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/9503007 · Science

Quantum Computers, Factoring, and Decoherence

Isaac L. Chuang, Raymond Laflamme, Peter W. Shor, Wojciech Hubert Zurek

Published Dec 8, 1995• 288 citations
arXivHTMLPDF

Abstract

It is known that quantum computers can dramatically speed up the task of finding factors of large numbers, a problem of practical significance for cryptographic applications. Factors of an L -digit number can be found in ∼ L 2 time [compared to ∼exp( L 1/3 ) time] by a quantum computer, which simultaneously follows all paths corresponding to distinct classical inputs, obtaining the solution from the coherent quantum interference of the alternatives. Here it is shown how the decoherence process degrades the interference pattern that emerges from the quantum factoring algorithm. For a quantum computer performing logical operations, an exponential decay of quantum coherence is inevitable. However, even in the presence of exponential decoherence, quantum computation can be useful as long as a sufficiently low decoherence rate can be achieved to allow meaningful results to be extracted from the calculation.

DOI
10.1126/science.270.5242.1633

Related papers

Quantum Physics and Computers

Adriano Barenco

Recent theoretical results confirm that quantum theory provides the possibility of new ways of performing efficient calculations. The most striking example is the factoring problem. It has recently been shown that computers that exploit quantum features could factor large composite integers This task is believed to be out of reach of classical computers as soon as the number of digits in the number to factor exceeds a certain limit. The additional power of quantum computers comes from the possibility of employing a superposition of states, of following many distinct computation paths and of producing a final output that depends on the interference of all of them. This "quantum parallelism" outstrips by far any parallelism that can be thought of in classical computation and is responsible for the "exponential" speed-up of computation. Experimentally, however, it will be extremely difficult to "decouple" a quantum computer from its environment. Noise fluctuations due to the outside world...

•Jan 1, 1996•98 citations

Quantum physics and computers

Adriano Barenco

Recent theoretical results confirm that quantum theory provides the possibility of new ways of performing efficient calculations. The most striking example is the factoring problem. It has recently been shown that computers that exploit quantum features could factor large composite integers. This task is believed to be out of reach of classical computers as soon as the number of digits in the number to factor exceeds a certain limit. The additional power of quantum computers comes from the possibility of employing a superposition of states, of following many distinct computation paths and of producing a final output that depends on the interference of all of them. This ‘quantum parallelism’ outstrips by far any parallelism that can be thought of in classical computation and is responsible for the ‘exponential’ speed-up of computation. Experimentally, however, it will be extremely difficult to ‘decouple’ a quantum computer from its environment. Noise fluctuations due to the outside world, no matter how little, are sufficient to drastically reduce the performance of these new computing devices. To control the nefarious effects of this environmental noise, one needs to implement efficient error-correcting techniques.

arXiv:quant-ph/9612014•Contemporary Physics•Sep 1, 1996•120 citations

Complexity and algorithms in quantum computation

Richard Jozsa

In 1982 Feynman noted that the simulation of a quantum process on a classical computer appears to generally involve an exponential slowdown in running time compared to the evolution of the process itself. More recently in quantum computation this effect has been turned around and exploited for computational advantage by formulating quantum processes (quantum algorithms) whose evolution corresponds to the performance of useful computational tasks. Thus the quantum algorithm will engender an exponential speedup in computing time. The most famous such quantum algorithm is Shor's algorithm for factorising whole numbers. It factorises a number having d digits in a time of order less than d whereas no known classical algorithm can perform factorisation in a time bounded by any polynomial in d. In theoretical computer science, computational complexity is generally characterised by classifying computational tasks into complexity classes such as P, BPP and NP. It therefore appears that quantum computers will be able to transgress some of the boundaries set by these classes. (2 pages)

