Toffoli¶
The Toffoli gate is a doubly-controlled NOT: it flips the target qubit only when both control qubits are \( |1\rangle \). It's the quantum version of a classical AND gate (the target ends up holding the AND of the two controls, if it started at \( |0\rangle \)).
| Qubits | 3 (two controls, one target) |
| Parameters | None |
| Inverse | Itself (Toffoli² = I) |
| Also called | CCX, CCNOT, doubly-controlled NOT |
What it does¶
Toffoli acts as the identity on every basis state except the one where both controls are \( |1\rangle \); there, it flips the target - swapping \( |110\rangle \) with \( |111\rangle \) (using the convention that the first two qubits are the controls and the third is the target). Every other basis state is unchanged.
Action on basis states¶
(controls first and second, target third)
- \( |110\rangle \rightarrow |111\rangle \)
- \( |111\rangle \rightarrow |110\rangle \)
- every other 3-qubit basis state is left unchanged
Example¶
With q[0] and q[1] both set to \( |1\rangle \) (e.g. with two X gates) and q[2] at \( |0\rangle \), a Toffoli from controls q[0], q[1] to target q[2] flips q[2] to \( |1\rangle \) - same as classical AND(1, 1) = 1.
Other notations¶
Naming conventions across ecosystems; exact syntax can vary by library version, and this does not claim to be Qompile's generated output unless otherwise noted.
| Language | Typical form |
|---|---|
| OpenQASM 2.0 | ccx q[0], q[1], q[2]; |
| Qiskit | circuit.ccx[q[0], q[1], q[2]] |
| Cirq | cirq.TOFFOLI(q[0], q[1], q[2]) |
| Q# | CCNOT(q[0], q[1], q[2]); |
Notes¶
Toffoli's full matrix is 8×8 - identity everywhere except a single 2×2 X block in the corner covering \( |110\rangle \) and \( |111\rangle \), matching the basis-state mapping above.