — only the relative phase changes.

06 Hadamard Gate Most Important Superposition

The Hadamard gate is the most commonly used gate in quantum computing. It creates an equal superposition of $|0\rangle$ and $|1\rangle$.

$$H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$$

H|0⟩ = |+⟩

$$H|0\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}} = |+\rangle$$

Equal superposition (positive phase)

H|1⟩ = |−⟩

$$H|1\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}} = |-\rangle$$

Equal superposition (negative phase)

Key Properties of H Gate:

Property	Expression
Self-inverse	$H = H^\dagger = H^{-1}$
Double application	$H^2 = I$
Conjugation: X → Z	$HXH = Z$
Conjugation: Z → X	$HZH = X$
Basis switch	Converts between Z-basis and X-basis

💡 Bloch Sphere: H rotates 180° around the axis halfway between X and Z (the X+Z diagonal axis).

07 Phase Gates (S & T)

S Gate π/2 Phase

$$S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}$$

$S|0\rangle = |0\rangle$

$S|1\rangle = i|1\rangle$

Key: $S^2 = Z$, $S^\dagger = \begin{pmatrix} 1 & 0 \\ 0 & -i \end{pmatrix}$

T Gate π/4 Phase

$$T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}$$

$T|0\rangle = |0\rangle$

$T|1\rangle = e^{i\pi/4}|1\rangle$

Key: $T^2 = S$, $T^4 = Z$

🔗 Phase Gate Hierarchy:

$$T \xrightarrow{T^2} S \xrightarrow{S^2} Z$$

Applying T twice gives S. Applying S twice gives Z. Applying T four times gives Z.

14 Rotation Gates (R_x, R_y, R_z)

Rotation gates allow arbitrary-angle rotations around each axis of the Bloch sphere. They are the most general single-qubit gates.

R_x(θ)

Around X-axis

$$\begin{pmatrix} \cos\frac{\theta}{2} & -i\sin\frac{\theta}{2} \\ -i\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}$$

R_y(θ)

Around Y-axis

$$\begin{pmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\ \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}$$

R_z(θ)

Around Z-axis

$$\begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}$$

Special Cases:

Rotation	Equivalent To	Note
$R_x(\pi)$	$X$ (with global phase)	180° around X
$R_y(\pi)$	$Y$ (with global phase)	180° around Y
$R_z(\pi)$	$Z$ (with global phase)	180° around Z
$R_z(\pi/2)$	$S$ (with global phase)	90° around Z
$R_z(\pi/4)$	$T$ (with global phase)	45° around Z

08 Quick Reference Table

Gate	Matrix	Key Action	Self-Inverse?
I	$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$	Identity (nothing)	✅ Yes
X	$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$	Bit flip (NOT)	✅ Yes
Y	$\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$	Bit + Phase flip	✅ Yes
Z	$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$	Phase flip	✅ Yes
H	$\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$	Superposition	✅ Yes
S	$\begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}$	π/2 phase shift	❌ No ($S^\dagger$)
T	$\begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}$	π/4 phase shift	❌ No ($T^\dagger$)

09 Bloch Sphere Visualization

Every single-qubit state (up to a global phase) can be represented as a point on the Bloch Sphere:

$$|\psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\phi}\sin\frac{\theta}{2}|1\rangle$$

Parameter	Range	Description
θ	$[0, \pi]$	Polar angle from Z-axis (|0⟩)
φ	$[0, 2\pi)$	Azimuthal angle from X-axis in XY-plane

Key States on Bloch Sphere:

• North Pole: $|0\rangle$ — θ=0

• South Pole: $|1\rangle$ — θ=π

• +X axis: $|+\rangle$ — θ=π/2, φ=0

• −X axis: $|-\rangle$ — θ=π/2, φ=π

• +Y axis: $|{+i}\rangle$ — θ=π/2, φ=π/2

• −Y axis: $|{-i}\rangle$ — θ=π/2, φ=3π/2

Gate	Rotation Axis	Angle
X	X-axis	180° (π)
Y	Y-axis	180° (π)
Z	Z-axis	180° (π)
H	X+Z diagonal axis	180° (π)
S	Z-axis	90° (π/2)
T	Z-axis	45° (π/4)

