Gate Matrices

To work confidently in quantum computing, one must be fluent in the mathematical language of gate matrices. Every single-qubit gate is a $2 \times 2$ complex unitary matrix, and applying a gate is simply matrix-vector multiplication. This topic serves as a comprehensive mathematical reference and deep dive into the algebraic properties of the standard single-qubit gates we have encountered. By consolidating these matrices and their properties in one place, we establish a rigorous foundation for analyzing and designing quantum circuits.

Imagine a translation dictionary. On one side, you have English words; on the other, you have their Spanish equivalents. A gate matrix is a translation dictionary between physical actions and linear algebra. When we say 'flip the qubit,' the dictionary translates this to the matrix [[0, 1], [1, 0]]. When we say 'create a superposition,' it translates to the Hadamard matrix. By mastering this mathematical dictionary, we can translate any physical quantum circuit into a precise algebraic equation that can be calculated, simulated, and verified.

Let's consolidate the standard single-qubit gate matrices. The Identity gate is $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. The Pauli matrices are $X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, $Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$, and $Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. The Hadamard gate is $H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$. The Phase gates are $S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}$ and $T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}$. The rotation gates are $R_x(\theta) = \begin{pmatrix} \cos(\theta/2) & -i\sin(\theta/2) \\ -i\sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$, $R_y(\theta) = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$, and $R_z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}$. All of these matrices are unitary ($U^\dagger U = I$). The Pauli matrices and the Hadamard matrix are also Hermitian ($U^\dagger = U$), which means they are their own inverses ($U^2 = I$). The Phase gates $S$ and $T$ are unitary but not Hermitian, meaning their inverses are their conjugate transposes ($S^\dagger$ and $T^\dagger$).