Distributed Shor's algorithm (pp0027-0044) 
         Ligang Xiao, Daowen Qiu, Le Luo and Paulo Mateus
            doi: https://doi.org/10.26421/QIC23.1-2-3
Abstracts: Shor's algorithm is one of the most important quantum algorithm proposed by Peter Shor [Proceedings of the 35th Annual Symposium on Foundations of Computer Science, 1994, pp. 124--134]. Shor's algorithm can factor a large integer with certain probability and costs  polynomial time in the length of the input integer. The key step of Shor's algorithm is the order-finding algorithm, the quantum part of which is to estimate $s/r$, where $r$ is the ``order" and $s$ is some natural number that less than $r$. {{Shor's algorithm requires lots of qubits and a deep circuit depth, which is unaffordable for current physical devices.}} In this paper, to reduce the number of qubits required and circuit depth, we propose a quantum-classical hybrid distributed order-finding algorithm for Shor's algorithm, which combines the advantages of both quantum processing and classical processing. {{ In our distributed order-finding algorithm, we use two quantum computers with the ability of quantum teleportation separately to estimate partial bits of $s/r$.}}  The measuring results will be processed through a classical algorithm to ensure the accuracy of the results. Compared with the traditional Shor's algorithm that uses multiple control qubits, our algorithm reduces nearly $L/2$ qubits for factoring an $L$-bit integer and reduces the circuit depth of each computer.
Key Words: Shor's algorithm, distributed Shor's algorithm, quantum-classcial hybrid, quantum teleportation, circuit depth