The Integral Decimation Method for Quantum Dynamics and Statistical Mechanics
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AbstractThe solutions to many problems in the mathematical, computational, and physical sciences often involve multidimensional integrals. A direct numerical evaluation of the integral incurs a computational cost that is exponential in the number of dimensions, a phenomenon called the curse of dimensionality. The problem is so substantial that one usually employs sampling methods, like Monte Carlo, to avoid integration altogether. Here, we derive and implement a quantum-inspired algorithm to decompose a multidimensional integrand into a product of matrix-valued functions – a spectral tensor train – changing the computational complexity of integration from exponential to polynomial. The algorithm constructs a spectral tensor train representation of the integrand by applying a sequence of quantum gates, where each gate corresponds to an interaction that involves increasingly more degrees of freedom in the action. Because it allows for the systematic elimination of small contributions to the integral through decimation, we call the method integral decimation. The functions in the spectral basis are analytically differentiable and integrable, and in applications to the partition function, integral decimation numerically factorizes an interacting system into a product of non-interacting ones. We illustrate integral decimation by evaluating the absolute free energy and entropy of a chiral XY model as a continuous function of temperature. We also compute the nonequilibrium time-dependent reduced density matrix of a quantum chain with between two and forty levels, each coupled to colored noise. When other methods provide numerical solutions to these models, they quantitatively agree with integral decimation. When conventional methods become intractable, integral decimation can be a powerful alternative.Featured image: An illustration of the integral decimation method that computes a high-dimensional integral by transforming the integrand to a quantum circuit, decimating to yield a tensor network, and then integrating the result.Popular summaryIntegral decimation (ID) computes classical and quantum path integrals by mapping the Hamiltonian or action of the system to a quantum circuit that generates a continuous tensor network representation of the integrand. The resulting representation is numerically exact and separable, facilitating analytical integration, evaluation, and differentiation at linear cost in the number of degrees of freedom. Among other applications, ID makes it possible to simulate numerically exact dynamics of very large open quantum systems, exceeding forty levels, with no short-memory approximation.► BibTeX data@article{Grimm2026integraldecimation, doi = {10.22331/q-2026-04-13-2064}, url = {https://doi.org/10.22331/q-2026-04-13-2064}, title = {The {I}ntegral {D}ecimation {M}ethod for {Q}uantum {D}ynamics and {S}tatistical {M}echanics}, author = {Grimm, Ryan T. and Staat, Alexander J. and Eaves, Joel D.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2064}, month = apr, year = {2026} }► References [1] Philipp Seitz, Ismael Medina, Esther Cruz, Qunsheng Huang, and Christian B Mendl. ``Simulating quantum circuits using tree tensor networks''. 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Copyright remains with the original copyright holders such as the authors or their institutions. AbstractThe solutions to many problems in the mathematical, computational, and physical sciences often involve multidimensional integrals. A direct numerical evaluation of the integral incurs a computational cost that is exponential in the number of dimensions, a phenomenon called the curse of dimensionality. The problem is so substantial that one usually employs sampling methods, like Monte Carlo, to avoid integration altogether. Here, we derive and implement a quantum-inspired algorithm to decompose a multidimensional integrand into a product of matrix-valued functions – a spectral tensor train – changing the computational complexity of integration from exponential to polynomial. The algorithm constructs a spectral tensor train representation of the integrand by applying a sequence of quantum gates, where each gate corresponds to an interaction that involves increasingly more degrees of freedom in the action. Because it allows for the systematic elimination of small contributions to the integral through decimation, we call the method integral decimation. The functions in the spectral basis are analytically differentiable and integrable, and in applications to the partition function, integral decimation numerically factorizes an interacting system into a product of non-interacting ones. We illustrate integral decimation by evaluating the absolute free energy and entropy of a chiral XY model as a continuous function of temperature. We also compute the nonequilibrium time-dependent reduced density matrix of a quantum chain with between two and forty levels, each coupled to colored noise. When other methods provide numerical solutions to these models, they quantitatively agree with integral decimation. When conventional methods become intractable, integral decimation can be a powerful alternative.Featured image: An illustration of the integral decimation method that computes a high-dimensional integral by transforming the integrand to a quantum circuit, decimating to yield a tensor network, and then integrating the result.Popular summaryIntegral decimation (ID) computes classical and quantum path integrals by mapping the Hamiltonian or action of the system to a quantum circuit that generates a continuous tensor network representation of the integrand. The resulting representation is numerically exact and separable, facilitating analytical integration, evaluation, and differentiation at linear cost in the number of degrees of freedom. Among other applications, ID makes it possible to simulate numerically exact dynamics of very large open quantum systems, exceeding forty levels, with no short-memory approximation.