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A general scheme is presented to decompose a d-by-d unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in d - 1 classes, where each class is isomorphic to the group of 2 × 2 unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on n-qubit registers can be expressed as no more than 2^n-1(2ⁿ-1) fully controlled single-qubit gates chosen from 2ⁿ-1 classes, where the quantum gates in each class share the same n - 1 control qubits. Moreover, it is shown that one can easily adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.

The main target of this document is to propose a new graphical method to find prime implicants, necessary prime implicants and minimum sum-of-product expressions for digital systems with any number of variables. Moreover, a simple example of application of the same method is included to be compared with Karnaugh's method. As a consequence of this proposed method, a few conditions are established and a fractal image is created, which explains why it is not possible to have more than one binary variable changing simultaneously and why the change between two non adjacent states has to be realized always going through an intermediate state.