Quantum Majority Rules Analysis Demonstrates Noise Resilience and Societal Ranking Shifts with Jensen-Shannon Divergence

Quantum voting, a field inspired by game theory, presents a fascinating challenge to traditional decision-making principles, and a team led by Gal Amit, Yuval Idan, and Michael Suleymanov from Bar-Ilan University, along with Luis Razo from the European Institute of Science in Management and Eliahu Cohen from Bar-Ilan University, now investigates how this concept performs in realistic conditions. The researchers evaluate a quantum majority rule constitution, revealing that it can circumvent limitations found in classical voting systems, and importantly, they demonstrate how noise inherent in current quantum computers affects the resulting outcomes. Their analysis, combining theoretical modelling with implementation on both simulated and real quantum hardware, shows that while moderate noise does not fundamentally alter the behaviour of the system, stronger noise can significantly shift societal rankings. This work connects abstract quantum principles to practical considerations for building future voting protocols and provides valuable insights into the resilience of quantum systems against the imperfections of noisy intermediate-scale quantum devices.

Quantum Voting and Arrow’s Impossibility Theorem

This document explores quantum voting, a field investigating whether quantum mechanics can offer advantages over classical voting systems. Arrow’s Impossibility Theorem states that no voting system can perfectly satisfy all desirable criteria, and quantum voting explores whether quantum principles, like entanglement and superposition, can potentially circumvent this theorem or offer benefits like enhanced security and expressiveness. Entanglement, a unique quantum phenomenon, links the fates of particles even when separated, and is utilized in some schemes to create correlations between voter choices. The research examines quantum voting schemes, including Quantum Pseudo-Telepathy (QPT) and Quantum Majority Rule (QMR), with variations like QMR2 emphasizing the most frequently occurring rankings and Schrödinger’s Ballot leveraging quantum superposition for nuanced preferences.

Key findings suggest some quantum schemes can potentially violate the conditions of Arrow’s theorem, offering enhanced expressiveness in capturing voter preferences, while acknowledging security concerns and practical challenges. The study utilizes Qiskit, an open-source quantum computing framework, to model and test these schemes, investigating how entanglement enables correlations between voters’ choices. The document details the workings of QMR, including Tarjan’s Algorithm for identifying preference patterns, and steps designed to ensure fair representation, ultimately exploring the theoretical possibility of overcoming the limitations of classical voting systems.

Quantum Voting Protocol Tested on Real Hardware

Researchers investigated a quantum-inspired voting system, meticulously evaluating its performance using both simulations and actual quantum hardware. They began by analytically examining the system’s behaviour with classical voting data, then implemented the final measurement stage as a quantum circuit, assessing the impact of realistic noise on the societal ranking produced by the voting protocol. The team quantified the effects of noise by tracking winner-agreement rates, measuring Condorcet-winner flips, and calculating the divergence between ranking distributions. To further explore the system’s capabilities, scientists developed an explicitly entanglement-based variant, termed QMR2, designed as a testbed for investigating multi-voter correlations under noisy conditions.

This variant utilizes GHZ-type and separable superpositions, allowing researchers to compare how these different quantum states respond to noise while maintaining identical expectation values. This precise control enabled the team to discern the distinct effects of noise on entangled versus separable states, revealing how correlations are affected in a multi-voter scenario. The experimental setup involved careful calibration of the quantum hardware and implementation of error mitigation techniques, demonstrating that moderate-to-high noise levels do not fundamentally alter the system’s qualitative behaviour, though strong noise can shift the distribution.

Quantum Voting Circumvents Classical Impossibility Theorem

Scientists investigated a quantum voting protocol, inspired by game theory, and demonstrated its behaviour on both simulated and real quantum hardware. The research centres on a quantum majority rule (QMR) constitution, which encodes voter preferences as quantum states and aggregates them using quantum operations. Experiments revealed that this QMR constitution violates a quantum analogue of Arrow’s impossibility theorem, suggesting a potential to circumvent limitations inherent in classical voting systems. The team quantified the impact of noise on the QMR constitution by measuring winner-agreement rates, Condorcet-winner flip rates, and divergence between societal ranking distributions.

Results demonstrate that moderate-to-high single-qubit noise does not qualitatively alter the behaviour of the QMR protocol, though strong noise shifts the resulting societal ranking distribution. Specifically, the team observed that as noise increases, the probability of selecting a different winner compared to the classical outcome also increases. Further exploration involved an explicitly entanglement-based variant of the QMR constitution, termed QMR2, proving that GHZ-type and separable superpositions yield identical expectation values but respond very differently to noise, highlighting the sensitivity of the protocol to quantum correlations.

Robust Voting Preserves Majority Preference Under Noise

This research demonstrates a quantum-inspired voting constitution, termed QMR, exhibits robust behaviour under realistic noise conditions, while also exploring the impact of entanglement on voting outcomes. By implementing the QMR constitution with five voters and three candidates, scientists observed that moderate to high levels of noise do not fundamentally alter the system’s ability to preserve the classical Condorcet winner and the underlying structure of majority preferences. Detailed analysis using metrics such as winner-agreement rates and divergence of ranking distributions revealed a smooth, predictable response to increasing noise, with the system converging towards a stable, reproducible outcome even under stress-test conditions. Further investigation involved an explicitly entanglement-based variant of the QMR constitution, designed to test the influence of multi-voter correlations. In ideal conditions, entangled voter blocks effectively eliminated draw outcomes, but this advantage proved fragile under even moderate levels of noise, highlighting the challenges of maintaining quantum advantages in noisy environments. The study acknowledges limitations stemming from the relatively small system size, and future work will likely focus on extending these findings to larger electorates.