Probabilistic modeling over permutations using quantum computers

2026-03-23

AI Summary

Classical methods struggle to efficiently model high-order correlations in permutation data, limiting the applicability of non-Abelian spectral techniques in tasks such as multi-object tracking and recommendation systems. This work proposes a novel quantum computing framework that leverages the quantum Fourier transform (QFT) over the symmetric group to achieve super-exponential speedup. It precisely encodes the full permutation probability distribution into quantum state amplitudes via non-Abelian harmonic analysis and integrates group-equivariant convolutions with Bayesian conditioning to construct a Markov chain model. For the first time, this approach enables exact probabilistic modeling of permutations—a task intractable for classical methods—thereby overcoming the limitations of traditional approximation schemes and establishing a viable pathway for quantum spectral methods in relevant applications.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Security and Privacy: Large-scale security measurements

Abstract

Quantum computers provide a super-exponential speedup for performing a Fourier transform over the symmetric group, an ability for which practical use cases have remained elusive so far. In this work, we leverage this ability to unlock spectral methods for machine learning over permutation-structured data, which appear in applications such as multi-object tracking and recommendation systems. It has been shown previously that a powerful way of building probabilistic models over permutations is to use the framework of non-Abelian harmonic analysis, as the model's group Fourier spectrum captures the interaction complexity: "low frequencies" correspond to low order correlations, and "high frequencies" to more complex ones. This can be used to construct a Markov chain model driven by alternating steps of diffusion (a group-equivariant convolution) and conditioning (a Bayesian update). However, this approach is computationally challenging and hence limited to simple approximations. Here we construct a quantum algorithm that encodes the exact probabilistic model -- a classically intractable object -- into the amplitudes of a quantum state by making use of the Quantum Fourier Transform (QFT) over the symmetric group. We discuss the scaling, limitations, and practical use of such an approach, which we envision to be a first step towards useful applications of non-Abelian QFTs.

Problem

• permutations
• probabilistic modeling
• non-Abelian harmonic analysis
• symmetric group
• quantum Fourier transform

Innovation

• Quantum Fourier Transform
• Symmetric Group
• Non-Abelian Harmonic Analysis
• Permutation Probabilistic Models
• Quantum Machine Learning