Potential cancellation of one additional CNOT in recursive CSD-based synthesis

Determine whether a circuit identity exists that cancels one further CNOT gate in the recursive cosine–sine decomposition (CSD)-based, parameter-optimal unitary synthesis circuit for n‑qubit unitaries—specifically in the decomposition employing multiplexed single‑qubit flags and a final diagonal—thereby yielding a total CNOT count smaller by one than the count derived here and consistent with the hint in Bergholm et al. (2005).

Background

The paper revisits the recursive cosine–sine decomposition (CSD) approach for synthesizing arbitrary n‑qubit unitaries into {Clifford + Rot} gates, obtaining parameter‑optimal rotation counts while analyzing the associated CNOT counts. The authors’ derivation leads to a CNOT count slightly above the best known results and asymptotically a factor of two above the lower bound.

In discussing this construction, the authors note that prior work (Bergholm et al., 2005) suggests a further cancellation of a single CNOT might be possible. However, they were unable to identify a concrete cancellation opportunity and therefore report a count larger by one. Resolving whether this last CNOT can be eliminated would settle the discrepancy and refine the exact optimal CNOT count for this recursive CSD-based scheme.