The State Preparation of Multivariate Normal Distributions using Tree Tensor Network

Hidetaka Manabe; Yuichi Sano

doi:10.22331/q-2025-05-28-1755

The State Preparation of Multivariate Normal Distributions using Tree Tensor Network

¹Graduate School of Engineering Science, Osaka University
²Department of Nuclear Engineering, Kyoto University

Published:	2025-05-28, volume 9, page 1755
Editor:	Yu Tong
Eprint:	arXiv:2412.12067v2
Doi:	https://doi.org/10.22331/q-2025-05-28-1755
Citation:	Quantum 9, 1755 (2025).

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Abstract

The quantum state preparation of probability distributions is an important subroutine for many quantum algorithms. When embedding $D$-dimensional multivariate probability distributions by discretizing each dimension into $2^n$ points, we need a state preparation circuit comprising a total of $nD$ qubits, which is often difficult to compile. In this study, we propose a scalable method to generate state preparation circuits for $D$-dimensional multivariate normal distributions, utilizing tree tensor networks (TTN). We establish theoretical guarantees that multivariate normal distributions with 1D correlation structures can be efficiently represented using TTN. Based on these analyses, we propose a compilation method that uses automatic structural optimization to find the most efficient network structure and compact circuit. We apply our method to state preparation circuits for various high-dimensional random multivariate normal distributions. The numerical results suggest that our method can dramatically reduce the circuit depth and CNOT count while maintaining fidelity compared to existing approaches.

Featured image: Initially, the multivariate normal distribution is represented as an MPS. By applying automatic structural optimization, the optimal tree-like structure is recovered.

Popular summary

Amplitude encoding of classical data is a key subroutine in quantum algorithms. In particular, preparing multivariate normal distributions is an important step in various practical quantum algorithms, such as quantum field theory and quantum finance. In quantum state preparation, we encode a $D$-dimensional function $f(x)$, discretizing each dimension into $2^n$ points, as an $n$-qubit quantum state. However, this generally requires a circuit with $nD$ qubits and a depth of $O(2^{nD})$, which can become a critical bottleneck in the execution of algorithms.

To address this, we propose a scalable method for generating state preparation circuits for $D$-dimensional multivariate normal distributions, utilizing tree tensor networks (TTNs). First, we prove that when variables exhibit a 1D correlation structure, such as exponentially decaying correlations, a multivariate normal distribution can be efficiently represented using a TTN. Our numerical simulations confirm this result and demonstrate that low-cost state preparation circuits for such distributions can be generated.

Second, we propose a compilation method based on automatic structural optimization of TTNs, a concept recently developed in quantum many-body physics. This technique optimizes the network structure by reconnecting the legs of local tensors based on entanglement entropy. This allows us to appropriately detect an efficient structure for the given distributions, as shown in the image. We numerically show that our method can greatly reduce the circuit cost by factors of 10 to 1000 compared to existing methods.

In summary, our work presents not only a practical compilation method for quantum state preparation but also highlights the power of tensor networks in representing high-dimensional functions and the utility of automatic structural optimization in computer science.

► BibTeX data

@article{Manabe2025statepreparationof,
  doi = {10.22331/q-2025-05-28-1755},
  url = {https://doi.org/10.22331/q-2025-05-28-1755},
  title = {The {S}tate {P}reparation of {M}ultivariate {N}ormal {D}istributions using {T}ree {T}ensor {N}etwork},
  author = {Manabe, Hidetaka and Sano, Yuichi},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1755},
  month = may,
  year = {2025}
}

► References

[1] Christian W. Bauer, Plato Deliyannis, Marat Freytsis, and Benjamin Nachman. Practical considerations for the preparation of multivariate Gaussian states on quantum computers. arXiv preprint arXiv:2109.10918, 2021. 10.48550/arXiv.2109.10918.
https://doi.org/10.48550/arXiv.2109.10918
arXiv:2109.10918

[2] Matan Ben-Dov, David Shnaiderov, Adi Makmal, and Emanuele G. Dalla Torre. Approximate encoding of quantum states using shallow circuits. npj Quantum Information, 10 (1): 1–8, 2024. ISSN 2056-6387. 10.1038/s41534-024-00858-1.
https://doi.org/10.1038/s41534-024-00858-1

