Quantum Gate Reference

Every quantum circuit is built from a small set of reusable gates. This lesson is a reference: each single-qubit gate is a $2\times2$ unitary matrix acting on the amplitude vector $\begin{bmatrix}\alpha\\\beta\end{bmatrix}$ of $\alpha|0\rangle + \beta|1\rangle$, and each multi-qubit gate is a permutation (or sign change) on the computational basis states.

Pick a gate and an input below to see its matrix and the resulting amplitudes. Watch how $Y$, $S$, and $T$ paint a phase onto an amplitude without changing its probability.

gate:XYZHST

input:|0⟩|1⟩|+⟩

H1/√2 ·

[

111−1

]

H |0⟩ =

(0.707) |0⟩ + (0.707) |1⟩

|0⟩

50.0%

|1⟩

50.0%

Hadamard: maps |0⟩→|+⟩ and |1⟩→|−⟩, creating equal superposition.

All amplitudes here are real, so the bars fully describe the state up to a sign.

Single-qubit gates

Pauli gates

The three Pauli gates are the fundamental single-qubit errors and also useful operations in their own right.

• $X$ (NOT / bit flip): swaps the amplitudes of $|0\rangle$ and $|1\rangle$.

$X = \begin{bmatrix}0&1\\1&0\end{bmatrix}, \qquad X|0\rangle = |1\rangle, \quad X|1\rangle = |0\rangle$

• $Y$ (bit-and-phase flip): combines a bit flip with a phase, using imaginary entries.

$Y = \begin{bmatrix}0&-i\\i&0\end{bmatrix}, \qquad Y|0\rangle = i|1\rangle, \quad Y|1\rangle = -i|0\rangle$

• $Z$ (phase flip): leaves $|0\rangle$ alone and negates the $|1\rangle$ amplitude.

$Z = \begin{bmatrix}1&0\\0&-1\end{bmatrix}, \qquad Z|0\rangle = |0\rangle, \quad Z|1\rangle = -|1\rangle$

Hadamard

The Hadamard gate $H$ creates and destroys superposition. It maps the basis states to the even mixes $|+\rangle$ and $|-\rangle$:

$H = \tfrac{1}{\sqrt2}\begin{bmatrix}1&1\\1&-1\end{bmatrix}, \qquad H|0\rangle = \tfrac{1}{\sqrt2}\big(|0\rangle + |1\rangle\big) = |+\rangle, \quad H|1\rangle = \tfrac{1}{\sqrt2}\big(|0\rangle - |1\rangle\big) = |-\rangle$

Because $H^2 = I$, applying it twice returns the original state — the two branches interfere and recombine.

Phase gates: S and T

The $S$ and $T$ gates rotate the phase of the $|1\rangle$ amplitude. They do nothing visible to a single measurement, but they are essential for interference and for building arbitrary rotations.

$S = \begin{bmatrix}1&0\\0&i\end{bmatrix}, \qquad T = \begin{bmatrix}1&0\\0&e^{i\pi/4}\end{bmatrix}$

Note the relationships $S = T^2$ and $Z = S^2 = T^4$: $T$ is a $45^\circ$ phase, $S$ a $90^\circ$ phase, and $Z$ a $180^\circ$ phase. The set $\{H, S\}$ generates the Clifford group; adding the non-Clifford $T$ gate makes the set universal for single-qubit operations.

A global phase like $e^{i\theta}$ multiplying the whole state is undetectable. Only a relative phase — a different phase on $|1\rangle$ than on $|0\rangle$ — changes physics, and it does so by steering later interference.

Multi-qubit gates

Multi-qubit gates act on $2^n$ basis states. We order two-qubit states as $|00\rangle, |01\rangle, |10\rangle, |11\rangle$, with the left bit as the control where relevant.

CNOT (controlled-NOT)

$\mathrm{CNOT}$ flips the target qubit when the control is $|1\rangle$, and does nothing when the control is $|0\rangle$:

$\mathrm{CNOT} = \begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}$

input	output
$	00\rangle$
$	01\rangle$
$	10\rangle$
$	11\rangle$

It is the entangling gate behind the Bell state: $\mathrm{CNOT}\,(H\otimes I)|00\rangle = \tfrac{1}{\sqrt2}(|00\rangle + |11\rangle)$.

CZ (controlled-Z)

$\mathrm{CZ}$ applies a $Z$ to the target when the control is $|1\rangle$ — in practice it just flips the sign of $|11\rangle$. It is symmetric: control and target are interchangeable.

$\mathrm{CZ} = \begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\end{bmatrix}$

input	output
$	00\rangle$
$	01\rangle$
$	10\rangle$
$	11\rangle$

Conjugating with Hadamards on the target turns one into the other: $\mathrm{CNOT} = (I\otimes H)\,\mathrm{CZ}\,(I\otimes H)$.

SWAP

$\mathrm{SWAP}$ exchanges the states of the two qubits, leaving the matching basis states $|00\rangle$ and $|11\rangle$ fixed:

$\mathrm{SWAP} = \begin{bmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{bmatrix}$

input	output
$	00\rangle$
$	01\rangle$
$	10\rangle$
$	11\rangle$

A $\mathrm{SWAP}$ decomposes into three CNOTs in alternating directions.

Toffoli (CCNOT)

The Toffoli gate is a doubly-controlled NOT on three qubits ($8\times8$): it flips the target only when both controls are $|1\rangle$. Every basis state is unchanged except the last two, which are swapped:

input	output
$	110\rangle$
$	111\rangle$
all others	unchanged

$\mathrm{CCNOT}\,|a, b, c\rangle = |a,\, b,\, c \oplus (a \wedge b)\rangle$

Toffoli is universal for classical reversible computation — it can build AND, NAND, and fan-out — so it bridges classical logic into the reversible quantum world.

Takeaways

• Single-qubit gates are $2\times2$ unitaries: $X$ flips bits, $Z$ flips phase, $Y$ does both, and $H$ creates superposition.
• $S$ and $T$ add $90^\circ$ and $45^\circ$ phases to $|1\rangle$; with $\{H, S, T\}$ you can approximate any single-qubit gate, and $T$ is the non-Clifford piece.
• A relative phase is invisible to one measurement but governs interference; a global phase is never observable.
• CNOT and CZ are the entangling two-qubit gates (related by Hadamards on the target), SWAP exchanges two qubits, and Toffoli (CCNOT) is universal for reversible classical logic.

References

• Nielsen & Chuang, Quantum Computation and Quantum Information — §4.2–4.3 (single- and multi-qubit gates)
• Qiskit Textbook — Single-qubit and multi-qubit gates