quantum gate directory

The orthogonal group consists of the real unitaries — gates whose matrices have no imaginary entries (the homepage filter calls this group "Real"). 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.

$$\mathsf{O}(n) := \left\{ O \in \mathrm{GL}(n, \mathbb{R}) \mid O^{\mathsf{T}} O = I \right\} = \mathsf{U}(n) \cap \mathrm{GL}(n, \mathbb{R})$$

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.