Quantum Circuits for Sparse Isometries

We consider the task of breaking down a quantum computation given as an isometry into C-nots and single-qubit gates, while keeping the number of C-not gates small. Although several decompositions are known for general isometries, here we focus on a method based on Householder reflections that adapts well in the case of sparse isometries. We show how to use this method to decompose an arbitrary isometry before illustrating that the method can lead to significant improvements in the case of sparse isometries. We also discuss the classical complexity of this method and illustrate its effectiveness in the case of sparse state preparation by applying it to randomly chosen sparse states.

Automated summary

This study focuses on a method based on Householder reflections for decomposing a quantum computation into C-not gates and single-qubit gates. It shows that this method performs well with sparse isometries, leading to significant reductions in the number of C-not gates required. The classical complexity of the method is also discussed, along with its effectiveness in sparse state preparation.