Score-based Variational Inference via Quantum Maximally Mixed States

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. Existing eigenvalue-based score-VI methods face two high-dimensional obstacles: exponential parameter growth and instability under degenerate or nearly degenerate low-energy eigenspaces. We propose QuanVI, a scalable quantum-inspired algorithm that combines a mixed-state density-operator formulation with an MPO-motivated quantum tensor network (QTN) parameterization. The former represents degenerate low-energy eigenspaces by their maximally mixed state, avoiding arbitrary eigenvector selection; the latter realizes a compact density operator without explicitly constructing exponentially large matrices. Experiments and ablations show that QuanVI preserves low-dimensional accuracy while scaling to high-dimensional synthetic and Bayesian posterior-approximation benchmarks, including challenging non-Gaussian targets.