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The controlled-NOT gate is a two-qubit unitary operator that flips a target qubit when its control is in $|1⟩$. In the ordered labels below, $a$ denotes the control, $b$ the target, and $\oplus$ is addition modulo two.

The controlled-NOT gate maps $|a,b⟩$ to $|a,b\oplus a⟩$.

The basis rule extends linearly to quantum superpositions. For example, it sends $|+⟩|0⟩$ to $(|00⟩+|11⟩)/\sqrt{2}$, a Bell state. The control is not first measured: its quantum coherence can become entanglement between the two qubits. Applying the controlled-NOT gate twice returns the original state. ^[ref]

A controlled unitary gate applies a unitary operator $U$ to a target system when a control qubit is $|1⟩$, and leaves the target unchanged when the control is $|0⟩$:For a coherent control, linearity givesThe control has not been measured; the output may be entangled. The controlled-unitary entangling criterion states exactly when some product input becomes entangled.

The global phase of a controlled quantum gate matters: if $U=e^{i\theta }I$, then $C(U)$ acts as a phase gate on the control. A phase invisible on the isolated target is a relative phase between control branches. ^[ref]

A graph state assigns one qubit to each vertex of an undirected graph. Begin with all qubits in $|+⟩$ and apply a Controlled-Z gate for every edge.

For an undirected graph $G=(V,E)$, the graph state is

The Controlled-Z gates commute, so their ordering does not change the result. On a computational-basis state $|x⟩$, each occupied edge contributes $(-1{)}^{x_a{x}_b}$; the amplitudes retain equal magnitude but acquire graph-dependent phases. An isolated vertex remains in the product state $|+⟩$, while edges can create entanglement. ^[ref]

The Hadamard basis is an orthonormal basis of one qubit consisting of the two eigenstates of the Pauli X gate.

The qubit Hadamard basis consists of $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$ and $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$.

The two states have the same computational-basis measurement probabilities, but opposite relative signs. A Hadamard gate maps them to $|0⟩$ and $|1⟩$, so applying it before a computational-basis measurement distinguishes the states. Relative phase is therefore physically accessible even when one chosen measurement has identical outcome probabilities. ^[ref]

The SWAP gate exchanges the states of two qubits, including any correlations they have with other systems. Its action on the computational basis fixes the operation on every quantum superposition by linearity.

The SWAP gate maps $|x⟩|y⟩$ to $|y⟩|x⟩$.

In particular, $\mathrm{SWAP}(|\psi ⟩\otimes |\varphi ⟩)=|\varphi ⟩\otimes |\psi ⟩$. This transfers rather than copies the two inputs, so it does not violate the no-cloning theorem. The Three-CNOT decomposition of the SWAP gate implements the same operation using alternating control directions. ^[ref]

The Toffoli gate is a reversible three-qubit quantum gate with two controls and one target. It flips the target only when both controls are $1$.

The Toffoli gate maps $|x_1{x}_{2}y⟩$ to $|x_1{x}_2,y\oplus x_1{x}_2⟩$.

For a target initially in $|0⟩$, the output target records the classical AND $x_1{x}_2$ while both input labels remain available. Applying the Toffoli gate again erases that computed target reversibly. The same rule acts linearly on quantum superpositions, so it can be used inside a coherent quantum circuit. ^[ref]