3  Lecture 3: Single-Qubit Gates

Class date: September 9, 2026

The previous lecture introduced quantum states, unitary evolution, Hermitian observables, and projective measurement. We now apply these postulates to a single qubit. The two-dimensional state space makes the main gates explicit, while retaining the algebraic structure used for larger quantum systems.

3.1 Single-qubit states and gates

A single-qubit pure state is a normalized vector in \(\mathbb{C}^2\),

\[ \lvert\psi\rangle=\alpha\lvert0\rangle+\beta\lvert1\rangle =\begin{pmatrix}\alpha\\\beta\end{pmatrix}, \qquad \alpha,\beta\in\mathbb{C}, \qquad \lvert\alpha\rvert^2+\lvert\beta\rvert^2=1. \tag{3.1}\]

A single-qubit gate is a \(2\times2\) unitary matrix \(U\). Its action on a state is \(\lvert\psi\rangle\mapsto U\lvert\psi\rangle\). Unitarity, \(U^\dagger U=I\), ensures that the transformed vector remains normalized.

3.2 Pauli gates

The three Pauli gates are

\[ X=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \qquad Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. \tag{3.2}\]

Their actions on the computational basis expose their distinct roles. The \(X\) gate exchanges the two basis states,

\[ X\lvert0\rangle=\lvert1\rangle, \qquad X\lvert1\rangle=\lvert0\rangle. \tag{3.3}\]

The \(Y\) gate also exchanges the basis states, but introduces phases,

\[ Y\lvert0\rangle=i\lvert1\rangle, \qquad Y\lvert1\rangle=-i\lvert0\rangle. \tag{3.4}\]

The \(Z\) gate preserves the computational-basis labels and changes the relative sign,

\[ Z\lvert0\rangle=\lvert0\rangle, \qquad Z\lvert1\rangle=-\lvert1\rangle. \tag{3.5}\]

The sign in Equation 3.5 is invisible when \(\lvert1\rangle\) is considered alone because it is a global phase. It becomes observable inside a superposition, where \(Z\) changes the relative phase between the \(\lvert0\rangle\) and \(\lvert1\rangle\) components.

Theorem 3.1 (Properties of the Pauli gates) Each Pauli matrix \(\sigma\in\{X,Y,Z\}\) is Hermitian and unitary. Moreover,

\[ \sigma^2=I. \tag{3.6}\]

Consequently, every eigenvalue of a Pauli matrix is either \(+1\) or \(-1\).

Hermiticity and unitarity follow directly from Equation 3.2. If \(\sigma\lvert v\rangle=\lambda\lvert v\rangle\), then Equation 3.6 implies \(\lambda^2=1\), which gives the stated eigenvalues.

3.3 The Hadamard gate

The Hadamard gate connects the computational basis, which diagonalizes \(Z\), to the eigenbasis of \(X\):

\[ \mathrm{H}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}. \tag{3.7}\]

It creates the states

\[ \lvert+\rangle=\mathrm{H}\lvert0\rangle=\frac{\lvert0\rangle+\lvert1\rangle}{\sqrt{2}}, \qquad \lvert-\rangle=\mathrm{H}\lvert1\rangle=\frac{\lvert0\rangle-\lvert1\rangle}{\sqrt{2}}. \tag{3.8}\]

These states satisfy \(X\lvert+\rangle=\lvert+\rangle\) and \(X\lvert-\rangle=-\lvert-\rangle\). Thus

\[ X=HZH, \qquad Z=HXH, \tag{3.9}\]

where we used \(\mathrm{H}^\dagger=\mathrm{H}\) and \(\mathrm{H}^2=I\). The gate also maps the \(X\) eigenbasis back to the computational basis:

\[ \mathrm{H}\lvert+\rangle=\lvert0\rangle, \qquad \mathrm{H}\lvert-\rangle=\lvert1\rangle. \tag{3.10}\]

3.4 Rotation gates

Continuous single-qubit transformations are generated by matrix exponentials. For a square matrix \(A\),

\[ e^A=I+A+\frac{A^2}{2!}+\frac{A^3}{3!}+\cdots =\sum_{n=0}^{\infty}\frac{A^n}{n!}. \tag{3.11}\]

Lemma 3.1 (Exponential of a Pauli matrix) For \(\sigma\in\{X,Y,Z\}\) and \(\theta\in\mathbb{R}\),

\[ e^{-i\theta\sigma/2} =\cos\left(\frac{\theta}{2}\right)I -i\sin\left(\frac{\theta}{2}\right)\sigma. \tag{3.12}\]

Using \(\sigma^2=I\), the even powers in Equation 3.11 are multiples of \(I\), whereas the odd powers are multiples of \(\sigma\). Separating the two series gives the cosine and sine terms in Equation 3.12.

The rotations about the three coordinate axes are therefore

\[ R_x(\theta)=e^{-i\theta X/2} =\begin{pmatrix} \cos(\theta/2)&-i\sin(\theta/2)\\ -i\sin(\theta/2)&\cos(\theta/2) \end{pmatrix}, \tag{3.13}\]

\[ R_y(\theta)=e^{-i\theta Y/2} =\begin{pmatrix} \cos(\theta/2)&-\sin(\theta/2)\\ \sin(\theta/2)&\cos(\theta/2) \end{pmatrix}, \tag{3.14}\]

and

\[ R_z(\theta)=e^{-i\theta Z/2} =\begin{pmatrix} e^{-i\theta/2}&0\\ 0&e^{i\theta/2} \end{pmatrix}. \tag{3.15}\]

For example,

\[ R_x(\theta)\lvert0\rangle =\cos\left(\frac{\theta}{2}\right)\lvert0\rangle -i\sin\left(\frac{\theta}{2}\right)\lvert1\rangle. \tag{3.16}\]

Each rotation gate is unitary, and

\[ R_a^\dagger(\theta)=R_a(-\theta)=R_a^{-1}(\theta), \qquad a\in\{x,y,z\}. \tag{3.17}\]

Unlike the Pauli and Hadamard gates, a rotation gate is not Hermitian for a general value of \(\theta\).

3.5 Single-qubit circuit notation

A quantum circuit reads from left to right. A single line carries one qubit, a box on the line represents a quantum gate, and a measurement converts the quantum state into a classical outcome. A double line after measurement denotes classical information.

As a basic example, prepare \(\lvert0\rangle\), apply \(\mathrm{H}\), and measure in the computational basis. Using Equation 3.8, the pre-measurement state is \(\lvert+\rangle\). The Born rule then gives

\[ \Pr(0)=\lvert\langle0\vert+\rangle\rvert^2=\frac12, \qquad \Pr(1)=\lvert\langle1\vert+\rangle\rvert^2=\frac12. \tag{3.18}\]

3.6 Summary

The Pauli and Hadamard gates provide discrete transformations of a single qubit, while exponentials of the Pauli matrices generate continuous rotations. Circuit notation composes these transformations in time and connects unitary evolution to measurement outcomes. The next lecture extends this framework to multi-qubit systems.