[3] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. Quantum machine learning. Nature, 549 (7671): 195–202, 2017. ISSN 1476-4687. 10.1038/nature23474.
https://doi.org/10.1038/nature23474

[4] Daniele Bigoni, Allan P. Engsig-Karup, and Youssef M. Marzouk. Spectral tensor-train decomposition. SIAM Journal on Scientific Computing, 38 (4): A2405–A2439, 2016. ISSN 1064-8275, 1095-7197. 10.1137/15M1036919.
https://doi.org/10.1137/15M1036919

[5] A. Yu. Bogdanov, Yu. I. Bogdanov, and K. A. Valiev. Schmidt information and entanglement of quantum systems. Moscow University Computational Mathematics and Cybernetics, 31 (1): 33–42, 2007. ISSN 1934-8428. 10.3103/S0278641907010074.
https://doi.org/10.3103/S0278641907010074

[6] Yu I. Bogdanov, N. A. Bogdanova, D. V. Fastovets, and V. F. Luckichev. Schmidt decomposition and multivariate statistical analysis. In International Conference on Micro- and Nano-Electronics 2016, volume 10224, pages 706–715. SPIE, 2016. 10.1117/12.2266891.
https://doi.org/10.1117/12.2266891

[7] Jacob C. Bridgeman and Christopher T. Chubb. Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks. Journal of Physics A: Mathematical and Theoretical, 50 (22): 223001, 2017. ISSN 1751-8113, 1751-8121. 10.1088/1751-8121/aa6dc3.
https://doi.org/10.1088/1751-8121/aa6dc3

[8] Almudena Carrera Vazquez and Stefan Woerner. Efficient State Preparation for Quantum Amplitude Estimation. Physical Review Applied, 15 (3): 034027, 2021. ISSN 2331-7019. 10.1103/PhysRevApplied.15.034027.
https://doi.org/10.1103/PhysRevApplied.15.034027

[9] Anirban Narayan Chowdhury and Rolando D. Somma. Quantum algorithms for Gibbs sampling and hitting-time estimation. Quantum Info. Comput., 17 (1-2): 41–64, 2017. ISSN 1533-7146. 10.26421/QIC17.1-2-3.
https://doi.org/10.26421/QIC17.1-2-3

[10] Daniel J. Egger, Ricardo García Gutiérrez, Jordi Cahué Mestre, and Stefan Woerner. Credit Risk Analysis Using Quantum Computers. IEEE Transactions on Computers, 70 (12): 2136–2145, 2021. ISSN 1557-9956. 10.1109/TC.2020.3038063.
https://doi.org/10.1109/TC.2020.3038063

[11] Paula García-Molina, Javier Rodríguez-Mediavilla, and Juan José García-Ripoll. Quantum Fourier analysis for multivariate functions and applications to a class of Schrödinger-type partial differential equations. Physical Review A, 105 (1): 012433, 2022. 10.1103/PhysRevA.105.012433.
https://doi.org/10.1103/PhysRevA.105.012433

[12] Javier Gonzalez-Conde, Thomas W. Watts, Pablo Rodriguez-Grasa, and Mikel Sanz. Efficient quantum amplitude encoding of polynomial functions. Quantum, 8: 1297, 2024. ISSN 2521-327X. 10.22331/q-2024-03-21-1297.
https://doi.org/10.22331/q-2024-03-21-1297

[13] Alex Gorodetsky, Sertac Karaman, and Youssef Marzouk. A continuous analogue of the tensor-train decomposition. Computer Methods in Applied Mechanics and Engineering, 347: 59–84, 2019. ISSN 0045-7825. 10.1016/j.cma.2018.12.015.
https://doi.org/10.1016/j.cma.2018.12.015

[14] Johnnie Gray. Quimb: A python package for quantum information and many-body calculations. Journal of Open Source Software, 3 (29): 819, 2018. ISSN 2475-9066. 10.21105/joss.00819.
https://doi.org/10.21105/joss.00819

