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Tensor-programmable quantum circuits for solving differential equations

Pia Siegl1,2,*,†, Greta Sophie Reese1,3,4,*,‡, Tomohiro Hashizume1,4, Nis-Luca van Hülst1, and Dieter Jaksch1,4,5

  • *These authors contributed equally to this work.
  • †Contact author: pia.siegl@dlr.de
  • ‡Contact author: greta.reese@uni-hamburg.de
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Phys. Rev. Research 8, 013052 – Published 20 January, 2026

DOI: https://doi.org/10.1103/2qzh-yf49

Abstract

We present a quantum solver for partial differential equations based on a flexible matrix product operator representation. Utilizing midcircuit measurements and a state-dependent norm correction, this scheme overcomes the restriction of unitary operators. Hence, it allows for the direct implementation of a broad class of differential equations governing the dynamics of classical and quantum systems. The capabilities of the framework are demonstrated for linear and nonlinear partial differential equations using the example of the linearized Euler equations with absorbing boundaries and the nonlinear Burgers’ equation. For a turbulence data set, we demonstrate potential advantages of the quantum-tensor scheme over its classical counterparts.

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References (100)

  1. F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ. 81, 637 (1973).
  2. S. T. Lee and H.-W. Sun, Fourth-order compact scheme with local mesh refinement for option pricing in jump-diffusion model, Numer. Methods Partial 28, 1079 (2012).
  3. X. Bian, C. Kim, and G. E. Karniadakis, 111 years of Brownian motion, Soft Matter 12, 6331 (2016).
  4. X. Wu, Y. Zhang, and S. Mao, Learning the physics-consistent material behavior from measurable data via PDE-constrained optimization, Comput. Methods Appl. Mech. Eng. 437, 117748 (2025).
  5. R. Courant, K. Friedrichs, and H. Lewy, On the partial difference equations of mathematical physics, IBM J. Res. Dev. 11, 215 (1967).
  6. J. H. Ferziger and M. Perić, Computational Methods for Fluid Dynamics, 3rd ed. (Springer, Berlin, 2002).
  7. P. Wu, Recent advances in the application of computational fluid dynamics in the development of rotary blood pumps, Med. Nov. Technol. Devices 16, 100177 (2022).
  8. J. P. Slotnick, A. Khodadoust, J. J. Alonso, D. L. Darmofal, W. Gropp, E. A. Lurie, and D. J. Mavriplis, CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences (National Aeronautics and Space Administration, Langley Research Center, Hampton, Virginia, 2014).
  9. F. R. Menter, Two-equation Eddy-viscosity turbulence models for engineering applications, AIAA J. 32, 1598 (1994).
  10. D. Wilcox, Turbulence modeling for CFD, DCW Industries, La Canada (DCW Industries, Inc., Canada, 1998).
  11. S. B. Pope, Turbulent Flows (Cambridge University Press, Cambridge, 2000).
  12. P. Sagaut, Large Eddy Simulation for Incompressible Flows: An Introduction (Springer, Berlin, Heidelberg, 2005).
  13. M. Germano, U. Piomelli, P. Moin, and W. H. Cabot, A dynamic subgrid-scale Eddy viscosity model, Phys. Fluids A: Fluid Dyn. 3, 1760 (1991).
  14. G.-L. Long and Y. Sun, Efficient scheme for initializing a quantum register with an arbitrary superposed state, Phys. Rev. A 64, 014303 (2001).
  15. N. Gourianov, M. Lubasch, S. Dolgov, Q. Y. van den Berg, H. Babaee, P. Givi, M. Kiffner, and D. Jaksch, A quantum-inspired approach to exploit turbulence structures, Nat. Comput. Sci. 2, 30 (2022).
  16. D. Jaksch, P. Givi, A. J. Daley, and T. Rung, Variational quantum algorithms for computational fluid dynamics, AIAA J. 61, 1885 (2023).
  17. P. Givi, A. J. Daley, D. Mavriplis, and M. Malik, Quantum speedup for aeroscience and engineering, AIAA J. 58, 3715 (2020).
  18. A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, Phys. Rev. Lett. 103, 150502 (2009).
  19. A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations, arXiv:1010.4458.
  20. A. M. Childs, R. Kothari, and R. D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision, SIAM J. Comput. 46, 1920 (2017).
  21. J. Penuel, A. Katabarwa, P. D. Johnson, C. Farquhar, Y. Cao, and M. C. Garrett, Feasibility of accelerating incompressible computational fluid dynamics simulations with fault-tolerant quantum computers, arXiv:2406.06323.
