Bloch Sphere Rotations of Single-Qubit Gates

Hello! Welcome to the second lesson in our exploration of quantum computing foundations.

In our previous lesson, we established the density matrix formalism as a universal tool for describing quantum states, distinguishing between pure states (on which we have complete knowledge) and mixed states (which represent statistical uncertainty). We also introduced the von Neumann entropy to quantify this uncertainty.

Today, we will build a powerful geometric intuition for the state of a single qubit and the operations that act upon it. Our learning outcome is to analyze single-qubit gates as rotations on the Bloch sphere, identifying the rotation axis and angle for each standard gate (Pauli, Hadamard, phase).

We will cover:

• The Bloch sphere representation for single-qubit states, connecting it to the density matrix formalism.
• The fundamental principle that single-qubit gates are rotations.
• A detailed analysis of the rotational action of the Pauli (X, Y, Z), phase (S, T, Rz), and Hadamard (H) gates.

This geometric picture is invaluable for visualizing and reasoning about single-qubit algorithms and the effects of noise, which we will encounter later in the course.

1. The Bloch Sphere: A Geometric Canvas for Qubits

While a qubit state is mathematically a vector in a 2D complex Hilbert space, we can visualize it as a vector in 3D real space using the Bloch sphere.

A general pure state of a single qubit can be written as:

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

where $\theta \in [0,\pi ]$ and $\varphi \in [0,2\pi )$ are real parameters. These two angles define a point on the surface of a unit sphere, where $\theta$ is the polar angle (colatitude) and $\varphi$ is the azimuthal angle (longitude).

• The state $|0⟩$ corresponds to $\theta =0$ (the North Pole).
• The state $|1⟩$ corresponds to $\theta =\pi$ (the South Pole).
• Superpositions with equal amplitudes, like $|+⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$, lie on the equator ($\theta =\pi /2$).

This representation connects directly to the density matrix we studied. Any single-qubit density matrix $\rho$ can be expressed as:

$$\rho =\frac{1}{2}(I+\vec{r}\cdot \vec{\sigma })$$

where $I$ is the identity matrix, $\vec{\sigma }=({\sigma }_x,{\sigma }_y,{\sigma }_z)$ is the vector of Pauli matrices, and $\vec{r}$ is a real 3D vector called the Bloch vector.

• For pure states, the vector has unit length, $|\vec{r}|=1$, so they lie on the surface of the sphere.
• For mixed states, the vector has a length less than one, $|\vec{r}|<1$, so they lie in the interior of the sphere. The maximally mixed state $\rho =\frac{1}{2}I$ corresponds to $\vec{r}=\vec{0}$, the center of the sphere.

The length of the Bloch vector is related to the purity we discussed: $\text{tr}({\rho }^2)=\frac{1}{2}(1+|\vec{r}{|}^2)$. You can see that $\text{tr}({\rho }^2)=1$ implies $|\vec{r}|=1$ (pure) and $\text{tr}({\rho }^2)<1$ implies $|\vec{r}|<1$ (mixed).

To solidify your understanding of this representation, please read the following section.

2. Single-Qubit Gates as Rotations

The evolution of a closed quantum system is described by unitary operators. For a single qubit, these are 2x2 unitary matrices. The crucial insight is that every single-qubit unitary operation corresponds to a rotation of the Bloch sphere.

This correspondence arises from a deep connection in group theory: the group of 2x2 special unitary matrices, $SU(2)$, is the double cover of the group of 3D rotations, $SO(3)$. Your background in physics might make you familiar with this relationship from the context of spin-1/2 particles. A key consequence is that a rotation of the state vector in the 2D complex Hilbert space by an angle $\alpha /2$ results in a rotation of the Bloch vector in 3D real space by an angle $\alpha$.

The general operator for a rotation by angle $\alpha$ about an axis defined by the unit vector $\vec{n}$ is given by the matrix exponential:

$$R_{\vec{n}}(\alpha )=e^{-i\frac{\alpha }{2}(\vec{n}\cdot \vec{\sigma })}=\cos (\alpha /2)I-i\sin (\alpha /2)(\vec{n}\cdot \vec{\sigma })$$

This formula is the key to analyzing any single-qubit gate.

3. Analyzing the Standard Gates

Let's now apply this framework to the most common single-qubit gates. The following text provides a comprehensive reference for this analysis. We will walk through it gate by gate.

For a visual walkthrough of these concepts, the following video is an excellent companion.

3.1. Pauli Gates: The Half-Turns

The Pauli gates $X$, $Y$, and $Z$ are the fundamental building blocks. They correspond to rotations of $\pi$ radians (180°) about their respective axes.

• Pauli-X Gate (Bit-Flip): $X={\sigma }_x$. This is a rotation by $\pi$ around the x-axis.

  • $X=R_x(\pi )$ (up to a global phase of $-i$).
  • This is why it flips $|0⟩$ (north pole) to $|1⟩$ (south pole) by rotating the sphere around the x-axis.

• Pauli-Y Gate: $Y={\sigma }_y$. This is a rotation by $\pi$ around the y-axis.

  • $Y=R_y(\pi )$ (up to a global phase of $-i$).

