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Chinese Journal of Quantum Electronics

Quantum computational characteristics of a classical coupled harmonic oscillator system

Li Sun

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Wave feature of a classical physical system demonstrates more efficiently than its relevant particle feature during information processing procedure.By analysis of wave and energy features of a classical coupled harmonic oscillator system,it was proved that Grover's quantum unsorted search algorithm can be implemented in such a classical system,which had almost the same efficiency as the fully quantum ones.Meanwhile,the better stability of wave implementation of information process compared to quantum computation was discussed.Furthermore,the application ofenergy focusingfeature of the classical coupled harmonic oscillator system was prospected.

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Wave feature of a classical physical system demonstrates more efficiently than its relevant particle feature during information processing procedure.By analysis of wave and energy features of a classical coupled harmonic oscillator system,it was proved that Grover's quantum unsorted search algorithm can be implemented in such a classical system,which had almost the same efficiency as the fully quantum ones.Meanwhile,the better stability of wave implementation of information process compared to quantum computation was discussed.Furthermore,the application ofenergy focusingfeature of the classical coupled harmonic oscillator system was prospected.
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 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, 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
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 investigate the quantum algorithm of Babbush et al. (arXiv:2303.13012v3) for simulating coupled harmonic oscillators, which promises exponential speedups over classical methods. Focusing on linearly connected oscillator chains, we bridge the gap between theory and implementation by developing and comparing three concrete realizations of the algorithm. First, we implement a sparse initial state preparation combined with product-formula (Suzuki-Trotter) Hamiltonian simulation. Second, we implement a fully quantum, oracle-based framework in which classical data are accessed via oracles, the Hamiltonian is block-encoded, and time evolution is performed using QSVT-based Hamiltonian simulation. Third, we propose an efficient alternative that combines the sparse state-preparation routine of the first approach with the oracle and block-encoding-based simulation pipeline of the second. We provide these implementations on Classiq, a high-level quantum design platform and provide appropriate resource benchmarks. Our simulation results show that the complex initial state preparation proposed by Babbush et al. can be circumvented at least in the linear-chain case. Finally, we illustrate two physical applications-extracting normal modes and simulating coarse-grained energy propagation-demonstrating how the algorithm connects to measurable observables. Our results clarify the resource requirements of the algorithm and provide concrete pathways toward practical quantum advantage.