Quantum Circuits for partial differential equations via Schrödingerisation

Junpeng Hu; Shi Jin; Nana Liu; Lei Zhang

doi:10.22331/q-2024-12-12-1563

Quantum Circuits for partial differential equations via Schrödingerisation

Junpeng Hu¹, Shi Jin^1,2,4, Nana Liu^2,3,4, and Lei Zhang^1,2,4

¹School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China
²Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China
³University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai, 200240, China
⁴MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China

Published:	2024-12-12, volume 8, page 1563
Eprint:	arXiv:2403.10032v3
Doi:	https://doi.org/10.22331/q-2024-12-12-1563
Citation:	Quantum 8, 1563 (2024).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Quantum computing has emerged as a promising avenue for achieving significant speedup, particularly in large-scale PDE simulations, compared to classical computing. One of the main quantum approaches involves utilizing Hamiltonian simulation, which is directly applicable only to Schrödinger-type equations. To address this limitation, Schrödingerisation techniques have been developed, employing the warped transformation to convert general linear PDEs into Schrödinger-type equations. However, despite the development of Schrödingerisation techniques, the explicit implementation of the corresponding quantum circuit for solving general PDEs remains to be designed. In this paper, we present detailed implementation of a quantum algorithm for general PDEs using Schrödingerisation techniques. We provide examples of the heat equation, and the advection equation approximated by the upwind scheme, to demonstrate the effectiveness of our approach. Complexity analysis is also carried out to demonstrate the quantum advantages of these algorithms in high dimensions over their classical counterparts.

► BibTeX data

@article{Hu2024quantumcircuits,
  doi = {10.22331/q-2024-12-12-1563},
  url = {https://doi.org/10.22331/q-2024-12-12-1563},
  title = {Quantum {C}ircuits for partial differential equations via {S}chr{\"{o}}dingerisation},
  author = {Hu, Junpeng and Jin, Shi and Liu, Nana and Zhang, Lei},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1563},
  month = dec,
  year = {2024}
}

► References

[1] Richard P Feynman. ``Simulating physics with computers''. In Feynman and computation. Pages 133–153. CRC Press (2018).
https://doi.org/10.1201/9780429500459

[2] Michael A Nielsen and Isaac L Chuang. ``Quantum computation and quantum information''. Cambridge university press. (2010).
https://doi.org/10.1017/cbo9780511976667

[3] Shi Jin and Nana Liu. ``Analog quantum simulation of partial differential equations''. Quantum Science and Technology 9, 035047 (2024).
https://doi.org/10.1088/2058-9565/ad49cf

[4] Seth Lloyd. ``Universal quantum simulators''. Science 273, 1073–1078 (1996).
https://doi.org/10.1126/science.273.5278.1073

[5] Dorit Aharonov and Amnon Ta-Shma. ``Adiabatic quantum state generation and statistical zero knowledge''. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing. Pages 20–29. (2003).
https://doi.org/10.1145/780542.780546

[6] Andrew Macgregor Childs. ``Quantum information processing in continuous time''. PhD thesis. Massachusetts Institute of Technology. (2004). url: http://hdl.handle.net/1721.1/16663.
http://hdl.handle.net/1721.1/16663

[7] Dominic W Berry, Graeme Ahokas, Richard Cleve, and Barry C Sanders. ``Efficient quantum algorithms for simulating sparse hamiltonians''. Communications in Mathematical Physics 270, 359–371 (2007).
https://doi.org/10.1007/s00220-006-0150-x

[8] Nathan Wiebe, Dominic W Berry, Peter Høyer, and Barry C Sanders. ``Simulating quantum dynamics on a quantum computer''. Journal of Physics A: Mathematical and Theoretical 44, 445308 (2011).
https://doi.org/10.1088/1751-8113/44/44/445308

[9] Andrew M Childs. ``On the relationship between continuous-and discrete-time quantum walk''. Communications in Mathematical Physics 294, 581–603 (2010).
https://doi.org/10.1007/s00220-009-0930-1

