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arXiv:quant-ph/0505057

Macroscopic entanglement in Quantum Computation

Akihisa Ukena, Shimizu, Akira

Published May 9, 2005• 4 citations
arXivHTMLPDF

Abstract

We investigate macroscopic entanglement of quantum states in quantum computers, where we say a quantum state is entangled macroscopically if the state has superposition of macroscopically distinct states. The index $p$ of the macroscopic entanglement is calculated as a function of the step of the computation, for Grover's quantum search algorithm and Shor's factoring algorithm. It is found that whether macroscopically entangled states are used or not depends on the numbers and properties of the solutions to the problem to be solved. When the solutions are such that the problem becomes hard in the sense that classical algorithms take more than polynomial steps to find a solution, macroscopically entangled states are always used in Grover's algorithm and almost always used in Shor's algorithm. Since they are representative algorithms for unstructured and structured problems, respectively, our results support strongly the conjecture that quantum computers utilize macroscopically entangled states when they solve hard problems much faster than any classical algorithms.

DOI
10.48550/arxiv.quant-ph/0505057

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We investigate macroscopic entanglement of quantum states in quantum computers, where we say a quantum state is entangled macroscopically if the state has superposition of macroscopically distinct states. The index ppp of the macroscopic entanglement is calculated as a function of the step of the computation, for Grover's quantum search algorithm and Shor's factoring algorithm. It is found that whether macroscopically entangled states are used or not depends on the numbers and properties of the solutions to the problem to be solved. When the solutions are such that the problem becomes hard in the sense that classical algorithms take more than polynomial steps to find a solution, macroscopically entangled states are always used in Grover's algorithm and almost always used in Shor's algorithm. Since they are representative algorithms for unstructured and structured problems, respectively, our results support strongly the conjecture that quantum computers utilize macroscopically entangled states when they solve hard problems much faster than any classical algorithms.
For quantum computation, we investigate the conjecture that the superposition of macroscopically distinct states is necessary for a large quantum speedup. Although this conjecture was supported for a circuit-based quantum computer performing Shor's factoring algorithm [A. Ukena and A. Shimizu, Phys. Rev. A69 (2004) 022301], it needs to be generalized for it to be applicable to a large class of algorithms and/or other models such as measurement-based quantum computers. To treat such general cases, we first generalize the indices for the superposition of macroscopically distinct states. We then generalize the conjecture, using the generalized indices, in such a way that it is unambiguously applicable to general models if a quantum algorithm achieves exponential speedup. On the basis of this generalized conjecture, we further extend the conjecture to Grover's quantum search algorithm, whose speedup is large but quadratic. It is shown that this extended conjecture is also correct. Since Grover's algorithm is a representative algorithm for unstructured problems, the present result further supports the conjecture.
The power of quantum mechanics, that is too complex for conventional computers, can be solved by an innovative model of computing known as quantum computing. Quantum algorithms can provide exponential speedups for some types of problems, such as many difficult mathematical ones. In this paper, we review some of the most important quantum algorithms for hard mathematical problems. When factoring large numbers, Shor's algorithm is orders of magnitude faster than any other known classical algorithm. The Grover's algorithm, which searches unsorted databases much more quickly than conventional algorithms, is then discussed.
Quantum Algorithms have long captured the imagination of computer scientists and physicists primarily because of the speed up achieved by them over their classical counterparts using principles of quantum mechanics. Entanglement is believed to be the primary phenomena behind this speed up. However their precise role in quantum algorithms is yet unclear. In this article, we explore the nature of entanglement in the Grover's search algorithm. This algorithm enables searching of elements from an unstructured database quadratically faster than the best known classical algorithm. Geometric measure of entanglement has been used to quantify and analyse entanglement across iterations of the algorithm. We reveal how the entanglement varies with increase in the number of qubits and also with the number of marked or solution states. Numerically, it is seen that the behaviour of the maximum value of entanglement is monotonous with the number of qubits. Also, for a given value of the number of qubits, a change in the marked states alters the amount of entanglement. The amount of entanglement in the final state of the algorithm has been shown to depend solely on the nature of the marked states. Explicit analytical expressions are given showing the variation of entanglement with the number of iterations and the global maximum value of entanglement attained across all iterations of the algorithm.
For decades, classical computers have been governed by predictable rules. But as we approach the physical limits of computation, a new paradigm is emerging. Quantum computing operates on the strange and powerful laws of quantum mechanics—superposition, entanglement, and interference—to tackle problems far beyond the reach of any classical machine. The true potential of this field was unlocked by two landmark discoveries: Shor’s algorithm, a revolutionary method for integer factorization that threatens to upend modern cryptography, and Grover’s algorithm, a versatile tool that offers a quadratic speedup for unstructured search problems. This paper serves as a guide to these foundational algorithms, detailing their inner workings and their profound impact. We also explore post-quantum cryptography, hybrid frameworks, and quantum machine learning—illustrating the evolving role of quantum algorithms in shaping the future of science and technology.
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%.