  22. S. Lloyd, G. D. Palma, C. Gokler, B. Kiani, Z.-W. Liu, M. Marvian, F. Tennie, and T. Palmer, Quantum algorithm for nonlinear differential equations, arXiv:2011.06571.
  23. P. Brearley and S. Laizet, Quantum algorithm for solving the advection equation using Hamiltonian simulation, Phys. Rev. A 110, 012430 (2024).
  24. P. Over, S. Bengoechea, P. Brearley, S. Laizet, and T. Rung, Quantum algorithm for the advection-diffusion equation with optimal success probability, Phys. Rev. A 112, L010401 (2025).
  25. A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, A variational eigenvalue solver on a photonic quantum processor, Nat. Commun. 5, 4213 (2014).
  26. A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature (London) 549, 242 (2017).
  27. M. Cerezo, K. Sharma, A. Arrasmith, and P. Coles, Variational quantum state eigensolver, npj Quantum Inf. 8, 113 (2022).
  28. G. Q. AI and Collaborators, Quantum error correction below the surface code threshold, Nature (London) 638, 920 (2025).
  29. B. W. Reichardt, D. Aasen, R. Chao, A. Chernoguzov, W. van Dam, J. P. Gaebler, D. Gresh, D. Lucchetti, M. Mills, S. A. Moses, B. Neyenhuis, A. Paetznick, A. Paz, P. E. Siegfried, M. P. da Silva, K. M. Svore, Z. Wang, and M. Zanner, Demonstration of quantum computation and error correction with a tesseract code, arXiv:2409.04628.
  30. W. van Dam, H. Liu, G. H. Low, A. Paetznick, A. Paz, M. Silva, A. Sundaram, K. Svore, and M. Troyer, End-to-end quantum simulation of a chemical system, arXiv:2409.05835.
  31. Quera computing releases a groundbreaking roadmap for advanced error-corrected quantum computers, pioneering the next frontier in quantum innovation, https://www.quera.com/press-releases/quera-computing-releases-a-groundbreaking-roadmap-for-advanced-error-corrected-quantum-computers-pioneering-the-next-frontier-in-quantum-innovation-0, visited on 2025-10-06.
  32. IBM technology atlas quantum roadmap, https://www.ibm.com/roadmaps/quantum.pdf, visited on 2024-10-10.
  33. M. Lubasch, J. Joo, P. Moinier, M. Kiffner, and D. Jaksch, Variational quantum algorithms for nonlinear problems, Phys. Rev. A 101, 010301 (2020).
  34. M. Schilling, F. Preti, M. M. Müller, T. Calarco, and F. Motzoi, Exponentiation of parametric Hamiltonians via unitary interpolation, Phys. Rev. Res. 6, 043278 (2024).
  35. W. J. Huggins, J. Lee, U. Baek, B. O’Gorman, and K. B. Whaley, A non-orthogonal variational quantum eigensolver, New J. Phys. 22, 073009 (2020).
  36. M. Umer, E. Mastorakis, S. Evangelou, and D. G. Angelakis, Probing the limits of variational quantum algorithms for nonlinear ground states on real quantum hardware: The effects of noise, Phys. Rev. A 111, 012626 (2025).
  37. A. J. Pool, A. D. Somoza, C. Mc Keever, M. Lubasch, and B. Horstmann, Nonlinear dynamics as a ground-state solution on quantum computers, Phys. Rev. Res. 6, 033257 (2024).
  38. Y. Shao, F. Wei, S. Cheng, and Z. Liu, Simulating noisy variational quantum algorithms: A polynomial approach, Phys. Rev. Lett. 133, 120603 (2024).
  39. E. Rosenberg, P. Ginsparg, and P. L. McMahon, Experimental error mitigation using linear rescaling for variational quantum eigensolving with up to 20 qubits, Quantum Sci. Technol. 7, 015024 (2022).
  40. B. Jaderberg, A. Agarwal, K. Leonhardt, M. Kiffner, and D. Jaksch, Minimum hardware requirements for hybrid quantum–classical DMFT, Quantum Sci. Technol. 5, 034015 (2020).
  41. B. Jaderberg, A. Eisfeld, D. Jaksch, and S. Mostame, Recompilation-enhanced simulation of electron–phonon dynamics on IBM quantum computers, New J. Phys. 24, 093017 (2022).
  42. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, and A. Kandala, Evidence for the utility of quantum computing before fault tolerance, Nature (London) 618, 500 (2023).
  43. I. V. Oseledets, Tensor-train decomposition, SIAM J. Sci. Comput. 33, 2295 (2011).
  44. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014).
  45. I. V. Oseledets, Constructive representation of functions in low-rank tensor formats, Constr. Approximation 37, 1 (2013).
  46. E. Ye and N. F. G. Loureiro, Quantum-inspired method for solving the Vlasov-Poisson equations, Phys. Rev. E 106, 035208 (2022).
