Quantum Gate Visualizer

Every single-qubit gate is a $2\times 2$ unitary matrix. Acting on the basis states $\lvert 0 \rangle$ and $\lvert 1 \rangle$, the columns of the matrix tell you exactly where each basis state goes.

Select a gate to see its matrix and its action:

Gate H

0.7070.7070.707−0.707

Hadamard — creates equal superposition.

Input	Output
|0⟩	(|0⟩+|1⟩)/√2
|1⟩	(|0⟩−|1⟩)/√2

The gate cheat-sheet

• X, Y, Z — the Pauli gates. $X$ is the quantum NOT, $Z$ flips the phase of $\lvert 1 \rangle$, and $Y = iXZ$ does both.
• H (Hadamard) — turns basis states into superpositions and back. It maps the $z$-axis to the $x$-axis on the Bloch sphere.
• S and T — phase gates. $S = \sqrt{Z}$ adds a $\pi/2$ phase; $T = \sqrt{S}$ adds $\pi/4$. The set $\{H, T, \text{CNOT}\}$ is universal — it can approximate any quantum computation.

Unitarity

A matrix $U$ is a valid quantum gate iff $U^\dagger U = I$. This guarantees the total probability stays $1$ and the operation is reversible.

Now build multi-qubit circuits in the Circuit Builder.