Multi-Controlled Gates

A controlled gate applies a single-qubit operation only when a designated control qubit is $|1\rangle$ . Extending this idea to two or more control qubits produces multi-controlled gates — gates that fire only when every control qubit is $|1\rangle$ . They are the quantum analogue of multi-input AND logic, and they are indispensable for constructing oracles, arithmetic circuits, and quantum error correction.

The Toffoli gate (CCX)

The most important multi-controlled gate is the Toffoli gate, also written CCX or doubly-controlled-NOT. It acts on three qubits: two controls $c_1$ , $c_2$ and one target $t$ .

\mathrm{CCX}\,|c_1, c_2, t\rangle = \begin{cases} |c_1, c_2, t \oplus 1\rangle & \text{if } c_1 = c_2 = 1, \\ |c_1, c_2, t\rangle & \text{otherwise.} \end{cases}

In words: flip the target bit if and only if both controls are $|1\rangle$ . The $8 \times 8$ unitary matrix of CCX is the identity on all computational basis states except the swap $|110\rangle \leftrightarrow |111\rangle$ (using the convention that qubit 0 is most significant). CCX is its own inverse: $\mathrm{CCX}^2 = I$ .

Controlled gates as conditional unitary maps

Any single-qubit gate $U$ can be lifted to a controlled version $C(U)$ on two qubits:

C(U) = |0\rangle\langle 0| \otimes I \;+\; |1\rangle\langle 1| \otimes U.

Extending to $n$ control qubits gives $C^n(U)$ , which applies $U$ to the target only when all $n$ controls are in state $|1\rangle$ . The simulator provides the most common instances directly:

c.cx(control, target) — CNOT, i.e. $C(X)$
c.cz(control, target) — $C(Z)$
c.ccx(c1, c2, target) — Toffoli, i.e. $C^2(X)$

Higher-degree multi-controlled gates can be synthesised from these primitives using ancilla qubits, a technique covered in the next lesson.

Doubly-controlled Z (CCZ)

A related gate, CCZ, flips the phase of $|111\rangle$ by $-1$ while leaving all other basis states unchanged:

\mathrm{CCZ}\,|c_1, c_2, t\rangle = \begin{cases} -|c_1, c_2, t\rangle & \text{if } c_1 = c_2 = t = 1, \\ |c_1, c_2, t\rangle & \text{otherwise.} \end{cases}

CCZ and CCX are related by a Hadamard on the target qubit:

\mathrm{CCX} = (I \otimes I \otimes H)\;\mathrm{CCZ}\;(I \otimes I \otimes H).

This conjugation identity is the multi-qubit generalisation of $HZH = X$ .

Try it

Build the Toffoli gate as a standalone 3-qubit circuit. The grader checks the full unitary matrix, so a do-nothing circuit will fail.

Run your code to see the quantum state.