3.2 Bloch sphere representation

The Bloch sphere is a powerful tool for visualizing qubit states and operations in quantum computing. It represents a single qubit's state space as points on a sphere, with the poles corresponding to the computational basis states |0⟩ and |1⟩.

Points on the Bloch sphere are described by a vector and angles, representing pure qubit states. Single-qubit operations can be visualized as rotations on the sphere. The Bloch sphere relates closely to Pauli matrices, forming a basis for qubit operations.

Qubit States and the Bloch Sphere

Bloch sphere for qubit representation

• Geometric representation of the state space of a single qubit
  • Qubit states represented as points on the surface of the sphere (pure states)
  • North and south poles correspond to the computational basis states $|0\rangle$ and $|1\rangle$
• Visualizes qubit states and their transformations
  • Any point on the surface represents a valid pure state of a qubit (superposition)
  • Mixed states represented by points inside the sphere (non-pure states)

Points on Bloch sphere

• Bloch vector $\vec{r}$ represents a qubit state
  • Vector points from the center of the sphere to a point on the surface
  • Parameterized by angles $\theta$ and $\phi$: $\vec{r} = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)$
• Qubit state corresponding to a point on the Bloch sphere: $|\psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\phi}\sin\frac{\theta}{2}|1\rangle$
  • $\theta$ represents the angle between the Bloch vector and the positive z-axis (latitude)
  • $\phi$ represents the angle between the projection of the Bloch vector onto the x-y plane and the positive x-axis (longitude)

Visualizing single-qubit operations

• Single-qubit gates visualized as rotations of the Bloch vector around the x, y, or z axes
  • Pauli-X gate ($\sigma_x$): rotation of $\pi$ radians around the x-axis (bit flip)
  • Pauli-Y gate ($\sigma_y$): rotation of $\pi$ radians around the y-axis (bit and phase flip)
  • Pauli-Z gate ($\sigma_z$): rotation of $\pi$ radians around the z-axis (phase flip)
• Hadamard gate (H): rotation of $\pi$ radians around the axis equally dividing the x and z axes (creates equal superposition)
• Phase shifts represented by rotations around the z-axis (changes relative phase)

Relationship between the Bloch Sphere and Pauli Matrices

Bloch sphere vs Pauli matrices

• Pauli matrices ($\sigma_x$, $\sigma_y$, $\sigma_z$) form a basis for single-qubit operations
  • Any single-qubit operation decomposed into a linear combination of Pauli matrices and the identity matrix
• Bloch vector $\vec{r}$ related to the density matrix $\rho$ of a qubit state: $\rho = \frac{1}{2}(I + \vec{r} \cdot \vec{\sigma})$
  • $I$ is the 2x2 identity matrix
  • $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$ is the vector of Pauli matrices
• Expectation values of the Pauli matrices for a given qubit state correspond to the coordinates of the Bloch vector
  • $\langle\sigma_x\rangle = \sin\theta\cos\phi$ (x-coordinate)
  • $\langle\sigma_y\rangle = \sin\theta\sin\phi$ (y-coordinate)
  • $\langle\sigma_z\rangle = \cos\theta$ (z-coordinate)