10 Important Relationships to Remember

🔄 Inverse Gates

$X^{-1} = X$

$Y^{-1} = Y$

$Z^{-1} = Z$

$H^{-1} = H$

$S^{-1} = S^\dagger = \begin{pmatrix} 1 & 0 \\ 0 & -i \end{pmatrix}$

$T^{-1} = T^\dagger = \begin{pmatrix} 1 & 0 \\ 0 & e^{-i\pi/4} \end{pmatrix}$

🔗 Useful Identities

$HXH = Z$

$HZH = X$

$S^2 = Z$

$T^2 = S$

$T^4 = Z$

$H^2 = I$

✖️ Pauli Multiplication Rules

$$XY = iZ$$

$$YZ = iX$$

$$ZX = iY$$

⚠️ Note: $XY \neq YX$ — Pauli matrices do not commute! (anti-commutation: $\{X,Y\} = 0$)

11 Worked Example

❓ Problem: Apply H gate followed by Z gate to |0⟩

Step 1: Apply H to |0⟩

$$H|0\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} = \frac{|0\rangle + |1\rangle}{\sqrt{2}} = |+\rangle$$

Step 2: Apply Z to the result

$$Z \cdot H|0\rangle = Z\left(\frac{|0\rangle + |1\rangle}{\sqrt{2}}\right) = \frac{Z|0\rangle + Z|1\rangle}{\sqrt{2}} = \frac{|0\rangle - |1\rangle}{\sqrt{2}} = |-\rangle$$

✅ Final Answer:

$$ZH|0\rangle = |-\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}$$

💡 Alternative: Using Matrix Multiplication

Combine into single matrix: $ZH = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}$

12 Key Takeaways

✅ Single qubit gates = 2×2 unitary matrices. Every gate must satisfy $U^\dagger U = I$.

✅ Pauli gates (X, Y, Z) are the fundamental building blocks — they rotate by 180° around their respective axes.

✅ Hadamard (H) creates equal superposition — the most frequently used gate in quantum algorithms.

✅ Phase gates (S, T) add relative phases between |0⟩ and |1⟩ without changing measurement probabilities.

✅ Rotation gates (Rx, Ry, Rz) allow arbitrary-angle rotations — any single-qubit operation can be decomposed into these.

✅ All quantum gates are reversible — this is a key difference from classical computing (e.g., NAND gate is not reversible).

✅ Bloch sphere is the geometric tool to visualize single-qubit states and understand gate actions as rotations.

✅ Global phases are physically irrelevant — only relative phases between |0⟩ and |1⟩ matter.

13 Common Exam Question Types

1 Matrix Multiplication

"Apply gate G to state |ψ⟩" → Write state as column vector, multiply by gate matrix.

💡 Tip: Always write the gate matrix on the LEFT, state vector on the RIGHT: G|ψ⟩

2 Sequential Gate Application

"What state results from applying H then X then Z to |0⟩?" → Multiply matrices right to left.

⚠️ Common mistake: For $G_3 G_2 G_1 |\psi\rangle$, multiply $G_3 \times G_2 \times G_1$ (right to left), NOT left to right!

3 Prove Unitarity

"Show that the Hadamard gate is unitary" → Compute $H^\dagger H$ and show it equals $I$.

💡 Tip: For real matrices, $U^\dagger = U^T$ (just transpose). For complex matrices, also conjugate each entry.

4 Gate Decomposition

"Express the S gate in terms of Z and T" → Use identities like $T^2 = S$ or $S^2 = Z$.

5 Bloch Sphere Analysis

"What rotation does the H gate perform?" → Identify the axis and angle of rotation on the Bloch sphere.

6 Measurement Probabilities

"What are the measurement probabilities after applying H to |0⟩?" → Square the absolute values of amplitudes.

💡 Example: $H|0\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle$ → P(0) = $|\frac{1}{\sqrt{2}}|^2$ = 0.5, P(1) = $|\frac{1}{\sqrt{2}}|^2$ = 0.5

🌟 PRO TIP — Order of Matrix Multiplication

For a circuit: |0⟩ —[G₁]—[G₂]—[G₃]— |ψ⟩

The mathematical expression is: $|\psi\rangle = G_3 \cdot G_2 \cdot G_1 \cdot |0\rangle$

Think of it as: the rightmost gate acts first on the state, just like function composition: $f(g(h(x)))$ means h acts first.

Quantum Computing — Single Qubit Gates

Study Notes • Topic 3.1 • 📚

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