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arXiv:0711.4733

A Coupled Oscillator Model for Grover's Quantum Database Search Algorithm

Aavishkar A. Patel

Published Nov 29, 2007• 3 citations

Abstract

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.

DOI: 10.48550/arxiv.0711.4733

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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.

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.

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.

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.

We present a quantum algorithm for simulating the classical dynamics of 2 n coupled oscillators (e.g., 2 n masses coupled by springs).Our approach leverages a mapping between the Schrödinger equation and Newton's equation for harmonic potentials such that the amplitudes of the evolved quantum state encode the momenta and displacements of the classical oscillators.When individual masses and spring constants can be efficiently queried, and when the initial state can be efficiently prepared, the complexity of our quantum algorithm is polynomial in n, almost linear in the evolution time, and sublinear in the sparsity.As an example application, we apply our quantum algorithm to efficiently estimate the kinetic energy of an oscillator at any time.We show that any classical algorithm solving this same problem is inefficient and must make 2 ΩðnÞ queries to the oracle, and when the oracles are instantiated by efficient quantum circuits, the problem is bounded-error quantum polynomial time complete.Thus, our approach solves a potentially practical application with an exponential speedup over classical computers.Finally, we show that under similar conditions our approach can efficiently simulate more general classical harmonic systems with 2 n modes.

In my project I would have considered how the old search algorithms that are used when working with large unstructured data set end up taking linear time (O(N)) that in effect is the time that scales with the size of the data. These old ways get gradually sluggish, as everything gets bigger. Then I read about quantum computing, and especially Search Algorithm that was developed by Grover which is capable of quadratic speed-up with a complexity of O(sqrt(N)) of unstructured data. I created a hybrid system which fuses classical and quantum search applied on Flask as the web front end and Qiskit as the quantum one. This will enable me to compare the simple linear algorithm of searching through the list item-by-item with the quantum algorithm of Grover which uses superposition, a marking oracle to identify the desired item, and multiple amplitude amplification diffusion operators. The web page also accepts a student ID, and actually searches both simultaneously and allows me to tweak a depolarizing noise model so as to more closely mimic real quantum hardware. I measured such variables as the number of calls to the oracle, the number of qubits that I require, the quantum run time, and the number of measurements that I receive out of the quantum measurement. As the results indicate, the algorithm developed by Grover reduces the number of operations significantly compared to the classical one, and it must be emphasized that the algorithm remains quite effective even in the case of simulated noise. On the whole, it demonstrates that the classical and quantum components may be an efficient method to deal with the large search in the database.