Group 25 gates
Orthogonal Group
The real unitaries. Circuits built from them keep every amplitude real, and that costs surprisingly little.
The orthogonal group consists of the real unitaries — gates whose matrices have no imaginary entries. Real gates compose to real gates, so circuits built from them keep all amplitudes real, and a surprising amount of quantum computing happens entirely inside this group.
$$ \begin{aligned} \mathsf{O}(n) &:= \left\{ O \in \mathrm{GL}(n, \mathbb{R}) \mid O^{\mathsf{T}} O = I \right\} \\ &= \mathsf{U}(n) \cap \mathrm{GL}(n, \mathbb{R}) \end{aligned} $$
Properties
- Every orthogonal matrix has $\det O = \pm 1$. The $+1$ component $\mathsf{SO}(n)$ contains the rotations, such as $R_y(\theta)$ and Givens rotations; determinant $-1$ elements are reflections, such as the Grover diffuser.
- Orthogonal gates are exactly the gates invariant under complex conjugation — quantum operations indistinguishable from their own "time-reversed-frame" copies.
- Permutation matrices form a finite subgroup: $X$, CNOT, Toffoli, SWAP, and the qudit shift all just shuffle basis states. All of classical reversible logic lives here.
Why real amplitudes suffice
Restricting to real amplitudes costs almost nothing: a single extra qubit can carry the role of the imaginary unit, and real gates on $n+1$ qubits then simulate arbitrary complex circuits on $n$ qubits (Rudolph and Grover showed a single two-qubit real gate suffices for universality). Many textbook algorithms — Grover search among them — never produce a complex amplitude in the first place.
Gates in this group
In the order of the table on the home page; the colour bar is the gate's family.
On the map
How the orthogonal group sits among the others: nested regions are subgroups, overlaps are shared gates, and the small labels are one example gate per region.