Bloch Sphere Representation

Bloch sphere is a geometric representation of single-qubit states as points on a 3D sphere of radius 1. Any pure single-qubit state $|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle$ maps to a unique point on the Bloch sphere with spherical coordinates $(\theta, \phi)$, where $\theta \in [0, \pi]$ and $\phi \in [0, 2\pi)$.

• North pole ($\theta = 0$): state $|0\rangle$ (Z eigenstate +1)
• South pole ($\theta = \pi$): state $|1\rangle$ (Z eigenstate -1)
• Positive x-axis ($\theta = \pi/2, \phi = 0$): state $|+\rangle$ (X eigenstate +1)
• Negative x-axis ($\theta = \pi/2, \phi = \pi$): state $|-\rangle$ (X eigenstate -1)
• Positive y-axis ($\theta = \pi/2, \phi = \pi/2$): state $|+i\rangle$ (Y eigenstate +1)
• Negative y-axis ($\theta = \pi/2, \phi = 3\pi/2$): state $|-i\rangle$ (Y eigenstate -1)

Single-qubit gates correspond to rotations of the Bloch sphere. The RX gate rotates around the x-axis, RY around the y-axis, RZ around the z-axis. The Hadamard swaps x and z axes.

Measuring a qubit in basis $\mathbf{n}$ projects the Bloch sphere point onto the axis defined by $\mathbf{n}$. The projection distance (how far the point is from the measurement axis) determines the outcome probability. A point exactly on the axis yields outcome +1 with probability 1.

Pure states lie on the surface of the sphere. Mixed states (statistical mixtures of pure states) lie in the interior. The center (origin) represents the maximally mixed state $I/2$. The distance from the center to a point measures the purity of the state.

The Bloch sphere applies only to single qubits. Multi-qubit states require higher-dimensional representations; their complexity grows exponentially with qubit count, making geometric visualization impractical for more than a few qubits.