[15] Prithvi Gundlapalli and Junyi Lee. Deterministic and Entanglement-Efficient Preparation of Amplitude-Encoded Quantum Registers. Physical Review Applied, 18 (2): 024013, 2022. ISSN 2331-7019. 10.1103/PhysRevApplied.18.024013.
https://doi.org/10.1103/PhysRevApplied.18.024013

[16] Kenji Harada, Tsuyoshi Okubo, and Naoki Kawashima. Tensor tree learns hidden relational structures in data to construct generative models. Machine Learning: Science and Technology, 6 (2): 025002, 2025. ISSN 2632-2153. 10.1088/2632-2153/adc2c7.
https://doi.org/10.1088/2632-2153/adc2c7

[17] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum Algorithm for Linear Systems of Equations. Physical Review Letters, 103 (15): 150502, 2009. ISSN 0031-9007, 1079-7114. 10.1103/PhysRevLett.103.150502.
https://doi.org/10.1103/PhysRevLett.103.150502

[18] Toshiya Hikihara, Hiroshi Ueda, Kouichi Okunishi, Kenji Harada, and Tomotoshi Nishino. Automatic structural optimization of tree tensor networks. Physical Review Research, 5 (1): 013031, 2023. ISSN 2643-1564. 10.1103/PhysRevResearch.5.013031.
https://doi.org/10.1103/PhysRevResearch.5.013031

[19] Adam Holmes and A. Y. Matsuura. Efficient Quantum Circuits for Accurate State Preparation of Smooth, Differentiable Functions. In 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 169–179. IEEE Computer Society, 2020. ISBN 978-1-72818-969-7. 10.1109/QCE49297.2020.00030.
https://doi.org/10.1109/QCE49297.2020.00030

[20] Jason Iaconis and Sonika Johri. Tensor Network Based Efficient Quantum Data Loading of Images. arXiv preprint arXiv:2310.05897, 2023. 10.48550/arXiv.2310.05897.
https://doi.org/10.48550/arXiv.2310.05897
arXiv:2310.05897

[21] Jason Iaconis, Sonika Johri, and Elton Yechao Zhu. Quantum state preparation of normal distributions using matrix product states. npj Quantum Information, 10 (1): 1–11, 2024. ISSN 2056-6387. 10.1038/s41534-024-00805-0.
https://doi.org/10.1038/s41534-024-00805-0

[22] Raban Iten, Roger Colbeck, Ivan Kukuljan, Jonathan Home, and Matthias Christandl. Quantum Circuits for Isometries. Physical Review A, 93 (3): 032318, 2016. ISSN 2469-9926, 2469-9934. 10.1103/PhysRevA.93.032318.
https://doi.org/10.1103/PhysRevA.93.032318

[23] Bernhard Jobst, Kevin Shen, Carlos A. Riofrío, Elvira Shishenina, and Frank Pollmann. Efficient MPS representations and quantum circuits from the Fourier modes of classical image data. Quantum, 8: 1544, 2024. 10.22331/q-2024-12-03-1544.
https://doi.org/10.22331/q-2024-12-03-1544

[24] Stephen P. Jordan, Keith S. M. Lee, and John Preskill. Quantum Algorithms for Quantum Field Theories. Science, 336 (6085): 1130–1133, 2012. ISSN 0036-8075, 1095-9203. 10.1126/science.1217069.
https://doi.org/10.1126/science.1217069

[25] Stephen P. Jordan, Keith S. M. Lee, and John Preskill. Quantum computation of scattering in scalar quantum field theories. Quantum Info. Comput., 14 (11-12): 1014–1080, 2014. ISSN 1533-7146. 10.48550/arXiv.1112.4833.
https://doi.org/10.48550/arXiv.1112.4833

[26] Raghav Jumade and Nicolas PD Sawaya. Data is often loadable in short depth: Quantum circuits from tensor networks for finance, images, fluids, and proteins. arXiv preprint arXiv:2309.13108, 2023. 10.48550/arXiv.2309.13108.
https://doi.org/10.48550/arXiv.2309.13108
arXiv:2309.13108

