bibkey: garioncross2020cnotdihedral
authors: Shelly Garion; Andrew W. Cross
year: 2020
title: Synthesis of CNOT-Dihedral circuits with optimal number of two qubit gates
doi: 10.22331/q-2020-12-07-369
url: https://quantum-journal.org/papers/q-2020-12-07-369/
claim: Equation 12 relates conjugation of controlled-S by a local X to controlled-S inverse times a phase gate on the other qubit.
strata_touched: []
license: citation-only
triage: anchor

Extracting a local phase from controlled-S

The published open PDF, Quantum 4, 369, §2, p. 4, equation (12), gives identities for conjugating controlled-S by local $X$ gates. The exact equation and the source’s gate definitions were inspected. Write $C=\mathrm{diag}(1,1,1,i)$ and $P=\mathrm{diag}(1,i)$ to distinguish its two-qubit $CS$ from its one-qubit $S$. The first two lines become

$$(X\otimes I)C(X\otimes I)=C^{†}(I\otimes P),(I\otimes X)C(I\otimes X)=C^{†}(P\otimes I).$$

Right multiplication by $C$, using commutation of the diagonal factors, gives

$$(X\otimes I)C(X\otimes I)C=I\otimes P,(I\otimes X)C(I\otimes X)C=P\otimes I.$$

Thus the local phase-extraction identity is an established relation. More generally, for $C_{\vartheta }|ab⟩=e^{i\vartheta ab}|ab⟩$ and $P_{\vartheta }=\mathrm{diag}(1,e^{i\vartheta })$, the same basis calculation follows from $ab+a(1-b)=a$. This elementary extension is not a novelty claim.

For a physical menu $R(t)=e^{-itX}$, the mutually inverse pulses $R(-\pi /2)=iX$ and $R(\pi /2)=-iX$ give the same conjugation with exact scalar cancellation. On a larger constrained space, one must still establish the pulse’s restriction, subspace invariance and its action on any orthogonal sector. Those model-specific checks, and the count of forward fixed-phase calls, are supplied in Static compatibility fibers §62. They are not supplied by the abstract two-qubit identity alone.

The article’s optimality results concern its declared CNOT-dihedral generating set. They do not establish optimal pulse counts for the AURIC menu. This citation-only note reproduces no article text, figures or PDF.