  Several counter-intuitive go-theorems have recently been shown for
transformations of unknown unitary operations; deterministic and exact complex
conjugation, inversion, and transposition of a general $d$-dimensional unknown
unitary operation are implementable with a finite number of queries of the
black-box unitary operation. However, the minimum numbers of the required
queries are not known except for $d=2$ unitary inversion and unitary
transposition (numerical) and unitary conjugation (analytic). In this work, we
derive complementary no-go theorems for deterministic and exact implementations
of inversion and transposition of a $d$-dimensional unknown unitary operation
under certain numbers of queries. The obtained no-go theorem indicates that the
analytical lower bound of the number of queries for unitary inversion is $d^2$
and that for unitary transposition is $4$ for $d=2$ and $d+3$ for $d \geq 3$.
We have developed a new framework that utilizes differentiation to obtain the
analytical lower bounds on the number of queries to the black-box unitary
operation required to implement a transformation given by a general
differentiable function mapping a unitary operation to another unitary
operation, which reproduces the lower bound of the number of queries for
unitary complex conjugation $d-1$. As a corollary, we show the relationship
between the tightness of the lower bounds and the existence of optimal
catalytic transformations, which is a new aspect recently identified in the
study of deterministic and exact unitary inversion. Furthermore, we extend our
framework to the probabilistic setting where the transformation is required to
succeed with a certain probability, thereby showing a possible tradeoff
relation between query numbers and the required success probability.
