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Journal of the Franklin Institute

Quantum mechanics experiment mimicked by stochastic calculations

Raul Rabinovici, Ahmad Alyan, Ronen Rabinovici

Published Mar 18, 2012• 0 citations
Publisher
DOI
10.1016/j.jfranklin.2012.03.003

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In spite of the macroscopic character of the fluctuation amplitudes, we show that the standard inflationary distribution of primordial density fluctuations still exhibits inherently quantum mechanical correlations (which cannot be mimicked by any classical stochastic ensemble). To this end, we propose a Gedanken experiment for which certain Bell inequalities are violated. We also compute the effect of decoherence and show that the violation persists provided that the decoherence lies below a certain non-vanishing threshold. Moreover, there exists a higher threshold above which no violation of any Bell inequalities can occur, so that the corresponding distributions can be interpreted as stochastic ensembles of classical fluctuations.
Quantum-inspired optimization represents a promising avenue for addressing computationally challenging optimization problems. This paper investigates the application of stochastic gradient descent (SGD) to emulate the quantum annealing process, aiming to improve the speed and accuracy of complex optimization tasks. We propose a novel algorithm that iteratively refines a solution based on the gradient of the objective function, mimicking the probabilistic exploration inherent in quantum annealing. This approach offers a potentially more efficient and scalable method compared to traditional optimization techniques, leveraging the strengths of stochasticity. The core mechanism centers around utilizing SGD to dynamically adjust the solution trajectory, effectively mimicking the quantum annealing process.
High-dimensional optimization problems pose significant challenges to classical stochastic search algorithms. The "curse of dimensionality" dramatically reduces the effectiveness of these methods as the number of variables increases. This paper proposes a novel stochastic search algorithm, Quantum-Inspired Stochastic Search (QISS), that draws inspiration from quantum mechanics to address this issue. QISS leverages the concepts of superposition and interference to efficiently explore the solution space. Each node in the search represents a potential solution existing in a probabilistic superposition of states. The algorithm simulates interference by dynamically adjusting the weighting of solution candidates based on their "quantum potential," a measure of uncertainty and promisingness. We demonstrate that QISS, by directly mimicking quantum phenomena, can potentially outperform traditional stochastic methods in complex, high-dimensional landscapes without the need for quantum computing resources. The core of the algorithm is formulated as follows: Let *N* be the number of nodes in the search space. Let *Si* represent the state of node *i*, where *i* = 1, 2, ..., *N*. The quantum potential of node *i*, denoted as *Pi*, is calculated as: *Pi* = ∑j=1N |*sij* - *sk*|2 / (Σj=1N |*sij* - *sk|2 + ε) where *sij* is the value of the *j*th variable in node *i*, *sk* is the average value of all nodes in the search space, and ε is a small constant to avoid division by zero. The weighting factor for node *i* is then given by: *Wi* = exp(-*Pi*) The algorithm iteratively updates the node states based on these weighted probabilities, effectively simulating quantum interference to guide the search toward promising regions of the solution space. The algorithm's performance is evaluated based on convergence rate and solution quality.