cs.LGSep 30, 2026

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

Authors: Yuchen Cong, Zerui Tao, Chao Li, Zhe Sun, Qibin Zhao

Organizations: Juntendo University · RIKEN-AIP

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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.

Figures & tables

Figure 1 : Graphical illustration of tensor-network notation and representations. (a) Basic tensor objects and tensor contraction. (b) A simple QTN example.
Figure 2 : Graphical representation of a local training contraction. The yellow tensors form the QTN representation of UΘU_{\Theta} and UΘ†U_{\Theta}^{\dagger} , while the blue tensor denotes a local score-based loss operator M(d)M^{(d)} . Red wires denote system wires and green wires denote ancilla wires. Open circles denote fixed boundary vectors used to close the finite QTN contraction.
DDTargetGSMBAMADVIEigenVIMoGQuanVI E0QuanVI E2
55Gaussian0.9520 ±1.32820.9520\,{\scriptstyle\pm 1.3282}≤1×10−2\bm{\leq 1{\times}10^{-2}}≤1×10−2\bm{\leq 1{\times}10^{-2}}≤1×10−2\bm{\leq 1{\times}10^{-2}}0.0152 ±0.00090.0152\,{\scriptstyle\pm 0.0009}≤1×10−2\bm{\leq 1{\times}10^{-2}}≤1×10−2\bm{\leq 1{\times}10^{-2}}
X-shape2.2494 ±0.49612.2494\,{\scriptstyle\pm 0.4961}3.0904 ±0.23733.0904\,{\scriptstyle\pm 0.2373}1.6816 ±0.07161.6816\,{\scriptstyle\pm 0.0716}0.4007 ±0.07430.4007\,{\scriptstyle\pm 0.0743}1.3039 ±0.02691.3039\,{\scriptstyle\pm 0.0269}0.2613 ±0.0035\bm{0.2613\,{\scriptstyle\pm 0.0035}}0.2661 ±0.00540.2661\,{\scriptstyle\pm 0.0054}
GMM31.6469 ±0.58951.6469\,{\scriptstyle\pm 0.5895}2.4417 ±0.22732.4417\,{\scriptstyle\pm 0.2273}1.1967 ±0.03131.1967\,{\scriptstyle\pm 0.0313}0.4454 ±0.06130.4454\,{\scriptstyle\pm 0.0613}0.4436 ±0.06170.4436\,{\scriptstyle\pm 0.0617}0.1608 ±0.01120.1608\,{\scriptstyle\pm 0.0112}0.1598 ±0.0042\bm{0.1598\,{\scriptstyle\pm 0.0042}}
Funnel3.3535 ±0.52413.3535\,{\scriptstyle\pm 0.5241}4.8814 ±0.20824.8814\,{\scriptstyle\pm 0.2082}2.8012 ±0.06532.8012\,{\scriptstyle\pm 0.0653}1.3435 ±0.10651.3435\,{\scriptstyle\pm 0.1065}0.8183 ±0.0742\bm{0.8183\,{\scriptstyle\pm 0.0742}}4.3944 ±1.82994.3944\,{\scriptstyle\pm 1.8299}2.6412 ±0.45792.6412\,{\scriptstyle\pm 0.4579}
Ring30.7895 ±8.822330.7895\,{\scriptstyle\pm 8.8223}39.8048 ±2.566439.8048\,{\scriptstyle\pm 2.5664}2.1094 ±0.02212.1094\,{\scriptstyle\pm 0.0221}1.7526 ±0.14541.7526\,{\scriptstyle\pm 0.1454}2.0528 ±0.22332.0528\,{\scriptstyle\pm 0.2233}0.8861 ±0.0048\bm{0.8861\,{\scriptstyle\pm 0.0048}}0.8913 ±0.01440.8913\,{\scriptstyle\pm 0.0144}
1010Gaussian≤1×10−2\bm{\leq 1{\times}10^{-2}}≤1×10−2\bm{\leq 1{\times}10^{-2}}≤1×10−2\bm{\leq 1{\times}10^{-2}}OOM0.0814 ±0.00240.0814\,{\scriptstyle\pm 0.0024}≤1×10−2\bm{\leq 1{\times}10^{-2}}≤1×10−2\bm{\leq 1{\times}10^{-2}}
Table 1 : Forward KL divergence over five runs, reported as mean and standard deviation. Lower values indicate better approximation quality. Best mean values are highlighted in bold font. OOM denotes out-of-memory, and “–” denotes unstable runs.
ModelGSMBAMADVIEigenVI
gpregr (dim=3)1.2107 ±0.03451.2107\,{\scriptstyle\pm 0.0345}1.1962 ±0.02931.1962\,{\scriptstyle\pm 0.0293}1.1509 ±0.01771.1509\,{\scriptstyle\pm 0.0177}0.0828 ±0.0681\mathbf{0.0828\,{\scriptstyle\pm 0.0681}}
hmm (dim=4)148.1225 ±2.5882148.1225\,{\scriptstyle\pm 2.5882}139.7349 ±3.9980139.7349\,{\scriptstyle\pm 3.9980}168.9795 ±32.6976168.9795\,{\scriptstyle\pm 32.6976}4.0295 ±1.5069\mathbf{4.0295\,{\scriptstyle\pm 1.5069}}
hmm_bball_0 (dim=6)749.1771 ±2.5298749.1771\,{\scriptstyle\pm 2.5298}19.9691 ±0.684919.9691\,{\scriptstyle\pm 0.6849}746.7151 ±2.0978746.7151\,{\scriptstyle\pm 2.0978}18.5057 ±3.032218.5057\,{\scriptstyle\pm 3.0322}
