Gate matrices represent quantum gates as unitary matrices. Understanding notation and matrix properties is essential for gate design and analysis.

A gate on $n$ qubits is a $2^n \times 2^n$ unitary matrix $U$ satisfying:

$$U^\dagger U = U U^\dagger = I$$

where $U^\dagger$ is the conjugate transpose. Unitarity preserves norm: $\langle \psi | U^\dagger U | \psi \rangle = \langle \psi | \psi \rangle = 1$ for all states.

Computational basis for $n$ qubits: $|0\rangle, |1\rangle, \ldots, |2^n - 1\rangle$ (binary labeling). For single qubit:

$$|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$

For two qubits (lexicographic order):

$$|00\rangle = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}, |01\rangle = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}, |10\rangle = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}, |11\rangle = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}$$

For composite systems, states and gates combine via tensor product $\otimes$:

$$|\psi_1\rangle \otimes |\psi_2\rangle = |\psi_1 \psi_2\rangle$$

$$U_1 \otimes U_2 = U_{12}$$

Example: X gate on qubit 1, identity on qubit 2:

$$X \otimes I = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}$$

Gate eigenvalues are phase factors (on the unit circle). For a gate $U$ with eigenvalue $e^{i\theta}$:

$$U|\psi\rangle = e^{i\theta}|\psi\rangle$$

Diagonal gates (like Z, S, T) have easily identifiable eigenvalues. Non-diagonal gates require diagonalization.

• $X, Y, Z$ or $\sigma_x, \sigma_y, \sigma_z$: Pauli gates
• $H$: Hadamard
• $R_X(\theta), R_Y(\theta), R_Z(\theta)$: Rotations
• $C_U$: Controlled-$U$
• $|0\rangle, |1\rangle$ or $\ket{0}, \ket{1}$: Ket notation