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Given an arbitrary $2^w \times 2^w$ unitary matrix $U$, a powerful matrixdecomposition can be applied, leading to four different syntheses of a$w$-qubit quantum circuit performing the unitary transformation. Thedemonstration is based on a recent theorem by F\"uhr and Rzeszotnik,generalizing the scaling of single-bit unitary gates ($w=1$) to gates witharbitrary value of~$w$. The synthesized circuit consists of controlled 1-qubitgates, such as NEGATOR gates and PHASOR gates. Interestingly, the approachreduces to a known synthesis method for classical logic circuits consisting ofcontrolled NOT gates, in the case that $U$ is a permutation matrix.

The block-ZXZ synthesis of an arbitrary quantum circuit