[27] Alexei Kitaev and William A. Webb. Wavefunction preparation and resampling using a quantum computer. arXiv preprint arXiv:0801.0342, 2009. 10.48550/arXiv.0801.0342.
https://doi.org/10.48550/arXiv.0801.0342
arXiv:0801.0342

[28] Xiangyu Li, Xiaolong Yin, Nathan Wiebe, Jaehun Chun, Gregory K. Schenter, Margaret S. Cheung, and Johannes Mülmenstädt. Potential quantum advantage for simulation of fluid dynamics. Physical Review Research, 7 (1): 013036, 2025. 10.1103/PhysRevResearch.7.013036.
https://doi.org/10.1103/PhysRevResearch.7.013036

[29] Gabriel Marin-Sanchez, Javier Gonzalez-Conde, and Mikel Sanz. Quantum algorithms for approximate function loading. Physical Review Research, 5 (3): 033114, 2023. ISSN 2643-1564. 10.1103/PhysRevResearch.5.033114.
https://doi.org/10.1103/PhysRevResearch.5.033114

[30] Sam McArdle, András Gilyén, and Mario Berta. Quantum state preparation without coherent arithmetic. arXiv preprint arXiv:2210.14892, 2022. 10.48550/arXiv.2210.14892.
https://doi.org/10.48550/arXiv.2210.14892
arXiv:2210.14892

[31] Ar A. Melnikov, A. A. Termanova, S. V. Dolgov, F. Neukart, and M. R. Perelshtein. Quantum state preparation using tensor networks. Quantum Science and Technology, 8 (3): 035027, 2023. ISSN 2058-9565. 10.1088/2058-9565/acd9e7.
https://doi.org/10.1088/2058-9565/acd9e7

[32] Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, and Keisuke Fujii. Quantum Circuit Learning. Physical Review A, 98 (3): 032309, 2018. ISSN 2469-9926, 2469-9934. 10.1103/PhysRevA.98.032309.
https://doi.org/10.1103/PhysRevA.98.032309

[33] Koichi Miyamoto. Quantum algorithm for calculating risk contributions in a credit portfolio. EPJ Quantum Technology, 9 (1): 1–16, 2022. ISSN 2196-0763. 10.1140/epjqt/s40507-022-00153-y.
https://doi.org/10.1140/epjqt/s40507-022-00153-y

[34] Ashley Montanaro. Quantum speedup of Monte Carlo methods. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471 (2181): 20150301, 2015. ISSN 1364-5021, 1471-2946. 10.1098/rspa.2015.0301.
https://doi.org/10.1098/rspa.2015.0301

[35] Mudassir Moosa, Thomas W. Watts, Yiyou Chen, Abhijat Sarma, and Peter L. McMahon. Linear-depth quantum circuits for loading Fourier approximations of arbitrary functions. Quantum Science and Technology, 9 (1): 015002, 2023. ISSN 2058-9565. 10.1088/2058-9565/acfc62.
https://doi.org/10.1088/2058-9565/acfc62

[36] Kohei Morimoto, Yusuke Takase, Kosuke Mitarai, and Keisuke Fujii. Continuous optimization by quantum adaptive distribution search. Physical Review Research, 6 (2): 023191, 2024. 10.1103/PhysRevResearch.6.023191.
https://doi.org/10.1103/PhysRevResearch.6.023191

[37] Mikko Möttönen, Juha J. Vartiainen, Ville Bergholm, and Martti M. Salomaa. Transformation of quantum states using uniformly controlled rotations. Quantum Info. Comput., 5 (6): 467–473, 2005. ISSN 1533-7146. 10.26421/QIC5.6-5.
https://doi.org/10.26421/QIC5.6-5

[38] Kouhei Nakaji, Shumpei Uno, Yohichi Suzuki, Rudy Raymond, Tamiya Onodera, Tomoki Tanaka, Hiroyuki Tezuka, Naoki Mitsuda, and Naoki Yamamoto. Approximate amplitude encoding in shallow parameterized quantum circuits and its application to financial market indicators. Physical Review Research, 4 (2): 023136, 2022. 10.1103/PhysRevResearch.4.023136.
https://doi.org/10.1103/PhysRevResearch.4.023136

