quantum gate directory

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: $$ \begin{align*} \mathrm{RCC}X = {} & R_y(-\tfrac{\pi}{4})_t \, \mathrm{CX}_{2,t} \, R_y(-\tfrac{\pi}{4})_t \, \mathrm{CX}_{1,t} \\ & \times R_y(\tfrac{\pi}{4})_t \, \mathrm{CX}_{2,t} \, R_y(\tfrac{\pi}{4})_t \end{align*} $$ (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.