The Bloch Sphere Representation

Visualizing a qubit state using only complex numbers and Dirac notation can be abstract. While the equation $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ describes the mathematics accurately, it does not provide an intuitive sense of how a qubit moves between states. To address this, physicists utilize a geometric model known as the Bloch sphere.

The Bloch sphere maps the state of a single qubit to a point on the surface of a unit sphere (a sphere with a radius of 1). This geometric representation transforms the abstract complex probability amplitudes $\alpha$ and $\beta$ into tangible coordinates. It allows you to visualize quantum gates not as matrix multiplications, but as rotations in 3D space.

The Geometry of the Sphere

In this model, the classical bits are placed at the poles of the sphere along the Z-axis.

• The North Pole represents the state $|0\rangle$. This corresponds to the standard basis vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$.
• The South Pole represents the state $|1\rangle$. This corresponds to the standard basis vector $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$.

Any point on the surface of the sphere that is not at the poles represents a superposition. The closer the point is to the North Pole, the higher the probability that the qubit will collapse to 0 upon measurement. Conversely, the closer it is to the South Pole, the higher the probability of finding it in state 1.

Geometric view of qubit states. The Z-axis represents the probability of measuring 0 or 1. The X and Y axes represent the phase of the qubit.

Spherical Coordinates

A point on a sphere is typically defined by two angles: latitude and longitude. In quantum mechanics, we use similar angles, denoted as $\theta$ (theta) and $\phi$ (phi), to describe the state vector $|\psi\rangle$.

We can rewrite the state equation using these angles:

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

This formula introduces two distinct parameters that control the behavior of the qubit.

The Role of Theta ($\theta$)

The angle $\theta$ measures the distance from the North Pole (the Z-axis). It determines the probability amplitudes of the state.

• When $\theta = 0$, the state is $|0\rangle$.
• When $\theta = \pi$ (180 degrees), the state is $|1\rangle$.
• When $\theta = \pi/2$ (90 degrees), the state is on the equator.

The use of $\theta/2$ in the equation is mathematically significant. In vector space, orthogonal vectors (like $|0\rangle$ and $|1\rangle$) are 90 degrees apart. However, on the Bloch sphere, we want them to be on opposite poles (180 degrees apart) to represent their distinct binary nature. The division by 2 maps the 180-degree physical separation on the sphere to the 90-degree orthogonality in the vector space.

The Role of Phi ($\phi$)

The angle $\phi$ represents the longitude, or the rotation around the Z-axis. This is known as the relative phase of the qubit.

Changing $\phi$ rotates the vector along the latitude lines (horizontal circles) of the sphere. This rotation does not change the distance from the poles. Therefore, changing the phase $\phi$ does not alter the probability of measuring a 0 or a 1. However, phase is essential for quantum algorithms because it dictates how qubits interfere with one another, leading to constructive or destructive interference patterns.

Visualizing Superposition

The Bloch sphere provides an excellent visualization for superposition. Any state that lies on the equator of the sphere implies that the vector is exactly equidistant from the North and South poles.

Mathematically, if $\theta = \pi/2$, then $\cos(\theta/2)$ and $\sin(\theta/2)$ are both equal to $1/\sqrt{2}$. This results in a $50\%$ probability of measuring 0 and a $50\%$ probability of measuring 1.

There are two prominent states on the equator located along the X-axis:

1. The $|+\rangle$ state: This point aligns with the positive X-axis ($\phi = 0$). $|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}$

2. The $|-\rangle$ state: This point aligns with the negative X-axis ($\phi = \pi$). $|-\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}$

While both states yield the same measurement probabilities, they are distinct quantum states due to their phase difference.

Limitations of this Model

The Bloch sphere is a powerful tool for single-qubit intuition, but it has boundaries. It effectively visualizes the state of one isolated qubit. However, when we begin working with multiple qubits and the phenomenon of entanglement in later chapters, the Bloch sphere representation breaks down. Entangled states cannot be described as points on separate spheres.

Despite this limitation, the Bloch sphere remains the standard interface for understanding single-qubit gates. When you apply a "Pauli X gate" or a "Hadamard gate" in the upcoming sections, you will essentially be performing rotations of this vector around the sphere's axes.