quantum gate directory

Qubits: $3$

hermitian

SDK Support

SDK	Name
Qiskit	qiskit.circuit.library.RCCXGate
PennyLane	—
Cirq	—
Q#	—
PyQuil	—
Braket	—
BQSKit	bqskit.ir.gates.MargolusGate ⓘ
Qibo	— ⓘ
pytket	—
Stim	—
OpenQASM	—

Groups

This gate is contained in the following groups:

• orthogonal

Reference

On the CNOT-cost of TOFFOLI gates (2008)

The Margolus gate (Qiskit's "relative-phase" or "simplified" Toffoli, RCCX) acts exactly like the Toffoli on every computational basis state except one, where it picks up a harmless-looking sign:

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

That is, $\mathrm{RCC}X = \mathrm{CC}X$ except $|101\rangle \mapsto -|101\rangle$.

Why it matters

An exact Toffoli requires six CNOTs; the Margolus gate needs only three (Shende and Markov proved both counts optimal). Whenever a Toffoli is used in a compute–uncompute pair — as ancilla logic almost always is — the stray phase cancels between the compute and uncompute halves, so the cheap version is safe and halves the entangling-gate cost.

Decompositions

• Three CNOTs and four rotations, all real: $R_y(-\tfrac{\pi}{4})t \, \mathrm{CX} \, R_y(-\tfrac{\pi}{4})t \, \mathrm{CX} \, R_y(\tfrac{\pi}{4})t \, \mathrm{CX} \, R_y(\tfrac{\pi}{4})_t$ (read right to left; subscripts denote control, target).

Properties

• Hermitian and self-inverse, like the Toffoli itself.
• Entirely real, so it lies in the orthogonal group.