Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/9602016 · Physical Review A

Efficient networks for quantum factoring

David Beckman, Amalavoyal N. Chari, Devabhaktuni Srikrishna, John P. Preskill

Published Aug 1, 1996• 268 citations
arXivHTMLPDF

Abstract

We consider how to optimize memory use and computation time in operating a quantum computer. In particular, we estimate the number of memory quantum bits (qubits) and the number of operations required to perform factorization, using the algorithm suggested by Shor [in Proceedings of the 35th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Computer Society, Los Alamitos, CA, 1994), p. 124]. A K-bit number can be factored in time of order ${\mathit{K}}^{3}$ using a machine capable of storing 5K+1 qubits. Evaluation of the modular exponential function (the bottleneck of Shor's algorithm) could be achieved with about 72${\mathit{K}}^{3}$ elementary quantum gates; implementation using a linear ion trap would require about 396${\mathit{K}}^{3}$ laser pulses. A proof-of-principle demonstration of quantum factoring (factorization of 15) could be performed with only 6 trapped ions and 38 laser pulses. Though the ion trap may never be a useful computer, it will be a powerful device for exploring experimentally the properties of entangled quantum states. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1034

Related papers

Quantum networks for elementary arithmetic operations

Vlatko Vedral, Adriano Barenco, ARTUR K. EKERT

Quantum computers require quantum arithmetic. We provide an explicit construction of quantum networks effecting basic arithmetic operations: from addition to modular exponentiation. Quantum modular exponentiation seems to be the most difficult (time and space consuming) part of Shor's quantum factorizing algorithm. We show that the auxiliary memory required to perform this operation in a reversible way grows linearly with the size of the number to be factorized. \textcopyright{} 1996 The American Physical Society.

arXiv:quant-ph/9511018•Physical Review A•Jul 1, 1996•811 citations

Factoring in a dissipative quantum computer

César Miquel, Juan Pablo Paz, R.P.J. Perazzo

We describe an array of quantum gates implementing Shor's algorithm [in Proceedings of the 35th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Computer Society, Los Alamitos, CA, 1994), p. 116; (unpublished); Phys. Rev. A 53, R2493 (1995)] for prime factorization in a quantum computer. The array includes a circuit for modular exponentiation with several subcomponents (such as controlled multipliers and adders) that are described in terms of elementary Toffoli gates. We present a simple analysis of the impact of losses and decoherence on the performance of this quantum factoring circuit. For that purpose, we simulate a quantum computer that is running the program to factor N=15 while interacting with a dissipative environment. As a consequence of this interaction, randomly selected quantum bits (qubits) may spontaneously decay. Using the results of our numerical simulations, we analyze the efficiency of some simple error correction techniques. \textcopyright{} 1996 The American Physical Society.

arXiv:quant-ph/9601021•Physical Review A•Oct 1, 1996•67 citations

Exact Analyses of Computational Time for Factoring in Quantum Computers

Noboru Kunihiro

We evaluate the exact number of gates for circuits of Shor's factoring algorithm. We estimate the running time for factoring a large composite such as 576 and 1024 bit numbers by appropriately setting gate operation time.For example, we show that on the condition that each elementary gate is operated within 50 μsec, the running time for factoring 576 bit number is 1 month even if the most effective circuit is adopted. Consequently, we find that if we adopt the long gate operation-time devices or qubit-saving circuits, factorization will not be completed within feasible time on the condition that a new efficient modular exponentiation algorithm will not be proposed. Furthermore, we point out that long gate operation time may become a new problem preventing a realization of quantum computers.

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences•Jan 1, 2005•13 citations

Circuit for Shor's algorithm using 2n+3 qubits

Stéphane Beauregard

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

arXiv:quant-ph/0205095•May 15, 2002•18 citations

Circuit for Shor's algorithm using 2n+3 qubits

Stéphane Beauregard

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.

Quantum Information and Computation•Mar 1, 2003•207 citations

Space-Efficient and Noise-Robust Quantum Factoring

Seyoon Ragavan, Vinod Vaikuntanathan

We provide two improvements to Regev's recent quantum factoring algorithm (Journal of the ACM 2025), addressing its space efficiency and its noise-tolerance. Our first contribution is to improve the quantum space efficiency of Regev's algorithm while keeping the circuit size the same. Our main result constructs a quantum factoring circuit using $O(n \log n)$ qubits and $O(n^{3/2} \log n)$ gates. We achieve the best of Shor and Regev (upto a logarithmic factor in the space complexity): on the one hand, Regev's circuit requires $O(n^{3/2})$ qubits and $O(n^{3/2} \log n)$ gates, while Shor's circuit requires $O(n^2 \log n)$ gates but only $O(n \log n)$ qubits. As with Regev, to factor an $n$-bit integer $N$, we run our circuit independently $O(\sqrt{n})$ times and apply Regev's classical postprocessing procedure. Our optimization is achieved by implementing efficient and reversible exponentiation with Fibonacci numbers in the exponent, rather than the usual powers of 2, adapting work by Kaliski (arXiv:1711.02491) from the classical reversible setting to the quantum setting. This technique also allows us to perform quantum modular exponentiation that is efficient in both space and size without requiring significant precomputation, a result that may be useful for other quantum algorithms. A key ingredient of our exponentiation implementation is an efficient circuit for a function resembling in-place quantum-quantum modular multiplication. Our second contribution is to show that Regev's classical postprocessing procedure can be modified to tolerate a constant fraction of the quantum circuit runs being corrupted by errors. In contrast, Regev's analysis of his classical postprocessing procedure requires all $\approx \sqrt{n}$ runs to be successful. In a nutshell, we achieve this using lattice reduction techniques to detect and filter out corrupt samples.

