Insider Brief
- Researchers tested a quantum majority-rule voting system on IBM quantum hardware and found that election outcomes can remain stable under moderate noise but become fragile near mathematical tipping points.
- Tests with five voters and three candidates showed that noise could distort underlying preference distributions without immediately changing the winning candidate.
- Entangled voter groups eliminated draws under ideal conditions, but the effect weakened under noise, highlighting quantum voting as a potential test bed for studying error mitigation and error correction.
- Photo by Element5 Digital on Pexels
When voters head to the polls, a ballot is designed to be as simple as possible. A voter makes a choice, the choice goes into the count and, whatever disputes follow, the mark on the ballot stays put.
In a quantum election, however, the quantum ballot is a much stranger, more complex proposition.
Instead of beginning with a single fixed preference, a quantum voting system can represent several possible rankings at once and, in some versions, link the preferences of different voters through quantum correlations. The final result emerges from a process governed partly by the rules of quantum mechanics.
Over the years, researchers haven’t been studying quantum voting systems as a futuristic way to tally election results, rather they are investigating how these systems respond to noise to create a useful test bed for quantum error correction. And that could help show how physical errors translate into changes in complex collective outcomes.
Now, researchers writing in Scientific Reports have put one version of quantum majority rule, or QMR, through that test, using simulations and IBM quantum hardware. They found that the system could preserve the same winning candidate under moderate levels of hardware noise, even as the underlying distribution of preferences began to change. But when voter preferences were already balanced near a mathematical tipping point, relatively small errors could have a much larger effect.
Researchers from Bar-Ilan University, the European Institute of Science in Management and Chapman University analyzed the QMR system mathematically and tested parts of its implementation using quantum simulators and IBM quantum processors.
The results suggest that moderate levels of hardware and measurement error often leave the winning candidate unchanged, even as the underlying distribution of voter preferences becomes distorted. At sufficiently high noise levels, however, outcomes can change sharply. The system also proved more fragile for some randomly generated electorates and for preference patterns close to voting cycles.
The researchers said the work should be viewed as a study of the stability of quantum voting rules rather than evidence that quantum computers offer a practical advantage for elections. Much of the QMR process can be calculated efficiently using conventional computers, while quantum hardware was used primarily to determine how physical noise affects the prescribed voting distributions.
Testing a Quantum Answer to a Classical Voting Problem
According to the study, a theorem developed by economist Kenneth Arrow — named, conveniently enough, Arrow’s impossibility theorem — describes an unavoidable problem in collective decision-making. The theorem essentially says that no voting system can turn everyone’s individual preferences into a group decision while meeting every reasonable standard of fairness.
Using a real-world example, in an election with three candidates, voters can collectively prefer A over B, B over C and C over A, creating a loop with no clear overall winner.
Quantum social-choice researchers have asked whether changing the mathematical framework can alter those constraints. A previous proposal by researchers Zhengfeng Bao and Nicole Yunger Halpern introduced QMR, which represents voter preferences using quantum states and constructs a collective preference distribution through a combination of quantum and classical operations.
Under its specifically defined quantum counterparts to the classical conditions, QMR can be non-dictatorial while satisfying requirements analogous to transitivity, unanimity and independence of irrelevant alternatives, thereby violating a quantum analogue of Arrow’s theorem.
The new study did not attempt to prove that result again. Instead, the researchers asked what happens when the abstract voting system encounters the imperfect hardware available today.
They modeled elections with five voters and three candidates, giving six possible strict rankings of the candidates. The QMR algorithm examines pairwise preferences, builds a directed graph showing the relationships among candidates and identifies groups involved in cycles using a standard graph-processing method known as Tarjan’s algorithm.
A Condorcet winner — a candidate that defeats every other candidate in head-to-head comparisons — provided one benchmark for judging whether noise changed an election result.
The researchers calculated the ideal QMR distributions analytically and then reproduced the state-preparation and measurement portions with quantum circuits. Tests included noiseless simulations, simulations containing controlled errors, IBM device models designed to reproduce realistic hardware behavior and runs on actual IBM superconducting quantum computers.
They focused particularly on readout errors, in which a quantum computer incorrectly records the state of a qubit when it is measured.
Winners Proved Stable — Until They Weren’t
The scientists report that two hand-crafted five-voter test cases showed substantial resilience.
In the first experiment, candidate C was the classical Condorcet winner, although only by relatively narrow margins. As simulated readout error increased, the overall distribution of rankings steadily moved away from its ideal form.
The winner, however, remained unchanged through a readout-error probability of 0.4. Only at the extreme level of 0.5 did the result abruptly deteriorate, with agreement with the classical winner falling to about 2%.
Tests using an IBM device model and real IBM hardware remained within the more stable region. In those cases, the QMR system continued to identify the same Condorcet winner despite modest changes in the broader ranking distribution.
The researchers write that this is important because a voting system can produce the same winner while becoming less stable beneath the surface. To capture that effect, the researchers tracked three measures, including how often the QMR winner agreed with the classical benchmark, how frequently the winner changed between repeated runs and how far the entire distribution of rankings moved from the ideal result.
