A distributed Shor’s algorithm is used to factorize a semiprime with two quantum computers using quantum teleportation to transport quantum states of qubits in parallel. We propose another distributed Shor’s algorithm with sequential quantum teleportation to improve the existing one. It explores how to employ distributed quantum computing through a quantum entanglement pair of qubits to transport quantum states of qubits sequentially. We combined two quantum computers to perform the quantum phase estimation (QPE) of Shor’s algorithm. This reduced the required number of qubits for a single quantum computer, decreased quantum circuit depth, mitigated the impact of insufficient decoherence time, and accelerated the process of breaking cryptographic systems such as Rivest-Shamir-Adleman (RSA). The designed quantum circuit was executed on the IBM Quantum Lab platform to verify its correctness. We analyzed and compared the required numbers of qubits to run different versions of Shor’s algorithm for individual quantum computers, as well as the numbers of qubits required to break different RSA cryptographic systems.
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.
Recently researchers have been explored quantum phenomena's for computation purpose and demonstrated important computational algorithms such as integer factorization using Shor's algorithm, unsorted date base search using Grover's algorithm, quantum cryptography, teleportation and many more. Many approaches have been investigated for implementing quantum algorithm physically. In this review quantum computer implementation techniques are introduced with an emphasize on linear optics quantum computing.