  In this paper, the problem of synthesizing a general Hermitian quantum gate
into a set of primary quantum gates is addressed. To this end, an extended
version of the Jacobi approach for calculating the eigenvalues of Hermitian
matrices in linear algebra is considered as the basis of the proposed synthesis
method. The quantum circuit synthesis method derived from the Jacobi approach
and its optimization challenges are described. It is shown that the proposed
method results in multiple-control rotation gates around the y axis,
multiple-control phase shift gates, multiple-control NOT gates and a middle
diagonal Hermitian matrix, which can be synthesized to multiple-control Pauli Z
gates. Using the proposed approach, it is shown how multiple-control U gates,
where U is a single-qubit Hermitian quantum gate, can be implemented using a
linear number of elementary gates in terms of circuit lines with the aid of one
auxiliary qubit in an arbitrary state.
