what is Bloch Sphere in quantum computing. explain with examples

In quantum computing, the Bloch Sphere is a geometric representation of the state of a single qubit.¹ While a classical bit is like a light switch (either on or off), a qubit is like a point on the surface of a globe, allowing it to exist in an infinite number of states between 0 and 1.²

1. The Geometry of a Qubit

The Bloch Sphere is a unit sphere (radius = 1) where every point on the surface represents a pure quantum state.³

• The North Pole (⁴$|0\rangle$): Represents the state "0".⁵

• The South Pole (⁶$|1\rangle$): Represents the state "1".⁷

• The Surface: Represents all possible superpositions. Any point that isn't a pole is a mix of $|0\rangle$ and $|1\rangle$.

• The Interior: Represents "mixed states," which occur when a qubit has lost its quantum properties due to noise (decoherence).⁸

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2. The Mathematical Foundation

A qubit state $|\psi\rangle$ is written using two angles, $\theta$ (theta) and $\phi$ (phi):

$$\ket{\psi} = \cos\left(\frac{\theta}{2}\right)\ket{0} + e^{i\phi}\sin\left(\frac{\theta}{2}\right)\ket{1}$$

• $\theta$ (Polar Angle): Determines the "latitude."⁹ It tells you the probability of measuring a 0 or a 1.¹⁰ If ¹¹$\theta = 0$, you are at the North Pole (¹²$|0\rangle$).¹³

• $\phi$ (Azimuthal Angle):¹⁴ Determines the "longitude." This represents the quantum phase, which is invisible during a standard measurement but crucial for quantum interference and algorithms.¹⁵

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3. Examples of States and Gates

Example 1: The Plus State ($|+\rangle$)

Imagine the vector pointing exactly at the "Equator" on the positive X-axis. This is the ¹⁶$|+\rangle$ state.¹⁷

• Position: $(\theta = \pi/2, \phi = 0)$.

• Meaning: It is an equal superposition. If you measure it, you have a 50% chance of getting 0 and a 50% chance of getting 1.¹⁸

Example 2: The Quantum "NOT" Gate (X-Gate)

In classical computing, a NOT gate flips a 0 to 1. On the Bloch Sphere, the X-Gate is a 180° rotation around the X-axis.¹⁹

• Action: If your qubit is at the North Pole (²⁰$|0\rangle$), the X-gate rotates it along the surface of the sphere down to the South Pole (²¹$|1\rangle$).²²

Example 3: The Hadamard Gate (H-Gate)

This is the "superposition creator."

• Action: It rotates the state 180° around an axis diagonal between X and Z. It effectively moves a qubit from the North Pole ($|0\rangle$) to the Equator ($|+\rangle$), putting it into a state of "maybe 0, maybe 1."

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Why is it useful?

The Bloch Sphere is the primary tool for visualizing quantum algorithms.²³ Instead of looking at complex matrices of numbers, researchers can "see" a quantum program as a series of rotations. If you want to change the probability of a result, you tilt the vector ($\theta$); if you want to change how the qubit interacts with others, you spin it ($\phi$).

Note: The Bloch Sphere only works for one qubit.²⁴ Because of entanglement, a system of two or more qubits is too complex to be represented by a simple 3D sphere!

Would you like me to explain how specific quantum gates, like the Phase gate, move the vector around this sphere?

The Bloch Sphere (simply explained)

This video provides a visual walkthrough of how a qubit's state maps to the sphere's surface and how gates act as physical rotations.