Given an arbitrary $2^w\times 2^w$ unitary matrix $U$, a powerful matrix decomposition can be applied, leading to four different syntheses of a $w$-qubit quantum circuit performing the unitary transformation. The demonstration is based on a recent theorem by H. Führ and Z. Rzeszotnik [Linear Algebra Its Appl. 484, 86 (2015)] generalizing the scaling of single-bit unitary gates ($w=1$) to gates with arbitrary value of $w$. The synthesized circuit consists of controlled one-qubit gates, such as negator gates and phasor gates. Interestingly, the approach reduces to a known synthesis method for classical logic circuits consisting of controlled not gates in the case that $U$ is a permutation matrix.

• Received 31 May 2016

DOI:https://doi.org/10.1103/PhysRevA.94.052317