The second main experiment used a stronger majority structure, with candidate A ranked first by four of five voters and defeating the other candidates in pairwise comparisons. It showed a similarly broad region in which noise changed the probability distribution without overturning the winner.
Additional tests, however, showed that those two examples understated the system’s sensitivity.
The researchers generated randomized electorates using probability distributions over the six possible candidate rankings. Some showed reduced agreement with the classical winner even at a readout-error probability of just 0.01. In one randomized case, winner agreement fell to 43%, while another group of randomized profiles averaged about 72%.
The difference suggests that noise tolerance depends not simply on the quantum hardware but also on the mathematical structure of the electorate. A voting profile already close to a majority-cycle boundary can be destabilized by comparatively small errors.
Tests comparing a cyclic preference profile with a nearly identical but slightly perturbed profile reinforced that result. The exact cycle remained structurally fragile, while a small change away from the cycle produced a much more ordered and stable result.
Limited tests on IBM‘s 156-qubit ibm_marrakesh processor produced results broadly consistent with that interpretation. A representative randomized profile recorded 80% winner agreement, the cyclic case recorded 70%, and the almost-cyclic case maintained 100% agreement.
Entanglement Changes Outcomes, but Noise Erases the Effect
The researchers separately explored whether quantum entanglement could change voting behavior.
This portion of the study used a simplified system inspired by another quantum voting proposal, rather than the full QMR protocol. Groups of voters were placed either in separable quantum superpositions or in GHZ-type entangled states.
GHZ states create strong correlations among multiple quantum systems. In the voting experiment, entangled voters were correlated so that measurement placed the entire group on one preference or its opposite, rather than allowing each voter to resolve independently.
The researchers ran 10,000 iterations of small voting rounds and compared entangled groups with otherwise equivalent separable groups.
Under ideal conditions, the entangled groups eliminated draws in the small test rounds because the voters in each block resolved collectively onto the same outcome. The individual voters could have the same local probability distributions in both cases, but the correlations among them changed the collective statistics.
That effect proved fragile, as seen when the researchers introduced local bit-flip errors, the distinctive behavior of the GHZ groups weakened. At moderate noise levels, winner frequencies and draw rates increasingly resembled those produced by separable or random voters. At a bit-flip probability of 0.5, the entangled system approached effectively random behavior.
The study also found that the entanglement effect largely disappeared when the researchers considered large populations in which only limited groups of voters could be entangled. That finding may present another obstacle to translating the small experimental effect into a practical large-scale voting system.
Not a Quantum Election System
The researchers identified several limitations, which would likely serve as areas for future research.
The main experiments involved only five voters and three candidates. Adding candidates rapidly increases the number of possible rankings, while adding voters raises both classical processing demands and the number of qubits required for a direct quantum implementation.
A general implementation with potentially entangled ballots would require a number of logical qubits that grows with both the electorate and the number of possible rankings. The researchers concluded that such an approach is unrealistic for real-world elections using today’s noisy intermediate-scale quantum computers.
The experiments also concentrated primarily on readout errors in relatively simple circuits. They did not comprehensively model problems such as crosstalk between qubits, leakage from computational states, long circuit depths or more complex time-dependent noise.
The entanglement experiments were also explicitly a simplified, QMR2-inspired test rather than an implementation of the full QMR2 constitution. In this case, the researchers did not claim that those experiments inherit the Arrow-related properties of the original QMR system.
It’s also important to acknowledge that the study demonstrate quantum computational advantage. The core QMR societal distribution was calculated analytically on classical computers. Quantum processors acted largely as physical sampling devices that allowed the team to measure how real hardware errors altered the results.
The system is also not a complete secure voting protocol. It does not by itself provide ballot secrecy, voter authentication, coercion resistance or protection against a malicious election authority. Those capabilities would require separate cryptographic or authentication systems.
There are, however, practical implications for the study that could lead to real-world innovations. The experiments, for example, provide a bridge between a largely theoretical branch of quantum social-choice research and physical quantum devices. They suggest that the stability of a quantum voting rule depends on both hardware quality and the underlying structure of voter preferences.
The work could go beyond social-choice research and impact quantum computing, itself. Future work, for example, could focus on applying measurement-error mitigation to reduce readout distortions and eventually test quantum error correction, according to the study. Researchers could encode voting registers in error-correcting codes and compare whether protected logical qubits produce more stable collective outcomes than raw physical qubits.
The researchers also identified a broader theoretical question on how Arrow-style impossibility results themselves should be formulated when voter preferences are subject to noise, missing information or physical errors.
Such work could help determine whether noise merely corrupts quantum voting systems or changes the mathematical boundaries governing collective choice.
The research team included: Gal Amit, Yuval Idan and Michael Suleymanov of the Faculty of Engineering and Institute of Nanotechnology and Advanced Materials at Bar-Ilan University in Israel; Luis Razo of the European Institute of Science in Management in Barcelona; and Eliahu Cohen of Bar-Ilan University and the Institute for Quantum Studies at Chapman University in California.
