Qubit Physics

The Bloch sphere is a geometric tool used to describe the state of a qubit. In a two-level qubit system, any quantum state can be expressed as a linear combination of the ground state \( \ket{0} \) and the excited state \( \ket{1} \):

\[ \ket{\psi} = \alpha \ket{0} + \beta \ket{1}, \] where \( \alpha \) and \( \beta \) are complex numbers satisfying the normalization condition \( |\alpha|^2 + |\beta|^2 = 1 \).

If we express \( \alpha \) and \( \beta \) using spherical coordinates, the state can be written as:

\[ \ket{\psi} = \cos\left(\frac{\theta}{2}\right)\ket{0} + e^{i\phi}\sin\left(\frac{\theta}{2}\right)\ket{1}, \] where \( \theta \) is the angle between the state vector and the \( z \)-axis, and \( \phi \) is the phase angle in the \( x \)-\( y \) plane.

This representation allows us to visualize the qubit state as a point on the surface of a unit sphere in three-dimensional space, known as the Bloch sphere. Some important points on the Bloch sphere include:

• North pole (\( \theta = 0 \)): corresponds to \( \ket{0} \)
• South pole (\( \theta = \pi \)): corresponds to \( \ket{1} \)
• Equator (\( \theta = \pi/2 \)): corresponds to equal-amplitude superposition states such as \( \frac{1}{\sqrt{2}}(\ket{0} + e^{i\phi}\ket{1}) \)

The three axes of the Bloch sphere correspond to the expectation values of the Pauli matrices:

\[ x = \langle \sigma_x \rangle, \quad y = \langle \sigma_y \rangle, \quad z = \langle \sigma_z \rangle, \] where: \[ \langle \sigma_x \rangle = 2\text{Re}(\alpha^*\beta), \quad \langle \sigma_y \rangle = 2\text{Im}(\alpha^*\beta), \quad \langle \sigma_z \rangle = |\alpha|^2 - |\beta|^2. \]

Through the Bloch sphere, we can intuitively visualize the evolution of a qubit state. For instance, when a qubit undergoes a rotation operation \( R_x(\theta) = e^{-i\theta \sigma_x / 2} \), its state rotates by angle \( \theta \) around the \( x \)-axis. This geometric view provides great convenience for controlling and measuring qubits in quantum computation.

The Bloch sphere is not only a tool for describing single-qubit states, but also a cornerstone for understanding quantum gates, entanglement, and coherence. In the upcoming sections, we will further explore how to use the Bloch sphere to design and simulate quantum algorithms.

Originally written in Chinese by the author, these articles are translated into English to invite cross-language resonance.