  47. M. Kiffner and D. Jaksch, Tensor network reduced order models for wall-bounded flows, Phys. Rev. Fluids 8, 124101 (2023).
  48. E. Kornev, S. Dolgov, K. Pinto, M. Pflitsch, M. Perelshtein, and A. Melnikov, Numerical solution of the incompressible Navier-Stokes equations for chemical mixers via quantum-inspired tensor train finite element method, arXiv:2305.10784.
  49. E. Ye and N. F. Loureiro, Quantized tensor networks for solving the Vlasov–Maxwell equations, J. Plasma Phys. 90, 805900301 (2024).
  50. R. D. Peddinti, S. Pisoni, A. Marini, P. Lott, H. Argentieri, E. Tiunov, and L. Aolita, Quantum-inspired framework for computational fluid dynamics, Commun. Phys. 7, 135 (2024).
  51. L. Hölscher, P. Rao, L. Müller, J. Klepsch, A. Luckow, T. Stollenwerk, and F. K. Wilhelm, Quantum-inspired fluid simulation of two-dimensional turbulence with GPU acceleration, Phys. Rev. Res. 7, 013112 (2025).
  52. N. Gourianov, P. Givi, D. Jaksch, and S. B. Pope, Tensor networks enable the calculation of turbulence probability distributions, Sci. Adv. 11, eads5990 (2025).
  53. J. Schachenmayer, B. P. Lanyon, C. F. Roos, and A. J. Daley, Entanglement growth in quench dynamics with variable range interactions, Phys. Rev. X 3, 031015 (2013).
  54. L. K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC’96 (Association for Computing Machinery, New York, 1996), pp. 212–219.
  55. S.-J. Ran, Encoding of matrix product states into quantum circuits of one- and two-qubit gates, Phys. Rev. A 101, 032310 (2020).
  56. D. Malz, G. Styliaris, Z.-Y. Wei, and J. I. Cirac, Preparation of matrix product states with log-depth quantum circuits, Phys. Rev. Lett. 132, 040404 (2024).
  57. K. C. Smith, A. Khan, B. K. Clark, S. Girvin, and T.-C. Wei, Constant-depth preparation of matrix product states with adaptive quantum circuits, PRX Quantum 5, 030344 (2024).
  58. M. Nibbi and C. B. Mendl, Block encoding of matrix product operators, Phys. Rev. A 110, 042427 (2024).
  59. A. Termanova, A. Melnikov, E. Mamenchikov, N. Belokonev, S. Dolgov, A. Berezutskii, R. Ellerbrock, C. Mansell, and M. Perelshtein, Tensor quantum programming, New J. Phys. 26, 123019 (2024).
  60. M. Goldack, Y. Atia, O. Alberton, and K. Jansen, Computing statistical properties of velocity fields on current quantum hardware, arXiv:2601.10166.
  61. F. Uchida, K. Miyamoto, S. Yamazaki, K. Fujisawa, and N. Yoshida, Quantum simulation of burgers turbulence: Nonlinear transformation and direct evaluation of statistical quantities, arXiv:2412.17206.
  62. E. Mastorakis, M. Umer, M. Guevara-Bertsch, J. Ulmanis, F. Rohde, and D. G. Angelakis, Resource-efficient Hadamard test circuits for nonlinear dynamics on a trapped-ion quantum computer, arXiv:2507.19250.
  63. A. Sarma, T. W. Watts, M. Moosa, Y. Liu, and P. L. McMahon, Quantum variational solving of nonlinear and multidimensional partial differential equations, Phys. Rev. A 109, 062616 (2024).
  64. P. Over, S. Bengoechea, T. Rung, F. Clerici, L. Scandurra, E. de Villiers, and D. Jaksch, Boundary treatment for variational quantum simulations of partial differential equations on quantum computers, Comput. Fluids 288, 106508 (2025).
  65. M. Schuld and F. Petruccione, Representing data on a quantum computer, in Machine Learning with Quantum Computers (Springer International Publishing, Cham, 2021), pp. 147–176.
  66. M. Ben-Dov, D. Shnaiderov, A. Makmal, and E. G. Dalla Torre, Approximate encoding of quantum states using shallow circuits, npj Quantum Inf. 10, 65 (2024).
  67. V. A. Kazeev and B. N. Khoromskij, Low-rank explicit QTT representation of the Laplace operator and its inverse, SIAM J. Matrix Anal. Appl. 33, 742 (2012).
  68. I. V. Oseledets, Approximation of 2d×2d matrices using tensor decomposition, SIAM J. Matrix Anal. Appl. 31, 2130 (2010).
  69. H. Zhao, M. Bukov, M. Heyl, and R. Moessner, Making trotterization adaptive and energy-self-correcting for NISQ devices and beyond, PRX Quantum 4, 030319 (2023).
  70. R. Puig, M. Drudis, S. Thanasilp, and Z. Holmes, Variational quantum simulation: A case study for understanding warm starts, PRX Quantum 6, 010317 (2025).