[10] Dominic W Berry and Andrew M Childs. ``Black-box hamiltonian simulation and unitary implementation''. Quantum Information & Computation 12, 29–62 (2012). url: https://dl.acm.org/doi/abs/10.5555/2231036.2231040.
https://dl.acm.org/doi/abs/10.5555/2231036.2231040

[11] Andrew M Childs and Nathan Wiebe. ``Hamiltonian simulation using linear combinations of unitary operations''. Quantum Information & Computation 12, 901–924 (2012). url: https://dl.acm.org/doi/abs/10.5555/2481569.2481570.
https://dl.acm.org/doi/abs/10.5555/2481569.2481570

[12] Dominic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D Somma. ``Simulating hamiltonian dynamics with a truncated taylor series''. Physical review letters 114, 090502 (2015).
https://doi.org/10.1103/PhysRevLett.114.090502

[13] Dominic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D Somma. ``Exponential improvement in precision for simulating sparse hamiltonians''. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing. Pages 283–292. (2014).
https://doi.org/10.1145/2591796.2591854

[14] Dominic W Berry, Andrew M Childs, and Robin Kothari. ``Hamiltonian simulation with nearly optimal dependence on all parameters''. In 2015 IEEE 56th annual symposium on foundations of computer science. Pages 792–809. IEEE (2015).
https://doi.org/10.1109/FOCS.2015.54

[15] Dominic W Berry, Andrew M Childs, Yuan Su, Xin Wang, and Nathan Wiebe. ``Time-dependent hamiltonian simulation with $ l^{1}$-norm scaling''. Quantum 4, 254 (2020).
https://doi.org/10.22331/q-2020-04-20-254

[16] Qi Zhao, You Zhou, Alexander F Shaw, Tongyang Li, and Andrew M Childs. ``Hamiltonian simulation with random inputs''. Physical Review Letters 129, 270502 (2022).
https://doi.org/10.1103/PhysRevLett.129.270502

[17] Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. ``Quantum algorithm for linear systems of equations''. Physical review letters 103, 150502 (2009).
https://doi.org/10.1103/PhysRevLett.103.150502

[18] Yudong Cao, Anmer Daskin, Steven Frankel, and Sabre Kais. ``Quantum circuit design for solving linear systems of equations''. Molecular Physics 110, 1675–1680 (2012).
https://doi.org/10.1080/00268976.2012.668289

[19] Andrew M Childs, Robin Kothari, and Rolando D Somma. ``Quantum algorithm for systems of linear equations with exponentially improved dependence on precision''. SIAM Journal on Computing 46, 1920–1950 (2017).
https://doi.org/10.1137/16M1087072

[20] Dominic W Berry. ``High-order quantum algorithm for solving linear differential equations''. Journal of Physics A: Mathematical and Theoretical 47, 105301 (2014).
https://doi.org/10.1088/1751-8113/47/10/105301

[21] Dominic W Berry, Andrew M Childs, Aaron Ostrander, and Guoming Wang. ``Quantum algorithm for linear differential equations with exponentially improved dependence on precision''. Communications in Mathematical Physics 356, 1057–1081 (2017).
https://doi.org/10.1007/s00220-017-3002-y

[22] Andrew M Childs, Jin-Peng Liu, and Aaron Ostrander. ``High-precision quantum algorithms for partial differential equations''. Quantum 5, 574 (2021).
https://doi.org/10.22331/q-2021-11-10-574

[23] Shi Jin, Nana Liu, and Yue Yu. ``Time complexity analysis of quantum difference methods for linear high dimensional and multiscale partial differential equations''. Journal of Computational Physics 471, 111641 (2022).
https://doi.org/10.1016/j.jcp.2022.111641

[24] Shi Jin and Nana Liu. ``Quantum algorithms for nonlinear partial differential equations''. Bulletin des Sciences Mathématiques 194, 103457 (2024).
https://doi.org/10.1016/j.bulsci.2024.103457

[25] Shi Jin, Nana Liu, and Yue Yu. ``Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations''. Journal of Computational Physics 487, 112149 (2023).
https://doi.org/10.1016/j.jcp.2023.112149