[39] I. V. Oseledets. Tensor-Train Decomposition. SIAM Journal on Scientific Computing, 33 (5): 2295–2317, 2011. ISSN 1064-8275, 1095-7197. 10.1137/090752286.
https://doi.org/10.1137/090752286

[40] I. V. Oseledets. Constructive Representation of Functions in Low-Rank Tensor Formats. Constructive Approximation, 37 (1): 1–18, 2013. ISSN 1432-0940. 10.1007/s00365-012-9175-x.
https://doi.org/10.1007/s00365-012-9175-x

[41] Ivan Oseledets and Eugene Tyrtyshnikov. TT-cross approximation for multidimensional arrays. Linear Algebra and its Applications, 432 (1): 70–88, 2010. ISSN 0024-3795. 10.1016/j.laa.2009.07.024.
https://doi.org/10.1016/j.laa.2009.07.024

[42] Martin Plesch and Časlav Brukner. Quantum-state preparation with universal gate decompositions. Physical Review A, 83 (3): 032302, 2011. ISSN 1050-2947, 1094-1622. 10.1103/PhysRevA.83.032302.
https://doi.org/10.1103/PhysRevA.83.032302

[43] Shi-Ju Ran. Encoding of Matrix Product States into Quantum Circuits of One- and Two-Qubit Gates. Physical Review A, 101 (3): 032310, 2020. ISSN 2469-9926, 2469-9934. 10.1103/PhysRevA.101.032310.
https://doi.org/10.1103/PhysRevA.101.032310

[44] Arthur G. Rattew and Bálint Koczor. Preparing Arbitrary Continuous Functions in Quantum Registers With Logarithmic Complexity. arXiv preprint arXiv:2205.00519, 2022. 10.48550/arXiv.2205.00519.
https://doi.org/10.48550/arXiv.2205.00519
arXiv:2205.00519

[45] Arthur G. Rattew, Yue Sun, Pierre Minssen, and Marco Pistoia. The Efficient Preparation of Normal Distributions in Quantum Registers. Quantum, 5: 609, 2021. ISSN 2521-327X. 10.22331/q-2021-12-23-609.
https://doi.org/10.22331/q-2021-12-23-609

[46] W. D. Ray. The Advanced Theory of Statistics, Vol. 3: Design and Analysis and Time Series. Royal Statistical Society. Journal. Series A: General, 147 (3): 523, 1984. ISSN 0035-9238. 10.2307/2981604.
https://doi.org/10.2307/2981604

[47] Paul B. Rohrbach, Sergey Dolgov, Lars Grasedyck, and Robert Scheichl. Rank Bounds for Approximating Gaussian Densities in the Tensor-Train Format. SIAM/ASA Journal on Uncertainty Quantification, 10 (3): 1191–1224, 2022. ISSN 2166-2525. 10.1137/20M1314653.
https://doi.org/10.1137/20M1314653

[48] Manuel S. Rudolph, Jacob Miller, Danial Motlagh, Jing Chen, Atithi Acharya, and Alejandro Perdomo-Ortiz. Synergy Between Quantum Circuits and Tensor Networks: Short-cutting the Race to Practical Quantum Advantage. arXiv preprint arXiv:2208.13673, 2023. 10.48550/arXiv.2208.13673.
https://doi.org/10.48550/arXiv.2208.13673
arXiv:2208.13673

[49] Yuichi Sano and Ikko Hamamura. Quantum State Preparation for Probability Distributions with Mirror Symmetry Using Matrix Product States. arXiv preprint arXiv:2403.16729, 2024. 10.48550/arXiv.2403.16729.
https://doi.org/10.48550/arXiv.2403.16729
arXiv:2403.16729

[50] Dmitry V. Savostyanov. Quasioptimality of maximum-volume cross interpolation of tensors. Linear Algebra and its Applications, 458: 217–244, 2014. ISSN 00243795. 10.1016/j.laa.2014.06.006.
https://doi.org/10.1016/j.laa.2014.06.006

