Shor's algorithm is one of the most significant quantum algorithms. Shor's algorithm can factor large integers with a certain success probability in polynomial time. However, Shor's algorithm requires an unbearable amount of qubits in the NISQ (Noisy Intermediate-scale Quantum) era. To reduce the resources required for Shor's algorithm, in this paper we first propose a new distributed phase estimation algorithm. Our distributed phase estimation algorithm does not require quantum communication and it reduces the number of qubits of a single node compared to the traditional phase estimation algorithm (non-iterative version). Then we apply our distributed phase estimation algorithm to form a distributed order-finding algorithm for Shor's algorithm. Compared with the traditional Shor's algorithm (non-iterative version), the maximum number of qubits required by a single node of our dristributed order-finding algorithm is reduced by $(2-\dfrac{2}{k})L-\log_2k-O(1)$ when factoring an $L$ -bit integer ( $k$ is the number of compute nodes). The communication complexity of our distributed order-finding algorithm is $O(kL)$ .Shor's algorithm is one of the most significant quantum algorithms. Shor's algorithm can factor large integers with a certain success probability in polynomial time. However, Shor's algorithm requires an unbearable amount of qubits in the NISQ (Noisy Intermediate-scale Quantum) era. To reduce the resources required for Shor's algorithm, in this paper we first propose a new distributed phase estimation algorithm. Our distributed phase estimation algorithm does not require quantum communication and it reduces the number of qubits of a single node compared to the traditional phase estimation algorithm (non-iterative version). Then we apply our distributed phase estimation algorithm to form a distributed order-finding algorithm for Shor's algorithm. Compared with the traditional Shor's algorithm (non-iterative version), the maximum number of qubits required by a single node of our dristributed order-finding algorithm is reduced by $(2-\dfrac{2}{k})L-\log_2k-O(1)$ when factoring an $L$-bit integer ($k$ is the number of compute nodes). The communication complexity of our distributed order-finding algorithm is $O(kL)$.

arXiv:2304.12100Apr 24, 20235 citations

Hybrid Quantum Computing: The Use Case of Shor’s Algorithm

Océane Koska, Arnaud Gazda, Maxime Remaud

Lecture notes in computer scienceJan 1, 20250 citations

Quantum Computing: A Mathematical Analysis of Shor's Algorithm

Stephen S. Clarke

Jan 1, 20200 citations

Near term implementation of Shor's Algorithm using Qiskit

Casimer M. DeCusatis, Emily Mcgettrick

Although the fundamental principles of quantum computing have been known for decades, it is only within the past few years that practical quantum computers have become available. Since these systems are limited to a small number of qubits, they cannot demonstrate quantum advantage for many practical problems. Accordingly, there has been an effort to develop near term implementations of algorithms purely for research and education. In this paper, we discuss a near term implementation of Shor's Algorithm using the Qiskit language on an IBM Q System One quantum computer. We extend currently available documentation by providing a full circuit diagram for Shor's Algorithm including gates specific to the implementation using Qiskit. We present an implementation capable of factoring small two-digit prime numbers, and discuss the limitations of noise when using real quantum computers vs. simulations.Although the fundamental principles of quantum computing have been known for decades, it is only within the past few years that practical quantum computers have become available. Since these systems are limited to a small number of qubits, they cannot demonstrate quantum advantage for many practical problems. Accordingly, there has been an effort to develop near term implementations of algorithms purely for research and education. In this paper, we discuss a near term implementation of Shor's Algorithm using the Qiskit language on an IBM Q System One quantum computer. We extend currently available documentation by providing a full circuit diagram for Shor's Algorithm including gates specific to the implementation using Qiskit. We present an implementation capable of factoring small two-digit prime numbers, and discuss the limitations of noise when using real quantum computers vs. simulations.

IEEE Annual Computing and Communication Workshop and Conference (CCWC)Jan 27, 20216 citations

Quantum Computing and Shor's Algorithm

Matt W. Hayward

Jan 1, 201519 citations

Distributed Shor's algorithm

Ligang Xiao, Daowen Qiu, Le Luo +1 more

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.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.

arXiv:2207.05976Quantum Information and ComputationJan 1, 202324 citations

(2-\dfrac{2}{k})L-\log_2k-O(1)

when factoring an

L

-bit integer (

k

is the number of compute nodes). The communication complexity of our distributed order-finding algorithm is

O(kL)

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.