QuanVI: Score-based Variational Inference via Quantum Maximally Mixed States

2026-09-30

AI Summary

This study addresses the exponential parameter explosion in eigenvalue-based methods and the non-uniqueness of eigenvectors within degenerate subspaces during high-dimensional variational inference. To overcome these challenges, we propose QuanVI, a variational framework built upon Fisher divergence that incorporates density operators and quantum tensor network representations. Specifically, QuanVI employs maximally mixed states to characterize degenerate subspaces, thereby eliminating solution non-uniqueness, and leverages matrix product operator (MPO) structures to compress density operators, circumventing traditional dimensional bottlenecks. Experimental results demonstrate that QuanVI achieves exact agreement with analytical solutions in low-dimensional settings while successfully scaling to high-dimensional synthetic data and Bayesian posterior benchmarks. By effectively approximating complex non-Gaussian target distributions, this work establishes QuanVI as an efficient and scalable approach for high-dimensional variational inference.

Abstract

Score-based variational inference (VI) provides an alternative to Kullback--Leibler (KL)-based VI by minimizing the Fisher divergence between the variational distribution and the target. A prior score-VI approach formulates this optimization as an eigenvalue problem, with the variational distribution constructed from low-energy eigenstates. However, this eigenvalue-based formulation faces two high-dimensional obstacles: an intractably large parameter count due to exponential scaling and non-uniqueness of individual eigenvectors in degenerate or nearly degenerate low-energy subspaces. We propose QuanVI, a scalable quantum-inspired algorithm that combines a mixed-state density-operator formulation with a quantum tensor network (QTN) parameterization using the matrix product operator (MPO) structure. In degenerate low-energy subspaces, the density-operator formulation represents the subspace by its maximally mixed state rather than relying on a non-unique individual eigenvector, while the QTN parameterization compresses the density operator to avoid exponential parameter growth. Experiments and ablations show that QuanVI agrees with exact solutions in low dimensions and scales to high-dimensional synthetic and Bayesian posterior-approximation benchmarks, including challenging non-Gaussian targets.

Problem

• score-based variational inference
• Fisher divergence
• eigenvalue problem
• high-dimensional scalability
• degenerate subspace

Innovation

• Score-based Variational Inference
• Quantum Tensor Network
• Matrix Product Operator
• Density Operator
• Maximally Mixed States