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arXiv:2411.13593

On quantum models for opinion and voting intention polls

François Dubois

Published Nov 18, 2024• 0 citations
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Abstract

In this contribution, we construct a connection between two quantum voting models presented previously. We propose to try to determine the result of a vote from associated given opinion polls. We introduce a density operator relative to the family of all candidates to a particular election. From an hypothesis of proportionality between a family of coefficients which characterize the density matrix and the probabilities of vote for all the candidates, we propose a numerical method for the entire determination of the density operator. This approach is a direct consequence of the Perron-Frobenius theorem for irreductible positive matrices. We applyour algorithm to synthetic data and to operational results issued from the French presidential election of April 2012.

DOI
10.48550/arxiv.2411.13593

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In this contribution, we construct a connection between two quantum voting models presented previously. We propose to try to determine the result of a vote from associated given opinion polls. We introduce a density operator relative to the family of all candidates to a particular election. From an hypothesis of proportionality between a family of coefficients which characterize the density matrix and the probabilities of vote for all the candidates, we propose a numerical method for the entire determination of the density operator. This approach is a direct consequence of the Perron-Frobenius theorem for irreductible positive matrices. We applyour algorithm to synthetic data and to operational results issued from the French presidential election of April 2012.
Quantum voting protocols aim to offer ballot secrecy and publicly verifiable tallies using physical guarantees from quantum mechanics, rather than relying solely on computational hardness. This article surveys whether such quantum voting protocols are practical. We begin by outlining core mathematical ideas such as the superposition principle, the no-cloning theorem, and quantum entanglement. We then define a common system and threat model, identifying key actors, trust assumptions, and security goals. Representative protocol families are reviewed, including entanglement-based schemes with central tallying, self-tallying designs that enable public verification, and authority-minimized approaches that certify untrusted devices through observable correlations. Finally, we evaluate implementation challenges, including loss, noise, device imperfections, scalability, and coercion resistance, and discuss realistic near-term deployment scenarios for small-scale elections.
Politics is regarded as a vital area of public choice theory, and it is strongly relying on the assumptions of voters’ rationality and as such, stability of preferences. However, recent opinion polls and real election outcomes in the USA have shown that voters often engage in ‘ticket splitting’, by exhibiting contrasting party support in Congressional and Presidential elections (cf. Khrennikova 2014Phys. ScriptaT163, 014010 ( doi:10.1088/0031-8949/2014/T163/014010 ); Khrennikova & Haven 2016Phil. Trans. R. Soc. A374, 20150106 ( doi:10.1098/rsta.2015.0106 ); Smithet al.1999Am. J. Polit. Sci.43, 737–764 ( doi:10.2307/2991833 )). Such types of preference reversals cannot be mathematically captured via the formula of total probability, thus showing that voters’ decision making is at variance with the classical probabilistic information processing framework. In recent work, we have shown that quantum probability describes well the violation of Bayesian rationality in statistical data of voting in US elections, through the so-calledinterference effectsof probability amplitudes. This paper is proposing a novel generalized observables framework of voting behaviour, by using the statistical data collected and analysed in previous studies by Khrennikova (Khrennikova 2015Lect. Notes Comput. Sci.8951, 196–209) and Khrennikova & Haven (Khrennikova & Haven 2016Phil. Trans. R. Soc. A374, 20150106 ( doi:10.1098/rsta.2015.0106 )). This framework aims to overcome the main problems associated with the quantum probabilistic representation of psychological data, namely thenon-double stochasticityof transition probability matrices. We develop a simplified construction of generalized positive operator valued measures by formulating special non-orthonormal bases with respect to these operators. This article is part of the themed issue ‘Second quantum revolution: foundational questions’.
The statistical operator of quantum theory may be determined empirically by computations based upon the measured mean values of a set of observables we have called a quorum. The requirement that a statistical operator be positive semidefinite is then used to generate a family of inequalities connecting these quorum means. Like the simpler uncertainty relations, these inequalities are universal, valid for all quantum states. In the special case of pure states, the method yields a family of equalities.
Electronic voting is a very useful but challenging internet-based protocol that despite many theoretical approaches and various implementations with different degrees of success, remains a contentious topic due to issues in reliability and security. Here we present a quantum protocol that exploits an untrusted source of multipartite entanglement to carry out an election without relying on election authorities, simultaneous broadcasting or computational assumptions, and whose result is publicly verifiable. The level of security depends directly on the fidelity of the shared multipartite entangled quantum state, and the protocol can be readily implemented for a few voters with state-of-the-art photonic technology.
Accurate, reliable sampling from fully-connected graphs with arbitrary correlations is a difficult problem. Such sampling requires knowledge of the probabilities of observing every possible state of a graph. As graph size grows, the number of model states becomes intractably large and efficient computation requires full sampling be replaced with heuristics and algorithms that are only approximations of full sampling. This work investigates the potential impact of adiabatic quantum computation for sampling purposes, building on recent successes training Boltzmann machines using a quantum device. We investigate the use case of quantum computation to train Boltzmann machines for predicting the 2016 Presidential election.