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Distributed quantum computing: A distributed Shor algorithm

Anocha Yimsiriwattana, Samuel J. Lomonaco Jr. A

Published Jan 1, 2004• 52 citations
OpenAlex

Abstract

We present a distributed implementation of Shor’s quantum factoring algorithm on a distributed quantum network model. This model provides a means for small capacity quantum computers to work together in such a way as to simulate a large capacity quantum computer. In this paper, entanglement is used as a resource for implementing non-local operations between two or more quantum computers. These non-local operations are used to implement a distributed factoring circuit with polynomially many gates. This distributed version of Shor’s algorithm requires an additional overhead of O((log N) 2) communication complexity, where N denotes the integer to be factored. Keywords: Shor’s algorithm, factoring algorithm, distributed quantum algorithms, quantum circuit. 1.

Related papers

Distributed quantum computing: a distributed Shor algorithm

Anocha Yimsiriwattana, Samuel J. Lomonaco Jr.

We present a distributed implementation of Shor's quantum factoring algorithm on a distributed quantum network model. This model provides a means for small capacity quantum computers to work together in such a way as to simulate a large capacity quantum computer. In this paper, entanglement is used as a resource for implementing non-local operations between two or more quantum computers. These non-local operations are used to implement a distributed factoring circuit with polynomially many gates. This distributed version of Shor's algorithm requires an additional overhead of O((log N)^2) communication complexity, where N denotes the integer to be factored.

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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.

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Quantum Information and Computation•Mar 1, 2003•207 citations
We present a distributed implementation of Shor’s quantum factoring algorithm on a distributed quantum network model. This model provides a means for small capacity quantum computers to work together in such a way as to simulate a large capacity quantum computer. In this paper, entanglement is used as a resource for implementing non-local operations between two or more quantum computers. These non-local operations are used to implement a distributed factoring circuit with polynomially many gates. This distributed version of Shor’s algorithm requires an additional overhead of O((log N) 2) communication complexity, where N denotes the integer to be factored. Keywords: Shor’s algorithm, factoring algorithm, distributed quantum algorithms, quantum circuit. 1.
We present a distributed implementation of Shor's quantum factoring algorithm on a distributed quantum network model. This model provides a means for small capacity quantum computers to work together in such a way as to simulate a large capacity quantum computer. In this paper, entanglement is used as a resource for implementing non-local operations between two or more quantum computers. These non-local operations are used to implement a distributed factoring circuit with polynomially many gates. This distributed version of Shor's algorithm requires an additional overhead of O((log N)^2) communication complexity, where N denotes the integer to be factored.
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 (2−2k)L−log⁡2k−O(1)(2-\dfrac{2}{k})L-\log_2k-O(1)(2−k2​)L−log2​k−O(1) when factoring an LLL-bit integer (kkk is the number of compute nodes). The communication complexity of our distributed order-finding algorithm is O(kL)O(kL)O(kL).
We try to minimize the number of qubits needed to factor an integer of n bits using Shor's algorithm on a quantum computer. We introduce a circuit which uses 2n+3 qubits and O(n^3 lg(n)) elementary quantum gates in a depth of O(n^3) to implement the factorization algorithm. The circuit is computable in polynomial time on a classical computer and is completely general as it does not rely on any property of the number to be factored. Keywords: Factorization, quantum circuits, modular arithmetics
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/rs/rs/r, where rrr is the ``order" and sss is some natural number that less than rrr. {{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/rs/rs/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/2L/2L/2 qubits for factoring an LLL-bit integer and reduces the circuit depth of each computer.
This paper presents the generalized implementation of the quantum circuit utilized in Shor’s algorithm, which was simulated using the quantum simulator offered by IBM. Shor’s Algorithm is a ground-breaking Quantum Algorithm that works on Integer Factorization; for a classical computer, the following task would require an exponential amount of time, whereas a Quantum Computer running Shor’s Algorithm would only require Polynomial-time. Using the principles of Quantum Superposition and Quantum Entanglement, the algorithm demonstrates the potential of quantum computing to solve complex mathematical problems efficiently. Experimental results show the effectiveness of the implemented algorithm in Integer Factorization and discrete logarithm, disturbing the safety of major encryption algorithms. This paper contributes to advancing the understanding and practical application of quantum algorithms in real-world scenarios.