arXiv:2310.00899•Oct 2, 2023•2 citations
We consider how to optimize memory use and computation time in operating a quantum computer. In particular, we estimate the number of memory quantum bits (qubits) and the number of operations required to perform factorization, using the algorithm suggested by Shor [in Proceedings of the 35th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Computer Society, Los Alamitos, CA, 1994), p. 124]. A K-bit number can be factored in time of order K3{\mathit{K}}^{3}K3 using a machine capable of storing 5K+1 qubits. Evaluation of the modular exponential function (the bottleneck of Shor's algorithm) could be achieved with about 72K3{\mathit{K}}^{3}K3 elementary quantum gates; implementation using a linear ion trap would require about 396K3{\mathit{K}}^{3}K3 laser pulses. A proof-of-principle demonstration of quantum factoring (factorization of 15) could be performed with only 6 trapped ions and 38 laser pulses. Though the ion trap may never be a useful computer, it will be a powerful device for exploring experimentally the properties of entangled quantum states. \textcopyright{} 1996 The American Physical Society.
Quantum computers require quantum arithmetic. We provide an explicit construction of quantum networks effecting basic arithmetic operations: from addition to modular exponentiation. Quantum modular exponentiation seems to be the most difficult (time and space consuming) part of Shor's quantum factorizing algorithm. We show that the auxiliary memory required to perform this operation in a reversible way grows linearly with the size of the number to be factorized. \textcopyright{} 1996 The American Physical Society.
We describe an array of quantum gates implementing Shor's algorithm [in Proceedings of the 35th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Computer Society, Los Alamitos, CA, 1994), p. 116; (unpublished); Phys. Rev. A 53, R2493 (1995)] for prime factorization in a quantum computer. The array includes a circuit for modular exponentiation with several subcomponents (such as controlled multipliers and adders) that are described in terms of elementary Toffoli gates. We present a simple analysis of the impact of losses and decoherence on the performance of this quantum factoring circuit. For that purpose, we simulate a quantum computer that is running the program to factor N=15 while interacting with a dissipative environment. As a consequence of this interaction, randomly selected quantum bits (qubits) may spontaneously decay. Using the results of our numerical simulations, we analyze the efficiency of some simple error correction techniques. \textcopyright{} 1996 The American Physical Society.
We evaluate the exact number of gates for circuits of Shor's factoring algorithm. We estimate the running time for factoring a large composite such as 576 and 1024 bit numbers by appropriately setting gate operation time.For example, we show that on the condition that each elementary gate is operated within 50 μsec, the running time for factoring 576 bit number is 1 month even if the most effective circuit is adopted. Consequently, we find that if we adopt the long gate operation-time devices or qubit-saving circuits, factorization will not be completed within feasible time on the condition that a new efficient modular exponentiation algorithm will not be proposed. Furthermore, we point out that long gate operation time may become a new problem preventing a realization of quantum computers.
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
We provide two improvements to Regev's recent quantum factoring algorithm (Journal of the ACM 2025), addressing its space efficiency and its noise-tolerance. Our first contribution is to improve the quantum space efficiency of Regev's algorithm while keeping the circuit size the same. Our main result constructs a quantum factoring circuit using O(nlog⁡n)O(n \log n)O(nlogn) qubits and O(n3/2log⁡n)O(n^{3/2} \log n)O(n3/2logn) gates. We achieve the best of Shor and Regev (upto a logarithmic factor in the space complexity): on the one hand, Regev's circuit requires O(n3/2)O(n^{3/2})O(n3/2) qubits and O(n3/2log⁡n)O(n^{3/2} \log n)O(n3/2logn) gates, while Shor's circuit requires O(n2log⁡n)O(n^2 \log n)O(n2logn) gates but only O(nlog⁡n)O(n \log n)O(nlogn) qubits. As with Regev, to factor an nnn-bit integer NNN, we run our circuit independently O(n)O(\sqrt{n})O(n​) times and apply Regev's classical postprocessing procedure. Our optimization is achieved by implementing efficient and reversible exponentiation with Fibonacci numbers in the exponent, rather than the usual powers of 2, adapting work by Kaliski (arXiv:1711.02491) from the classical reversible setting to the quantum setting. This technique also allows us to perform quantum modular exponentiation that is efficient in both space and size without requiring significant precomputation, a result that may be useful for other quantum algorithms. A key ingredient of our exponentiation implementation is an efficient circuit for a function resembling in-place quantum-quantum modular multiplication. Our second contribution is to show that Regev's classical postprocessing procedure can be modified to tolerate a constant fraction of the quantum circuit runs being corrupted by errors. In contrast, Regev's analysis of his classical postprocessing procedure requires all ≈n\approx \sqrt{n}≈n​ runs to be successful. In a nutshell, we achieve this using lattice reduction techniques to detect and filter out corrupt samples.