•Jan 1, 1997•15 citations

EXPLAINING THE BASICS OF QUANTUM MECHANICS FOR COMPUTING

Sanjay Tiwari, Neeraj Bhat, Varun Sharma +1 more

Quantum computers operate on the principles of quantum logic, a fundamentally distinct paradigm from classical Boolean logic. This disparity leads to quantum computation's enhanced efficiency compared to classical computing. In this comprehensive review, we demystify the fundamental concepts of quantum computation, covering the creation of elementary gates and networks. We highlight the capabilities of quantum algorithms by examining the simple Deutsch problem and, in straightforward terms, dissect the renowned Shor algorithm for factoring large numbers into primes. Furthermore, we delve into the realm of physical quantum computer implementations, with a particular focus on tlo he linear ion trap approach. Here, we shed light on the primary challenge hindering the realization of practical quantum computers: the issue of decoherence. Nonetheless, we demonstrate that this hurdle can be overcome through the application of quantum error correction methods

International Journal of Psychosocial Rehabilitation•Apr 1, 2020•0 citations

Quantum computers and dissipation

G. Massimo Palma, Kalle‐Antti Suominen, ARTUR K. EKERT

We analyse dissipation in quantum computation and its destructive impact on efficiency of quantum algorithms. Using a general model of decoherence, we study the time evolution of a quantum register of arbitrary length coupled with an environment of arbitrary coherence length. We discuss relations between decoherence and computational complexity and show that the quantum factorization algorithm must be modified in order to be regarded as efficient and realistic.

arXiv:quant-ph/9702001•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Dec 31, 1996•898 citations

Penerapan Komputasi Kuantum dalam Kriptografi Modern dan Sistem Keamanan Digital

Rudolf Sinaga, Uswatun Kasanah

Quantum computing has emerged as a revolutionary paradigm, holding immense potential to solve complex problems that classical computing struggles to address. This study explores the application of quantum computing in cryptography, with a specific focus on two major quantum algorithms: Shor’s algorithm for large number factorization and Grover’s algorithm for unstructured database searching. The main objective of this research is to compare the performance of these quantum algorithms with classical cryptographic methods in terms of computational efficiency and time. Shor’s algorithm, which can factorize large numbers in polynomial time, presents a significant threat to the security of public-key cryptosystems such as RSA, which rely on the difficulty of factoring large numbers. On the other hand, Grover’s algorithm offers a quadratic speedup for searching unstructured databases, making it highly relevant for symmetric key cryptography systems like AES. In this study, simulations of both algorithms were conducted using quantum simulators to assess their speed and effectiveness in solving cryptographic challenges. The results demonstrate that quantum algorithms significantly reduce the computation time compared to classical methods, with Shor’s algorithm efficiently solving factorization problems and Grover’s algorithm accelerating key searching processes. However, while these quantum algorithms show promise in improving cryptographic systems, the implementation of large-scale quantum computers remains a challenge. This research highlights the potential of quantum computing to revolutionize data security and underscores the need for further development in quantum algorithms and the transition to quantum-resistant cryptographic systems to safeguard against the threat posed by quantum computers.

