quantum gate directory

The diagonal group contains the gates that are diagonal in the computational basis: they never change which basis state you are in, only attach a phase to each one. Unitarity forces every diagonal entry onto the unit circle.

$$\mathsf{D}(n) := \left\{ \mathrm{diag}(\mathrm{e}^{i\theta_1}, \ldots, \mathrm{e}^{i\theta_n}) \mid \theta_j \in \mathbb{R} \right\} \subset \mathsf{U}(n)$$

As a group it is the $n$-torus $\mathsf{U}(1)^n$, and it is a maximal abelian subgroup of the unitary group (a maximal torus): you cannot enlarge it without adding gates that fail to commute.

Properties

• Action on basis states: $|x\rangle \mapsto \mathrm{e}^{i\theta_x}|x\rangle$. A phase oracle is nothing but a diagonal gate with $\pm 1$ entries.
• Abelian: diagonal gates all commute, so they can be freely reordered, merged, and canceled — this is what compiler "phase folding" passes exploit.
• Closed under adding controls: controlling a diagonal gate gives another diagonal gate. This generates the ladder $Z$ → $\mathrm{C}Z$ → $\mathrm{CC}Z$, and likewise $S$ → $\mathrm{C}S$, $P(\phi)$ → $\mathrm{C}P(\phi)$.
• Invisible to final measurements: diagonal gates commute with computational-basis projectors, so one applied just before measurement changes nothing and can be deleted.
• The diagonal gates at each level of the Clifford hierarchy form the sequence $Z$, $S$, $T$, ... — each the square root of the previous.
• Contained in the number-preserving group: a gate that fixes every basis state certainly preserves its excitation count.

Hardware

Diagonal $z$-axis rotations are special on most platforms: $R_z$ is implemented virtually, as a bookkeeping change to the phase of subsequent pulses, taking zero time and introducing zero error. Compilers therefore work hard to push phases into diagonal form.