# Quantum Circuit Decomposition

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Quantum computing uses quantum mechanical properties of matter to perform operations unavailable to classical computers. Two important representations for an n-qubit quantum algorithm are the matrix representation, which represents the quantum state as a complex unit vector and the algorithm as a unitary matrix, and the circuit representation, which represents the algorithm as a circuit of quantum gates. The matrix representation is mathematically useful, but the circuit representation is closer to what can be implemented on quantum hardware, so it is important to be able to convert between the two. While going from gates to matrices is trivially done by multiplying the matrix representations of the gates, going from a unitary to a quantum circuit is the subject of a significant field of study known as quantum circuit decomposition. Here, I present the decomposition algorithm presented by Cybenko. After a brief introduction to quantum computing and the two representations, I show how the QR decomposition can be implemented using Givens operations, and how applying it to unitary matrices results in a product of simple unitary matrices. I then explain the structure of quantum gate matrices, and show that although Givens operations do not always match it, they can be decomposed into a product of unitary matrices that do. I finally reduce the set of necessary gates to a small universal gate set.

@article{ecoffet2022quantum,