Improving photon detector efficiency using a high-fidelity optical CNOT gate

A significant problem for optical quantum computing is inefficient, or inaccurate photo-detectors. It is possible to use CNOT gates to improve a detector by making a large cat state then measuring every qubit in that state. In this paper we develop a code that compares five different schemes for making multiple measurements, some of which are capable of detecting loss and some of which are not. We explore how each of these schemes performs in the presence of different errors, and derive a formula to find at what probability of qubit loss is it worth detecting loss, and at what probability does this just lead to further errors than the loss introduces.