Implementation and Analysis of Quantum Majority Rules under Noisy Conditions

TL;DR

This work explores the quantum majority rule (QMR) constitution of Bao and Yunger Halpern as a testbed for quantum social choice, focusing on stability under realistic noise on NISQ devices. It provides an analytic implementation of QMR, complemented by circuit-level sampling on simulators, IBM FakeBackends, and real hardware, with metrics that separate benchmark agreement, stability, and full distribution deformation. A second, exploratory QMR2-inspired entanglement protocol examines how multi-voter quantum correlations (GHZ-type vs separable blocks) influence winner frequencies and draws under noise. The key findings show that QMR is robust to moderate readout and device noise, preserving the classical Condorcet winner and the majority digraph structure for representative profiles, while the QMR2 entanglement variant can produce draw-free outcomes in ideal conditions but degrades rapidly with noise. These results inform practical design considerations for quantum voting protocols on near-term devices and motivate future work on error-corrected voting circuits and noisy Arrow analyses.

Abstract

Quantum voting, inspired by quantum game theory, provides a framework in which the quantum majority rule (QMR) constitution of Bao and Yunger Halpern [Phys. Rev. A 95, 062306 (2017)] violates the quantum analogue of Arrow's impossibility theorem. We evaluate this QMR constitution analytically on classical profile data and implement its final measurement stage as a quantum circuit, running on both noiseless simulators and noisy IBM quantum hardware to map how realistic noise deforms the resulting societal ranking distribution. Moderate-high single-qubit noise does not change the qualitative behavior of QMR, whereas strong noise shifts the distribution toward other dominant winners than the classical one. We quantify this behavior using winner-agreement rates, Condorcet-winner flip rates, and Jensen-Shannon divergence between societal ranking distributions. In a second, exploratory component, we demonstrate an explicitly entanglement-based variant of the QMR constitution that serves as a testbed for multi-voter quantum correlations under noise, which we refer to as the QMR2-inspired variant. There, GHZ-type and separable superpositions over opposite rankings have the same expectation values but respond very differently to noise. Taken together, these two components connect the abstract QMR constitution to concrete implementations on noisy intermediate-scale quantum (NISQ) devices and highlight design considerations for future quantum voting protocols.

Figures (8)

• Figure 1: Quantum Majority rule. Streamlined version of the protocol proposed by Bao and Yunger Halpern.
• Figure 2: Winner agreement convergence. The winner agreement metrics convergence as a function of the number of shots in experiment 1, per readout noise amplitude.
• Figure 3: Experiment 1 -- Winner–agreement rate. Plot of$\Gamma_{\mathrm{win}}$ under readout noise.
• Figure 4: Experiment 1 -- Divergence. Plot of $\widehat{JS}_{\mathrm{div}}$ and the flip-rate $\tilde{\gamma}$ under readout noise.
• Figure 5: Experiment 2 -- Winner–agreement rate. Plot of $\Gamma_{\mathrm{win}}$ under readout noise.