Bloch Sphere

A geometric representation of a single qubit’s state as a point on (or inside) a unit sphere.

The Bloch sphere is the standard way to visualize single-qubit states. Every pure state of a qubit corresponds to a unique point on the surface of a unit sphere, making it invaluable for building intuition about quantum operations.

The Geometry

A general single-qubit pure state can be written as:

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

where:

• $\theta$ (theta): angle from the north pole (0 to π)
• $\varphi$ (phi): azimuthal angle around the equator (0 to 2π)

These angles define a point on the sphere:

• North pole ($\theta =0$): $|0⟩$
• South pole ($\theta =\pi$): $|1⟩$
• Equator ($\theta =\pi /2$): Equal superposition states like $|+⟩$, $|-⟩$, $|i⟩$, $|-i⟩$

Key Points on the Sphere

Gates as Rotations

Single-qubit gates correspond to rotations on the Bloch sphere:

• Pauli X: 180° rotation around X-axis (bit flip: $|0⟩\leftrightarrow |1⟩$)
• Pauli Y: 180° rotation around Y-axis
• Pauli Z: 180° rotation around Z-axis (phase flip)
• Hadamard: 180° rotation around the axis halfway between X and Z
• Phase gates: Rotations around Z-axis by various angles

Any single-qubit gate is a rotation on the Bloch sphere, and any rotation can be decomposed into rotations around X, Y, and Z axes.

Mixed States

For mixed states, points lie inside the sphere:

• Pure states: surface of the sphere (radius = 1)
• Mixed states: inside the sphere (radius < 1)
• Maximally mixed state ($\rho =I/2$): center of the sphere

The radius indicates the “purity” of the state.

Limitations

The Bloch sphere only works for single qubits. For multiple qubits, the state space is too high-dimensional to visualize this way. There’s no simple “Bloch sphere” for 2 or more qubits.

Why It Matters

The Bloch sphere is essential for:

• Building intuition about qubit states and gates
• Visualizing quantum operations
• Understanding errors (how states drift on the sphere)
• Teaching quantum computing concepts

See also: Qubit, Quantum State, Pauli Gates, Mixed State