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Score-based Variational Inference via Quantum Maximally Mixed States

NeurIPS 2026 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Variational inference theory and gradients

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Proposes QuanVI, a score-based variational inference method that replaces EigenVI's single-eigenvector rank-one solution with a mixed-state density operator parameterized by a complex-valued MPO-style quantum tensor network, evaluated on synthetic and PosteriorDB targets.

Problem

Score-based VI via eigenvalue problems (EigenVI) scales exponentially in dimension (K^D x K^D operator) and is unstable when low-energy eigenspaces are degenerate or nearly degenerate.

Approach

Generalize the rank-one (pure-state) eigenvector formulation to a mixed-state density operator rho = UU†/r (maximally mixed on degenerate subspaces), and parameterize U with a circuit-style MPO quantum tensor network exploiting Markov/path-graph locality, so the objective is a sum of local terms.

Key results

• On D=5 X-shape, QuanVI E2 reaches forward KL 0.2661 vs EigenVI 0.4007 and best among methods (E0 0.2613).
• At D=100, QuanVI E0/E2 achieve KL about 42.3/40.5 on X-shape vs ADVI 47.26 and MoG 72.76; EigenVI is OOM.
• On PosteriorDB hmm_bball_0 (dim 6), QuanVI E2 gets Fisher divergence 13.45 vs EigenVI 18.51, but on gpregr EigenVI (0.083) beats QuanVI (0.218).

Surprising findings

• On the posterior gpregr benchmark, EigenVI outperforms QuanVI, and on Funnel QuanVI is less competitive than MoG and EigenVI at low dimension.
• Normalizing flows were highly unstable, with KL values exploding to 1e28 or more in high dimensions, so they were excluded from the main table.

Limitations

• Relies on local (Markov/path-graph) structure; larger local windows L increase cost as K^L.
• Funnel target less competitive at default window size; EigenVI better on gpregr posterior.
• Results on posterior benchmarks are reported on only three models and the complexity analysis is approximate.

Experimental setup

Datasets: PosteriorDB (gpregr, hmm, hmm_bball_0) synthetic X-shape, Ring, Funnel, GMM3, Gaussian targets

Benchmarks: synthetic X-shape, Ring, Funnel, GMM3, Gaussian (D=5,10,20,100) PosteriorDB gpregr, hmm, hmm_bball_0

Metrics: forward KL divergence forward Fisher divergence

Baselines: GSM BaM ADVI EigenVI MoG normalizing flow

Builds on: EigenVI Batch-and-Match VI Gaussian score matching matrix product operator

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