► BibTeX data@article{Grimm2026integraldecimation, doi = {10.22331/q-2026-04-13-2064}, url = {https://doi.org/10.22331/q-2026-04-13-2064}, title = {The {I}ntegral {D}ecimation {M}ethod for {Q}uantum {D}ynamics and {S}tatistical {M}echanics}, author = {Grimm, Ryan T. and Staat, Alexander J. and Eaves, Joel D.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2064}, month = apr, year = {2026} }► References [1] Philipp Seitz, Ismael Medina, Esther Cruz, Qunsheng Huang, and Christian B Mendl. ``Simulating quantum circuits using tree tensor networks''. Quantum 7, 964 (2023). arXiv:2206.01000v3. https://doi.org/10.22331/q-2023-03-30-964 arXiv:2206.01000v3 [2] S R White. ``Density matrix formulation for quantum renormalization groups''. Phys. Rev. Lett. 69, 2863–2866 (1992). https://doi.org/10.1103/PhysRevLett.69.2863 [3] A Strathearn, P Kirton, D Kilda, J Keeling, and B W Lovett. ``Efficient non-Markovian quantum dynamics using time-evolving matrix product operators''. Nat. Commun. 9, 3322 (2018). https://doi.org/10.1038/s41467-018-05617-3 [4] Ryan T Grimm and Joel D Eaves. ``Direct Numerical Solutions to Stochastic Differential Equations with Multiplicative Noise''. Phys. Rev. Lett. 132, 267101 (2024). https://doi.org/10.1103/PhysRevLett.132.267101 [5] Nikita Gourianov, Peyman Givi, Dieter Jaksch, and Stephen B Pope. ``Tensor networks enable the calculation of turbulence probability distributions''. Sci. Adv. 11, eads5990 (2025). https://doi.org/10.1126/sciadv.ads5990 [6] Michael Levin and Cody P Nave. ``Tensor renormalization group approach to two-dimensional classical lattice models''. Phys. Rev. Lett. 99, 120601 (2007). https://doi.org/10.1103/PhysRevLett.99.120601 [7] Daniele Bigoni, Allan P Engsig-Karup, and Youssef M Marzouk. ``Spectral Tensor-Train Decomposition''. SIAM J. Sci. Comput. 38, A2405–A2439 (2016). https://doi.org/10.1137/15M1036919 [8] Ryan T Grimm and Joel D Eaves. ``Accurate numerical simulations of open quantum systems using spectral tensor trains''. J. Chem. Phys. 161, 234111 (2024). https://doi.org/10.1063/5.0228873 [9] Micheline B Soley, Paul Bergold, Alex A Gorodetsky, and Victor S Batista. ``Functional tensor-train Chebyshev method for multidimensional quantum dynamics simulations''. J. Chem. Theory Comput. 18, 25–36 (2022). https://doi.org/10.1021/acs.jctc.1c00941 [10] Yuriel Núñez Fernández, Marc K Ritter, Matthieu Jeannin, Jheng-Wei Li, Thomas Kloss, Thibaud Louvet, Satoshi Terasaki, Olivier Parcollet, Jan von Delft, Hiroshi Shinaoka, and Xavier Waintal. ``Learning tensor networks with tensor cross interpolation: New algorithms and libraries''. SciPost Phys. 18, 104 (2025). https://doi.org/10.21468/scipostphys.18.3.104 [11] Boris Khoromskij and Alexander Veit. ``Efficient computation of highly oscillatory integrals by using QTT tensor approximation''. J. Comput. Methods Appl. Math. 16, 145–159 (2016). https://doi.org/10.1515/cmam-2015-0033 [12] Nathan Ng, Gunhee Park, Andrew J Millis, Garnet Kin-Lic Chan, and David R Reichman. ``Real-time evolution of Anderson impurity models via tensor network influence functionals''. Phys. Rev. B 107, 125103 (2023). https://doi.org/10.1103/physrevb.107.125103 [13] F Verstraete, M M Wolf, D Perez-Garcia, and J I Cirac. ``Criticality, the area law, and the computational power of projected entangled pair states''. Phys. Rev. Lett. 96, 220601 (2006). https://doi.org/10.1103/PhysRevLett.96.220601 [14] Jacques Bloch, Raghav G Jha, Robert Lohmayer, and Maximilian Meister. ``Tensor renormalization group study of the three-dimensional O (2) model''. Phys. Rev. D 104, 094517 (2021). https://doi.org/10.1103/physrevd.104.094517 [15] Xiao Luo and Yoshinobu Kuramashi. ``Tensor renormalization group approach to ( 1+1 )-dimensional SU(2) principal chiral model at finite density''. Phys. Rev. D 107, 094509 (2023). https://doi.org/10.1103/physrevd.107.094509 [16] Feng-Feng Song and Guang-Ming Zhang. ``Tensor network approach to the two-dimensional fully frustrated XY model and a chiral ordered phase''. Phys. Rev. B 105, 134516 (2022). https://doi.org/10.1103/physrevb.105.134516 [17] Dian-Teng Chen, Phillip Helms, Ashlyn R Hale, Minseong Lee, Chenghan Li, Johnnie Gray, George Christou, Vivien S Zapf, Garnet Kin-Lic Chan, and Hai-Ping Cheng. ``Using hyperoptimized tensor networks and first-principles electronic structure to simulate the experimental properties of the giant Mn84 torus''. J. Phys. Chem. Lett. 13, 2365–2370 (2022). https://doi.org/10.1021/acs.jpclett.2c00354 [18] J F Yu, Z Y Xie, Y Meurice, Yuzhi Liu, A Denbleyker, Haiyuan Zou, M P Qin, J Chen, and T Xiang. ``Tensor renormalization group study of classical XY model on the square lattice''. Phys. Rev. E 89, 013308 (2014). https://doi.org/10.1103/PhysRevE.89.013308 [19] Yuriel Núñez Fernández, Matthieu Jeannin, Philipp T Dumitrescu, Thomas Kloss, Jason Kaye, Olivier Parcollet, and Xavier Waintal. ``Learning Feynman Diagrams with Tensor Trains''. Phys. Rev. X 12, 041018 (2022). https://doi.org/10.1103/PhysRevX.12.041018 [20] Sergey Dolgov and Dmitry Savostyanov. ``Parallel cross interpolation for high-precision calculation of high-dimensional integrals''. Comput. Phys. Commun. 246, 106869 (2020). https://doi.org/10.1016/j.cpc.2019.106869 [21] Ivan Oseledets and Eugene Tyrtyshnikov. ``TT-cross approximation for multidimensional arrays''.
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