[51] Ulrich Schollwoeck. The density-matrix renormalization group in the age of matrix product states. Annals of Physics, 326 (1): 96–192, 2011. ISSN 00034916. 10.1016/j.aop.2010.09.012.
https://doi.org/10.1016/j.aop.2010.09.012

[52] Y.-Y. Shi, L.-M. Duan, and G. Vidal. Classical simulation of quantum many-body systems with a tree tensor network. Physical Review A, 74 (2): 022320, 2006. ISSN 1050-2947, 1094-1622. 10.1103/PhysRevA.74.022320.
https://doi.org/10.1103/PhysRevA.74.022320

[53] Vladimir Skavysh, Sofia Priazhkina, Diego Guala, and Thomas R. Bromley. Quantum monte carlo for economics: Stress testing and macroeconomic deep learning. Journal of Economic Dynamics and Control, 153: 104680, 2023. ISSN 0165-1889. 10.1016/j.jedc.2023.104680.
https://doi.org/10.1016/j.jedc.2023.104680

[54] Nikitas Stamatopoulos and William J. Zeng. Derivative Pricing using Quantum Signal Processing. Quantum, 8: 1322, 2024. 10.22331/q-2024-04-30-1322.
https://doi.org/10.22331/q-2024-04-30-1322

[55] Queenie Sun, Nicholas Grablevsky, Huaizhang Deng, and Pooya Azadi. Quantum Computing for Multi Period Asset Allocation. arXiv preprint arXiv:2410.11997, 2024. 10.48550/arXiv.2410.11997.
https://doi.org/10.48550/arXiv.2410.11997
arXiv:2410.11997

[56] Joseph Tindall, Miles Stoudenmire, and Ryan Levy. Compressing multivariate functions with tree tensor networks. arXiv preprint arXiv:2410.03572, 2024. 10.48550/arXiv.2410.03572.
https://doi.org/10.48550/arXiv.2410.03572
arXiv:2410.03572

[57] Ryo Watanabe, Hidetaka Manabe, Toshiya Hikihara, and Hiroshi Ueda. TTNOpt: Tree tensor network package for high-rank tensor compression. arXiv preprint arXiv:2505.05908, 2025. 10.48550/arXiv.2505.05908.
https://doi.org/10.48550/arXiv.2505.05908
arXiv:2505.05908

[58] Sascha Wilkens and Joe Moorhouse. Quantum computing for financial risk measurement. Quantum Information Processing, 22 (1): 51, 2023. ISSN 1573-1332. 10.1007/s11128-022-03777-2.
https://doi.org/10.1007/s11128-022-03777-2

[59] Stefan Woerner and Daniel J. Egger. Quantum risk analysis. npj Quantum Information, 5 (1): 1–8, 2019. ISSN 2056-6387. 10.1038/s41534-019-0130-6.
https://doi.org/10.1038/s41534-019-0130-6

[60] Peng-Fei Zhou, Rui Hong, and Shi-Ju Ran. Automatically differentiable quantum circuit for many-qubit state preparation. Physical Review A, 104 (4): 042601, 2021. ISSN 2469-9926, 2469-9934. 10.1103/PhysRevA.104.042601.
https://doi.org/10.1103/PhysRevA.104.042601

[61] Elton Yechao Zhu, Sonika Johri, Dave Bacon, Mert Esencan, Jungsang Kim, Mark Muir, Nikhil Murgai, Jason Nguyen, Neal Pisenti, Adam Schouela, Ksenia Sosnova, and Ken Wright. Generative quantum learning of joint probability distribution functions. Physical Review Research, 4 (4): 043092, 2022. ISSN 2643-1564. 10.1103/PhysRevResearch.4.043092.
https://doi.org/10.1103/PhysRevResearch.4.043092

[62] Julien Zylberman and Fabrice Debbasch. Efficient Quantum State Preparation with Walsh Series. arXiv preprint arXiv:2307.08384, 2023. 10.48550/arXiv.2307.08384.
https://doi.org/10.48550/arXiv.2307.08384
arXiv:2307.08384

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