  71. K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Phys. Rev. A 98, 032309 (2018).
  72. M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Phys. Rev. A 99, 032331 (2019).
  73. J. Spall, An overview of the simultaneous perturbation method for efficient optimization, Johns Hopkins Apl Technical Digest 19, 482 (1998).
  74. A. J. Chorin and J. E. Marsden, A Mathematical Iintroduction to Fluid Mechanics edited by J.E. Marsden (Caltech), L. Sirovich (Brown University), M. Golubitsky (University of Houston) (Springer Science+Business Media, New York, 1979).
  75. R. Peric and M. Abdel-Maksoud, Reliable damping of free surface waves in numerical simulations, Sh. Technol. Res. 63 (2015).
  76. F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids 8, 2182 (1965).
  77. M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, 10th Anniversary ed. (Cambridge University Press, Cambridge, 2010).
  78. R. Haghshenas, J. Gray, A. C. Potter, and G. K.-L. Chan, Variational power of quantum circuit tensor networks, Phys. Rev. X 12, 011047 (2022).
  79. D. Wang, O. Higgott, and S. Brierley, Accelerated variational quantum eigensolver, Phys. Rev. Lett. 122, 140504 (2019).
  80. D. Grinko, J. Gacon, C. Zoufal, and S. Woerner, Iterative quantum amplitude estimation, npj Quantum Inf. 7, 52 (2021).
  81. Y. Li, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, and G. Eyink, A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence, J. Turbul. 9, N31 (2008).
  82. E. Perlman, R. Burns, Y. Li, and C. Meneveau, Data exploration of turbulence simulations using a database cluster, in Proceedings of the 2007 ACM/IEEE Conference on Supercomputing (SC '07), 1-11, https://api.semanticscholar.org/CorpusID:11021323.
  83. John Hopkins turbulence database—isotropic turbulence.
  84. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  85. F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Commun. Math. Phys. 346, 397 (2016).
  86. K. Goswami, P. Schmelcher, and R. Mukherjee, Qudit-based scalable quantum algorithm for solving the integer programming problem, arXiv:2508.13906.
  87. V. V. Shende, I. L. Markov, and S. S. Bullock, Minimal universal two-qubit controlled-not-based circuits, Phys. Rev. A 69, 062321 (2004).
  88. M. Lubasch, P. Moinier, and D. Jaksch, Multigrid renormalization, J. Comput. Phys. 372, 587 (2018).
  89. L. Slattery, B. Villalonga, and B. K. Clark, Unitary block optimization for variational quantum algorithms, Phys. Rev. Res. 4, 023072 (2022).
  90. A. Y. Kitaev, Quantum measurements and the Abelian stabilizer problem, arXiv:quant-ph/9511026.
  91. S. Fomichev, K. Hejazi, M. S. Zini, M. Kiser, J. Fraxanet, P. A. M. Casares, A. Delgado, J. Huh, A.-C. Voigt, J. E. Mueller, and J. M. Arrazola, Initial state preparation for quantum chemistry on quantum computers, PRX Quantum 5, 040339 (2024).
  92. P. Siegl, G. S. Reese, T. Hashizume, N.-L. van Hülst, and D. Jaksch, Dataset tensor-programmable quantum circuits for solving differential equations (version 3) (2025), http://doi.org/10.25592/uhhfdm.18138.
  93. V. Bergholm et al., PennyLane: Automatic differentiation of hybrid quantum-classical computations, arXiv:1811.04968.
  94. A. Paszke et al., PyTorch: An imperative style, high-performance deep learning library, arXiv:1912.01703.
  95. J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX: Composable transformations of Python + NumPy programs (2018), https://github.com/jax-ml/jax.
  96. I. A. Luchnikov, M. E. Krechetov, and S. N. Filippov, Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies, New J. Phys. 23, 073006 (2021).
  97. J. Li, F. Li, and S. Todorovic, Efficient Riemannian optimization on the Stiefel manifold via the Cayley transform, arXiv:2002.01113.
  98. G. Strang, Introduction to Linear Algebra (Wellesley, Cambridge, 2016).
  99. J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 4812 (2018).
  100. A. A. Mele, G. B. Mbeng, G. E. Santoro, M. Collura, and P. Torta, Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz, Phys. Rev. A 106, L060401 (2022).

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Phys. Rev. Research 8, 013052– Published 20 January, 2026
  • Received 12 February 2025
  • Accepted 8 December 2025
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