Algorithms for quantum computation: discrete logarithms and factoring

Shor, Peter W.

doi:10.1109/sfcs.1994.365700

Cited by 5,590 publications

(3,880 citation statements)

References 23 publications

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“…Most quantum algorithms are stochastic algorithms which can give the correct decision-making with probability 1-ǫ (ǫ > 0, close to 0) after several times of repeated computing [23], [25]. As for quantum reinforcement learning algorithms, optimal policies are acquired by the collapse of quantum system and we will analyze the optimality of these policies from two aspects as follows.…”

Section: B Optimality and Stochastic Algorithmmentioning

confidence: 99%

“…Some results have shown that quantum computation can more efficiently solve some difficult problems than the classical counterpart. Two important quantum algorithms, the Shor algorithm [23], [24] and the Grover algorithm [25], [26] have been proposed in 1994 and 1996, respectively. The Shor algorithm can give an exponential speedup for factoring large integers into prime numbers and it has been realized [27] for the factorization of integer 15 using nuclear magnetic resonance (NMR).…”

Section: Introductionmentioning

confidence: 99%

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Quantum Reinforcement Learning

Dong

Chen

2005

Lecture Notes in Computer Science

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Abstract-The key approaches for machine learning, especially learning in unknown probabilistic environments are new representations and computation mechanisms. In this paper, a novel quantum reinforcement learning (QRL) method is proposed by combining quantum theory and reinforcement learning (RL). Inspired by the state superposition principle and quantum parallelism, a framework of value updating algorithm is introduced. The state (action) in traditional RL is identified as the eigen state (eigen action) in QRL. The state (action) set can be represented with a quantum superposition state and the eigen state (eigen action) can be obtained by randomly observing the simulated quantum state according to the collapse postulate of quantum measurement. The probability of the eigen action is determined by the probability amplitude, which is parallelly updated according to rewards. Some related characteristics of QRL such as convergence, optimality and balancing between exploration and exploitation are also analyzed, which shows that this approach makes a good tradeoff between exploration and exploitation using the probability amplitude and can speed up learning through the quantum parallelism. To evaluate the performance and practicability of QRL, several simulated experiments are given and the results demonstrate the effectiveness and superiority of QRL algorithm for some complex problems. The present work is also an effective exploration on the application of quantum computation to artificial intelligence.Index Terms-quantum reinforcement learning, state superposition, collapse, probability amplitude, Grover iteration.

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Section: B Optimality and Stochastic Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantum Reinforcement Learning

Dong

Chen

2005

Lecture Notes in Computer Science

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“…However, since the computational hardness of the underlying problems in public-key cryptography have not been formally established, public-key cryptography is not immune to scenarios where an eavesdropper would possess some unexpectedly strong computational power or would know better cryptanalysis techniques than the best published ones. Moreover, most of the currently used public-key cryptographic schemes 2.3 Classical computationally secure symmetric-key cryptography and secret key agreement (for example RSA) could be cracked in polynomial time with a quantum computer: this results from Shor's algorithm for discrete log and factoring, that has a complexity of O(n 3 ) [21]. It however seems possible to build alternative public-key cryptographic schemes on problems that could resist polynomial cryptanalysis on a quantum computer, such as lattice shortest vector problem [22,23].…”

Section: Classical Public-key Cryptography and Secret Key Agreementmentioning

confidence: 99%

Using quantum key distribution for cryptographic purposes: A survey

Alléaume

Branciard

Bouda

et al. 2014

Theoretical Computer Science

164

105

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The appealing feature of quantum key distribution (QKD), from a cryptographic viewpoint, is the ability to prove the information-theoretic security (ITS) of the established keys. As a key establishment primitive, QKD however does not provide a standalone security service in its own: the secret keys established by QKD are in general then used by a subsequent cryptographic applications for which the requirements, the context of use and the security properties can vary. It is therefore important, in the perspective of integrating QKD in security infrastructures, to analyze how QKD can be combined with other cryptographic primitives.The purpose of this survey article, which is mostly centered on European research results, is to contribute to such an analysis. We first review and compare the properties of the existing key establishment techniques, QKD being one of them. We then study more specifically two generic scenarios related to the practical use of QKD in cryptographic infrastructures: 1) using QKD as a key renewal technique for a symmetric cipher over a point-to-point link ; 2) using QKD in a network containing many users with the objective of offering any-to-any key establishment service. We discuss the constraints as well as the potential interest of using QKD in these contexts. We finally give an overview of challenges relative to the development of QKD technology that also constitute potential avenues for cryptographic research.

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“…Recently, quantum computing has been hailed as the possible solution to some of the computationally hard classical problems [11]. Indeed, Grover's [7] and Shor's [12] algoritms provide such solutions to the problems of¯nding a given element in an unsorted set and the prime factorization of very large numbers, respectively. Here we present a solution to the continuous GOP in polynomial time, by developing a generalization of Grover's algorithm to continuous problems.…”

Section: Global Optimization Problemmentioning

confidence: 99%

Quantum Algorithm for Continuous Global Optimization

Protopopescu

Barhen

2005

Applied Optimization

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We investigate the entwined roles of information and quantum algorithms in reducing the complexity of the global optimization problem (GOP). We show that: (i) a modest amount of additional information is su±cient to map the general continuous GOP into the (discrete) Grover problem; (ii) while this additional information is actually available in some classes of GOPs, it cannot be taken advantage of within classical optimization algorithms; (iii) on the contrary, quantum algorithms o®er a natural framework for the e±cient use of this information resulting in a speed-up of the solution of the GOP.

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Algorithms for quantum computation: discrete logarithms and factoring

Cited by 5,590 publications

References 23 publications

Quantum Reinforcement Learning

Quantum Reinforcement Learning

Using quantum key distribution for cryptographic purposes: A survey

Quantum Algorithm for Continuous Global Optimization

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