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Computación y Sistemas

Remedies for the Inconsistences in the Times of Execution of the Unsorted Database Search Algorithm within the Wave Approach

Manuel Aoki Avila, José Benito Elizalde-Salas

Published Jan 1, 2018• 1 citation
Publisher

Abstract

The typical semiclassical wave version of the unsorted database search algorithm based on a system of coupled simple harmonic oscillators does not consider an important ingredient of Grover’s original algorithm as it is quantum entanglement. The role of entanglement in the wave version of the unsorted database search algorithm is explored and contradictions with the time of execution of Grover’s algorithm are found. We remedy the contradictions by employing two arguments, one of them qualitative and the other quantitative. For the qualitative argument we employ the probabilistic nature of a legitimate quantum algorithm and remedy the above inconsistence. Within the quantitative argument we identify a parameter in the wave version of the unsorted database search algorithm which is related to entanglement. The contradiction with the time of execution of Grover’s algorithm is solved by choosing an appropriate values of such a parameter which incorporates entanglement to the wave version of the unsorted database search algorithm. The utility of the present arguments are evident if the wave version of the unsorted data base search algorithm is experimentally implemented through a system of N quantum dots with a harmonic oscillator potential as a confinement potential for each of the quantum dots. Each of the above N vibrating quantum dots must be coupled∗vlkmanuel@uaemex.mx1to an extra single vibrating quantum dot which entangles to all of them. In order to obtain optimal results, the coupling constants of the mentioned quantum dots should be adjusted in the way described in the present work.

DOI
10.13053/cys-21-4-2367

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The typical semiclassical wave version of the unsorted database search algorithm based on a system of coupled simple harmonic oscillators does not consider an important ingredient of Grover’s original algorithm as it is quantum entanglement. The role of entanglement in the wave version of the unsorted database search algorithm is explored and contradictions with the time of execution of Grover’s algorithm are found. We remedy the contradictions by employing two arguments, one of them qualitative and the other quantitative. For the qualitative argument we employ the probabilistic nature of a legitimate quantum algorithm and remedy the above inconsistence. Within the quantitative argument we identify a parameter in the wave version of the unsorted database search algorithm which is related to entanglement. The contradiction with the time of execution of Grover’s algorithm is solved by choosing an appropriate values of such a parameter which incorporates entanglement to the wave version of the unsorted database search algorithm. The utility of the present arguments are evident if the wave version of the unsorted data base search algorithm is experimentally implemented through a system of N quantum dots with a harmonic oscillator potential as a confinement potential for each of the quantum dots. Each of the above N vibrating quantum dots must be coupled∗vlkmanuel@uaemex.mx1to an extra single vibrating quantum dot which entangles to all of them. In order to obtain optimal results, the coupling constants of the mentioned quantum dots should be adjusted in the way described in the present work.
Abstract. Grover’s database search algorithm, although discovered in the context of quantum computation, can be implemented using any physical system that allows superposition of states. A physical realization of this algorithm is described using coupled simple harmonic oscillators, which can be exactly solved in both classical and quantum domains. Classical wave algorithms are far more stable against decoherence compared to their quantum counterparts. In addition to providing convenient demonstration models, they may have a role in practical situations, such as catalysis. Keywords: Grover’s database search algorithm, Catalysis, Resonance
Grover’s database search algorithm, although discovered in the context of quantum computation, can be implemented using any physical system that allows superposition of states. A physical realization of this algorithm is described using coupled simple harmonic oscillators, which can be exactly solved in both classical and quantum domains. Classical wave algorithms are far more stable against decoherence compared to their quantum counterparts. In addition to providing convenient demonstration models, they may have a role in practical situations, such as catalysis.
Grover's database search algorithm is the optimal algorithm for finding a desired object from an unsorted collection of items. Although it was discovered in the context of quantum computation, it is simple and versatile enough to be implemented using any physical system that allows superposition of states, and several proposals have been made in the literature. I study a mechanical realisation of the algorithm using coupled simple harmonic oscillators, and construct its physical model for the simplest case of four identical oscillators. The identification oracle is implemented as an elastic reflection of the desired oscillator, and the overrelaxation operation is realised as evolution of the system by half an oscillation period. I derive the equations of motion, and solve them both analytically and by computer simulation. I extend the ideal case analysis and explore the sensitivity of the algorithm to changes in the initial conditions, masses of springs and damping. The amplitude amplification provided by the algorithm enhances the energy of the desired oscillator, while running the algorithm backwards spreads out the energy of the perturbed oscillator among its partners. The former (efficient focusing of energy into a specific oscillator) can have interesting applications in processes that need crossing of an energy threshold for completion, and can be useful in nanotechnological devices and catalysis. The latter (efficient redistribution of energy) can be useful in processes requiring rapid dissipation of energy, such as shock-absorbers and vibrational shielding. I present some tentative proposals.
The present study aims to examine the real-world implications of Grover's algorithm, specifically focusing on its quadratic speedup in contrast to classical linear search techniques employed for unsorted databases. The objective is to bridge the gap between theoretical potential and practical application in the field of quantum computing. This study conducts a comprehensive analysis of Grover's Algorithm and traditional linear search across databases of varying sizes, uniquely contributing by empirically validating Grover's Algorithm's superior performance and aligning it with theoretical expectations. Significantly, the study expands its analysis to encompass larger databases, further affirming the algorithm's utility in handling growing database sizes. This study significantly contributes to the progress of quantum computing through the practical implementation of theoretical principles as well as enhancing the scientific community's understanding of quantum search algorithms and their relative efficacy.
Grover's database search algorithm, although discovered in the context of quantum computation, can be implemented using any system that allows superposition of states. A physical realization of this algorithm is described using coupled simple harmonic oscillators, which can be exactly solved in both classical and quantum domains. Classical wave algorithms are far more stable against decoherence compared to their quantum counterparts. In addition to providing convenient demonstration models, they may have a role in practical situations, such as catalysis.
Quantum computers are proven to be more efficient at solving a specific class of problems compared to traditional digital computers. Superposition of states and quantum entanglement are the two key ingredients that make quantum computing so powerful. However, not all quantum algorithms require quantum entanglement (e.g., search through an unsorted database). Is it possible to utilize classical wave superposition to speed up database searching as much as by using quantum computers? There were several attempts to mimic quantum computers using classical waves. It was concluded that the use of classical wave superposition comes with the cost of an exponential increase in resources. In this work, we consider the feasibility of building classical wave-based devices able to provide fundamental speedup over digital counterparts without the exponential overhead. We present experimental data on database searching through a magnetic database using spin wave superposition. The results demonstrate the same speedup as expected for quantum computers. Also, we present examples of numerical modeling demonstrating classical wave interference for period finding. This approach may not compete with quantum computers with efficiency but outperform classical digital computers. We argue that classical wave-based devices can perform some of the quantum algorithms with the same efficiency as quantum computers as long as quantum entanglement is not required.