RCCX¶
RCCX ("relative-phase CCX", also called the Margolus gate) gives the same computational-basis result as a Toffoli gate, but can be built from fewer elementary gates - at the cost of introducing extra phases on some inputs.
| Qubits | 3 (two controls, one target) |
| Parameters | None |
| Inverse | Itself, on computational basis inputs |
| Also called | Margolus gate, simplified Toffoli |
What it does¶
On every computational basis state, RCCX flips the target exactly when both controls are \( |1\rangle \) - identical to Toffoli's action. The difference is not visible in the basis-state mapping: RCCX implements this using a cheaper sequence of gates by allowing a relative phase to appear on some intermediate/superposition inputs, which Toffoli does not introduce.
Action on basis states¶
(controls first and second, target third - identical to Toffoli on computational basis states)
- \( |110\rangle \rightarrow |111\rangle \)
- \( |111\rangle \rightarrow |110\rangle \)
- every other computational basis state is left unchanged
Example¶
RCCX is most useful as a cheaper building block inside larger circuits - e.g. constructing RC3X or other multi-controlled gates - where its relative-phase caveat doesn't matter because it's immediately uncomputed or only ever touches basis states.
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 | rccx q[0], q[1], q[2]; |
| Qiskit | circuit.rccx[q[0], q[1], q[2]] |
| Cirq | - |
| Q# | - |
Notes¶
Because RCCX only matches Toffoli on computational basis states - not as an exactly identical unitary - it's safe to use as a drop-in replacement when its inputs and outputs are always basis states (e.g. classical logic built from quantum gates), but it should not be substituted for a true Toffoli inside a larger circuit where the extra relative phase would affect interference between amplitudes.