Journal of New Trends in Sciences•Nov 30, 2023•0 citations
It is known that quantum computers can dramatically speed up the task of finding factors of large numbers, a problem of practical significance for cryptographic applications. Factors of an L -digit number can be found in ∼ L 2 time [compared to ∼exp( L 1/3 ) time] by a quantum computer, which simultaneously follows all paths corresponding to distinct classical inputs, obtaining the solution from the coherent quantum interference of the alternatives. Here it is shown how the decoherence process degrades the interference pattern that emerges from the quantum factoring algorithm. For a quantum computer performing logical operations, an exponential decay of quantum coherence is inevitable. However, even in the presence of exponential decoherence, quantum computation can be useful as long as a sufficiently low decoherence rate can be achieved to allow meaningful results to be extracted from the calculation.
Recent theoretical results confirm that quantum theory provides the possibility of new ways of performing efficient calculations. The most striking example is the factoring problem. It has recently been shown that computers that exploit quantum features could factor large composite integers This task is believed to be out of reach of classical computers as soon as the number of digits in the number to factor exceeds a certain limit. The additional power of quantum computers comes from the possibility of employing a superposition of states, of following many distinct computation paths and of producing a final output that depends on the interference of all of them. This "quantum parallelism" outstrips by far any parallelism that can be thought of in classical computation and is responsible for the "exponential" speed-up of computation. Experimentally, however, it will be extremely difficult to "decouple" a quantum computer from its environment. Noise fluctuations due to the outside world...
Recent theoretical results confirm that quantum theory provides the possibility of new ways of performing efficient calculations. The most striking example is the factoring problem. It has recently been shown that computers that exploit quantum features could factor large composite integers. This task is believed to be out of reach of classical computers as soon as the number of digits in the number to factor exceeds a certain limit. The additional power of quantum computers comes from the possibility of employing a superposition of states, of following many distinct computation paths and of producing a final output that depends on the interference of all of them. This ‘quantum parallelism’ outstrips by far any parallelism that can be thought of in classical computation and is responsible for the ‘exponential’ speed-up of computation. Experimentally, however, it will be extremely difficult to ‘decouple’ a quantum computer from its environment. Noise fluctuations due to the outside world, no matter how little, are sufficient to drastically reduce the performance of these new computing devices. To control the nefarious effects of this environmental noise, one needs to implement efficient error-correcting techniques.
In 1982 Feynman noted that the simulation of a quantum process on a classical computer appears to generally involve an exponential slowdown in running time compared to the evolution of the process itself. More recently in quantum computation this effect has been turned around and exploited for computational advantage by formulating quantum processes (quantum algorithms) whose evolution corresponds to the performance of useful computational tasks. Thus the quantum algorithm will engender an exponential speedup in computing time. The most famous such quantum algorithm is Shor's algorithm for factorising whole numbers. It factorises a number having d digits in a time of order less than d whereas no known classical algorithm can perform factorisation in a time bounded by any polynomial in d. In theoretical computer science, computational complexity is generally characterised by classifying computational tasks into complexity classes such as P, BPP and NP. It therefore appears that quantum computers will be able to transgress some of the boundaries set by these classes. (2 pages)
Quantum computers operate on the principles of quantum logic, a fundamentally distinct paradigm from classical Boolean logic. This disparity leads to quantum computation's enhanced efficiency compared to classical computing. In this comprehensive review, we demystify the fundamental concepts of quantum computation, covering the creation of elementary gates and networks. We highlight the capabilities of quantum algorithms by examining the simple Deutsch problem and, in straightforward terms, dissect the renowned Shor algorithm for factoring large numbers into primes. Furthermore, we delve into the realm of physical quantum computer implementations, with a particular focus on tlo he linear ion trap approach. Here, we shed light on the primary challenge hindering the realization of practical quantum computers: the issue of decoherence. Nonetheless, we demonstrate that this hurdle can be overcome through the application of quantum error correction methods
Quantum computing has emerged as a revolutionary paradigm, holding immense potential to solve complex problems that classical computing struggles to address. This study explores the application of quantum computing in cryptography, with a specific focus on two major quantum algorithms: Shor’s algorithm for large number factorization and Grover’s algorithm for unstructured database searching. The main objective of this research is to compare the performance of these quantum algorithms with classical cryptographic methods in terms of computational efficiency and time. Shor’s algorithm, which can factorize large numbers in polynomial time, presents a significant threat to the security of public-key cryptosystems such as RSA, which rely on the difficulty of factoring large numbers. On the other hand, Grover’s algorithm offers a quadratic speedup for searching unstructured databases, making it highly relevant for symmetric key cryptography systems like AES. In this study, simulations of both algorithms were conducted using quantum simulators to assess their speed and effectiveness in solving cryptographic challenges. The results demonstrate that quantum algorithms significantly reduce the computation time compared to classical methods, with Shor’s algorithm efficiently solving factorization problems and Grover’s algorithm accelerating key searching processes. However, while these quantum algorithms show promise in improving cryptographic systems, the implementation of large-scale quantum computers remains a challenge. This research highlights the potential of quantum computing to revolutionize data security and underscores the need for further development in quantum algorithms and the transition to quantum-resistant cryptographic systems to safeguard against the threat posed by quantum computers.