The goals of this paper are to show the following. First, Grover's algorithm can be viewed as a digital approximation to the analog quantum algorithm proposed in "An Analog Analogue of a Digital Quantum Computation", by E. Farhi and S. Gutmann, Phys.Rev. A 57, 2403 - 2406 (1998), quant-ph/9612026. We will call the above analog algorithm the Grover-Farhi-Gutmann or GFG algorithm. Second, the propagator of the GFG algorithm can be written as a sum-over-paths formula and given a sum-over-path interpretation, i.e., a Feynman path sum/integral. We will use nonstandard analysis to do this. Third, in the semi-classical limit $\hbar\to 0$ , both the Grover and the GFG algorithms (viewed in the setting of the approximation in this paper) must run instantaneously. Finally, we will end the paper with an open question. In "Semiclassical Shor's Algorithm", by P. Giorda, et al, Phys. Rev.A 70, 032303 (2004), quant-ph/0303037, the authors proposed building semi-classical quantum computers to run Shor's algorithm because the success probability of Shor's algorithm does not change much in the semi-classical limit. We ask the open questions: In the semi-classical limit, does Shor's algorithm have to run instantaneously?The goals of this paper are to show the following. First, Grover's algorithm can be viewed as a digital approximation to the analog quantum algorithm proposed in "An Analog Analogue of a Digital Quantum Computation", by E. Farhi and S. Gutmann, Phys.Rev. A 57, 2403 - 2406 (1998), quant-ph/9612026. We will call the above analog algorithm the Grover-Farhi-Gutmann or GFG algorithm. Second, the propagator of the GFG algorithm can be written as a sum-over-paths formula and given a sum-over-path interpretation, i.e., a Feynman path sum/integral. We will use nonstandard analysis to do this. Third, in the semi-classical limit $\hbar\to 0$, both the Grover and the GFG algorithms (viewed in the setting of the approximation in this paper) must run instantaneously. Finally, we will end the paper with an open question. In "Semiclassical Shor's Algorithm", by P. Giorda, et al, Phys. Rev.A 70, 032303 (2004), quant-ph/0303037, the authors proposed building semi-classical quantum computers to run Shor's algorithm because the success probability of Shor's algorithm does not change much in the semi-classical limit. We ask the open questions: In the semi-classical limit, does Shor's algorithm have to run instantaneously?

arXiv:quant-ph/0502137Feb 22, 20050 citations

Quantum Computing: A Mathematical Analysis of Shor's Algorithm

Stephen S. Clarke

Jan 1, 20200 citations

Quantum Cryptanalysis: Breaking Classical Encryption with Shor's and Grover's Algorithms

Godwin Olaoye

The advent of quantum computing presents a paradigm shift in the field of cryptanalysis, challenging the security of classical encryption schemes that underpin modern digital communications. This study explores the capabilities of quantum algorithms, particularly Shor's algorithm and Grover's algorithm, in undermining widely used cryptographic systems. Shor's algorithm poses a direct threat to publickey cryptosystems such as RSA and ECC by enabling efficient integer factorization and discrete logarithm computation, which are computationally infeasible for classical computers. Meanwhile, Grover's algorithm provides a quadratic speedup for bruteforce attacks on symmetric-key cryptosystems like AES, significantly reducing their effective security. Through theoretical analysis and simulation-based evaluation, this paper highlights the vulnerabilities of classical encryption in the face of quantum advances and underscores the urgent need for post-quantum cryptographic solutions. The findings emphasize the critical importance of transitioning to quantum-resistant algorithms to ensure long-term data confidentiality in a future dominated by quantum technologies.The advent of quantum computing presents a paradigm shift in the field of cryptanalysis, challenging the security of classical encryption schemes that underpin modern digital communications. This study explores the capabilities of quantum algorithms, particularly Shor's algorithm and Grover's algorithm, in undermining widely used cryptographic systems. Shor's algorithm poses a direct threat to publickey cryptosystems such as RSA and ECC by enabling efficient integer factorization and discrete logarithm computation, which are computationally infeasible for classical computers. Meanwhile, Grover's algorithm provides a quadratic speedup for bruteforce attacks on symmetric-key cryptosystems like AES, significantly reducing their effective security. Through theoretical analysis and simulation-based evaluation, this paper highlights the vulnerabilities of classical encryption in the face of quantum advances and underscores the urgent need for post-quantum cryptographic solutions. The findings emphasize the critical importance of transitioning to quantum-resistant algorithms to ensure long-term data confidentiality in a future dominated by quantum technologies.

Apr 10, 20253 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 Mechanical Algorithms for Solving Root Finding Problem

Sun Guo-don

Root finding problem is a hard problem in computational number theory. To improve the efficiency of solving the problem and expand the scope of applications of quantum computing,the quantum algorithms for solving the root finding problem are analyzed in this paper. Based on the two important quantum algorithms—Shor's algorithm and Grover's algorithm, two quantum mechanical algorithms are proposed to solve the root finding problem,which are named RF-Shor algorithm and RFGrover algorithm. According to the analysis,the RF-Shor algorithm can find all the solutions to the root finding problem with a probability closed to 100%,with a consumption of polynomial quantum gates; the RF-Grover algorithm does not involve any classical method to improve the search efficiency and one of the k solutions to the root finding problem can be obtained in O(M/k) steps with a probability of at least50%.Root finding problem is a hard problem in computational number theory. To improve the efficiency of solving the problem and expand the scope of applications of quantum computing,the quantum algorithms for solving the root finding problem are analyzed in this paper. Based on the two important quantum algorithms—Shor's algorithm and Grover's algorithm, two quantum mechanical algorithms are proposed to solve the root finding problem,which are named RF-Shor algorithm and RFGrover algorithm. According to the analysis,the RF-Shor algorithm can find all the solutions to the root finding problem with a probability closed to 100%,with a consumption of polynomial quantum gates; the RF-Grover algorithm does not involve any classical method to improve the search efficiency and one of the k solutions to the root finding problem can be obtained in O(M/k) steps with a probability of at least50%.

Journal of Beijing University of TechnologyJan 1, 20151 citation

\hbar\to 0

, both the Grover and the GFG algorithms (viewed in the setting of the approximation in this paper) must run instantaneously. Finally, we will end the paper with an open question. In "Semiclassical Shor's Algorithm", by P. Giorda, et al, Phys. Rev.A 70, 032303 (2004), quant-ph/0303037, the authors proposed building semi-classical quantum computers to run Shor's algorithm because the success probability of Shor's algorithm does not change much in the semi-classical limit. We ask the open questions: In the semi-classical limit, does Shor's algorithm have to run instantaneously?