• Pauli-Z Gate (Phase-Flip): $Z={\sigma }_z$. This is a rotation by $\pi$ around the z-axis.

  • $Z=R_z(\pi )$ (up to a global phase of $-i$).
  • This leaves $|0⟩$ and $|1⟩$ in place (they are on the axis of rotation) but flips the phase of states on the equator, e.g., $Z|+⟩=|-⟩$.

Please review Section 3.1 of the Crooks text for the matrix forms and diagrams.

3.2. Phase Gates: Rotations around Z

The Z-gate is a special case of a more general class of gates that implement rotations around the z-axis. These gates modify the relative phase $\varphi$ of the qubit state without changing the polar angle $\theta$, and thus do not affect the measurement probabilities in the computational basis.

The general z-rotation gate is $R_z(\alpha )$. Two important special cases form a universal gate set with the Hadamard gate:

• S Gate (Phase Gate): This is a rotation by $\pi /2$ around the z-axis.

  • $S=\left(\begin{array}{cc}1& 0\\ 0& i\end{array}\right)=R_z(\pi /2)$ (up to a phase). It is the "square root of Z", since $S^2=Z$.

• T Gate: This is a rotation by $\pi /4$ around the z-axis.

  • $T=\left(\begin{array}{cc}1& 0\\ 0& e^{i\pi /4}\end{array}\right)=R_z(\pi /4)$ (up to a phase). It is the "square root of S", since $T^2=S$.

You can find these gates in Sections 3.4 and 3.7 of the Crooks text.

3.3. The Hadamard Gate: Changing Basis

The Hadamard gate is one of the most important gates in quantum computing. It creates superpositions and is key to many quantum algorithms.

• Hadamard Gate: $H=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$.
  • Its action maps the z-axis to the x-axis ($H|0⟩=|+⟩$) and the x-axis to the z-axis ($H|+⟩=|0⟩$).
  • This corresponds to a rotation by $\pi$ around the diagonal axis $\vec{n}=\frac{1}{\sqrt{2}}(\hat{x}+\hat{z})$.

This is a crucial example of a rotation that is not aligned with the cardinal x, y, or z axes. Please review Section 3.5 of the Crooks text for the derivation and diagrams.

4. Visual Summary and Consolidation

The image below provides a helpful visual summary of the rotational actions for several key gates.

Let's consolidate this with a summary table and a short exercise.

Gate	Matrix	Rotation Axis $\vec{n}$	Rotation Angle $\alpha$
Pauli-X	$\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$	$\hat{x}=(1,0,0)$	$\pi$
Pauli-Y	$\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$	$\hat{y}=(0,1,0)$	$\pi$
Pauli-Z	$\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$	$\hat{z}=(0,0,1)$	$\pi$
S Gate	$\left(\begin{array}{cc}1& 0\\ 0& i\end{array}\right)$	$\hat{z}=(0,0,1)$	$\pi /2$
T Gate	$\left(\begin{array}{cc}1& 0\\ 0& e^{i\pi /4}\end{array}\right)$	$\hat{z}=(0,0,1)$	$\pi /4$
Hadamard	$\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$	$\frac{1}{\sqrt{2}}(\hat{x}+\hat{z})=(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})$	$\pi$

Exercise:
The relationship between gates and rotations is powerful because it allows us to reason about sequences of gates geometrically. Consider the sequence $HSH$.

1. Describe the action of this sequence in words, thinking about it as a transformation of the coordinate system.
2. What single rotation gate is this sequence equivalent to (up to a global phase)?

Solution

1. The first $H$ gate transforms the coordinate system, effectively swapping the x and z axes. The $S$ gate then performs a $\pi /2$ rotation around the new z-axis, which is the original x-axis. The final $H$ gate transforms the coordinate system back.
2. Therefore, the sequence $HSH$ is equivalent to a rotation by $\pi /2$ around the x-axis, which is the $R_x(\pi /2)$ gate. Mathematically, this is an example of a similarity transform: $HS{H}^{-1}=H{R}_z(\pi /2)H$. Since $H$ swaps the X and Z operators ($HZH=X$), this transforms a Z-rotation into an X-rotation.

Conclusion

In this lesson, we have developed a geometric understanding of single-qubit states and operations.

Key Takeaways:

• Pure single-qubit states can be visualized as points on the surface of the Bloch sphere, while mixed states occupy its interior.
• Every single-qubit gate is equivalent to a rotation of the Bloch sphere.
• The Pauli gates (X, Y, Z) are rotations by $\pi$ (180°) around their respective axes.
• Phase gates (Rz, S, T) are rotations around the z-axis by varying angles.
• The Hadamard gate is a $\pi$ rotation around an axis diagonal to $\hat{x}$ and $\hat{z}$, which facilitates transformations between the computational ($Z$) basis and the Hadamard ($X$) basis.

This geometric intuition is a cornerstone of quantum information science. It provides a powerful way to reason about the design and analysis of quantum circuits.

Preview of the Next Lesson:

Our next topic is two-qubit gates. We will see how gates like CNOT, CZ, and SWAP operate on pairs of qubits. This will introduce the concept of entanglement and require us to move beyond the simple Bloch sphere picture. To describe multi-qubit systems, we will need a new mathematical tool: the tensor product.