Matrices as gates

February 3, 2025

Single-qubit gates are $2 \times 2$ unitaries. Common examples include

X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},\quad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},\quad H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.

If you keep seeing

\sigma_x, \sigma_y, \sigma_z

and wonder what they mean, read What do the Pauli matrices mean?.

Controlled operations

A CNOT flips the target conditioned on the control being $\lvert 1 \rangle$ . Its matrix is a permutation on the computational basis of two qubits.

Matrix dimensions grow as $2^n$ for

n

qubits. Simulators track the full state vector; real hardware executes the same mathematics with noise and limited connectivity.

Go to Single-qubit gates for a practical walkthrough, then Install Qiskit to run circuits locally.

Dirac notation Tensor Products and Partial Trace (With Intuition)

Matrices as gates

Controlled operations

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