ModelMoGNFQuanVI E0QuanVI E2
gpregr (dim=3)7.6146 ±3.99047.6146\,{\scriptstyle\pm 3.9904}3.3739 ±0.49733.3739\,{\scriptstyle\pm 0.4973}0.2180 ±0.04630.2180\,{\scriptstyle\pm 0.0463}0.2330 ±0.03690.2330\,{\scriptstyle\pm 0.0369}
hmm (dim=4)––5.0783 ±0.62145.0783\,{\scriptstyle\pm 0.6214}4.8047 ±0.44914.8047\,{\scriptstyle\pm 0.4491}
Table 2 : Forward Fisher divergence over four runs, reported as mean and standard deviation. Lower values indicate better approximation quality. Best mean values are highlighted in bold font, and “–” denotes unstable runs.
Figure 3 : Ablation studies of QuanVI.
Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4 : Synthetic target distributions used in our experiments.
Figure 5 : Sample visualization on the first two dimensions of the X-shape target with D=20D=20 , comparing all baselines with our method.
DDGaussianX-shapeGMM3FunnelRing
50.0109 (0/5)0.0319 (0/5)0.0297 (0/5)0.0778 (2/5)0.2682 (0/5)
100.0247 (0/5)4.36×10174.36\times 10^{17} (5/5)0.6085 (1/5)1.05×10151.05\times 10^{15} (4/5)6.25×10286.25\times 10^{28} (3/5)
200.0224 (0/5)184.8571 (1/5)5.06×10135.06\times 10^{13} (3/5)1.7613 (0/5)3.7114 (1/5)
1000.0774 (1/5)37.5930 (1/5)404.6059 (2/5)3.38×10233.38\times 10^{23} (4/5)44.0952 (0/5)
Appendix
Table 3 : Normalizing flow results on synthetic targets over five runs. Values report the median forward KL divergence, with the number of divergent runs in parentheses.
Figure 6 : Representative posterior visualization on hmm_bball_0 .
MethodMain hyperparametersInitialization / standardizationSpecial case
GSM1000010000 iter, batch size 1616NoneNone
BaM10001000 iter, batch size 1616GSM mean/covNone
ADVI10001000 iter, batch size 1616 , lr 10−410^{-4}GSM mean/covNone
NF88 layers, hidden dim 3232 , 10001000 iter, lr 10−310^{-3}NoneNone
EigenVIHermite basis, K=5K=5 , 6000060000 samplesGSM mean/covhmm_bball_0 uses BaM mean/cov
MoG5050 mixtures, 1000010000 iter, lr 10−410^{-4}GSM mean/cov, fallback to randomNone
Appendix
Table 4 : Summary of hyperparameter settings for posterior benchmark experiments. All targets share the same hyperparameters unless otherwise specified.
DDTargetL=1L=1L=2L=2L=3L=3L=4L=4L=5L=5
5X-shape6.7725±0.03996.7725\pm 0.03990.2645±0.00650.2645\pm 0.00650.2478±0.00500.2478\pm 0.00500.2552±0.00830.2552\pm 0.00830.2460±0.01200.2460\pm 0.0120
Ring11.0495±3.081811.0495\pm 3.08180.8787±0.01760.8787\pm 0.01760.8905±0.02500.8905\pm 0.02500.9186±0.02720.9186\pm 0.02720.8932±0.02440.8932\pm 0.0244
Funnel9.2988±2.14819.2988\pm 2.14812.8384±1.26152.8384\pm 1.26151.6063±0.21201.6063\pm 0.21201.3879±0.09261.3879\pm 0.09261.4809±0.26511.4809\pm 0.2651
GMM39.2656±4.61289.2656\pm 4.61280.1602±0.00380.1602\pm 0.00380.1560±0.00520.1560\pm 0.00520.1551±0.00690.1551\pm 0.00690.1550±0.00470.1550\pm 0.0047
10X-shape16.2028±1.614616.2028\pm 1.61463.2120±0.06863.2120\pm 0.06862.9558±0.05262.9558\pm 0.05262.8594±0.05302.8594\pm 0.05302.8685±0.05972.8685\pm 0.0597
Ring20.6691±3.089420.6691\pm 3.08943.8223±0.04663.8223\pm 0.04663.5803±0.07423.5803\pm 0.07423.4501±0.06023.4501\pm 0.06023.3433±0.06423.3433\pm 0.0642
Appendix
Table 5 : Forward KL divergence for different local window sizes L∈{1,2,3,4,5}L\in\{1,2,3,4,5\} , reported as the mean and standard deviation over five runs. A dash indicates that no final KL divergence was available.
Dimension DDLLTraining time
52≈5\approx 5 min
53≈20\approx 20 min
102≈15\approx 15 min
103>12>12 hours
202≈30\approx 30 min
1002≈5\approx 5 hours
Appendix
Table 6 : Training time for different local window sizes LL on the Funnel target. Larger local windows substantially increase the computational cost.

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