[26] Shi Jin, Nana Liu, and Yue Yu. ``Quantum simulation of partial differential equations via schrodingerisation'' (2022). arXiv:2212.13969.
arXiv:2212.13969

[27] Shi Jin, Nana Liu, and Yue Yu. ``Quantum simulation of partial differential equations: Applications and detailed analysis''. Physical Review A 108, 032603 (2023).
https://doi.org/10.1103/PhysRevA.108.032603

[28] Zhen Lu and Yue Yang. ``Quantum computing of reacting flows via hamiltonian simulation''. Proceedings of the Combustion Institute 40, 105440 (2024).
https://doi.org/10.1016/j.proci.2024.105440

[29] Shi Jin and Nana Liu. ``Quantum simulation of discrete linear dynamical systems and simple iterative methods in linear algebra''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 480, 20230370 (2024).
https://doi.org/10.1098/rspa.2023.0370

[30] Dong An, Jin-Peng Liu, and Lin Lin. ``Linear combination of hamiltonian simulation for nonunitary dynamics with optimal state preparation cost''. Physical Review Letters 131, 150603 (2023).
https://doi.org/10.1103/PhysRevLett.131.150603

[31] Yuki Sato, Ruho Kondo, Ikko Hamamura, Tamiya Onodera, and Naoki Yamamoto. ``Hamiltonian simulation for hyperbolic partial differential equations by scalable quantum circuits''. Physical Review Research 6, 033246 (2024).
https://doi.org/10.1103/PhysRevResearch.6.033246

[32] Qiskit contributors. ``Qiskit: An open-source framework for quantum computing'' (2023).

[33] Andrew M Childs, Yuan Su, Minh C Tran, Nathan Wiebe, and Shuchen Zhu. ``Theory of trotter error with commutator scaling''. Physical Review X 11, 011020 (2021).
https://doi.org/10.1103/PhysRevX.11.011020

[34] Rafaella Vale, Thiago Melo D Azevedo, Ismael CS Araújo, Israel F Araujo, and Adenilton J da Silva. ``Circuit decomposition of multi-controlled special unitary single-qubit gates''. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems (2023).
https://doi.org/10.1109/TCAD.2023.3327102

[35] Lin Lin. ``Lecture notes on quantum algorithms for scientific computation'' (2022). arXiv:2201.08309.
arXiv:2201.08309

[36] Shi Jin, Nana Liu, and Chuwen Ma. ``Schrödingerisation based computationally stable algorithms for ill-posed problems in partial differential equations'' (2024). arXiv:2403.19123.
arXiv:2403.19123

[37] Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, C J Carey, İlhan Polat, Yu Feng, Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. A. Quintero, Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa, Paul van Mulbregt, and SciPy 1.0 Contributors. ``SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python''. Nature Methods 17, 261–272 (2020).
https://doi.org/10.1038/s41592-019-0686-2

[38] Shi Jin, Nana Liu, and Chuwen Ma. ``On schrödingerization based quantum algorithms for linear dynamical systems with inhomogeneous terms'' (2024). arXiv:2402.14696.
arXiv:2402.14696

[39] Junpeng Hu. ``Quantum circuits for partial differential equations via schrodingerisation''. GitHub Repository (2024). url: https://github.com/hjp3268/Schrodingerisation-Circuits.
https://github.com/hjp3268/Schrodingerisation-Circuits

Cited by

[1] Yijin Mao and Yuwen Zhang, "Modeling Heat Conduction With an Internal Heat Source Using a Quantum Lattice Boltzmann Method", ASME Journal of Heat and Mass Transfer 148 2, 021403 (2026).

[2] Paul Over, Sergio Bengoechea, Peter Brearley, Sylvain Laizet, and Thomas Rung, "Quantum algorithm for the advection-diffusion equation by direct block encoding of the time-marching operator", Physical Review A 112 1, L010401 (2025).

[3] Sergio Bengoechea, Paul Over, Dieter Jaksch, and Thomas Rung, "Toward Variational Quantum Algorithms for Generalized Linear and Nonlinear Transport Phenomena", AIAA Journal 64 2, 585 (2026).

[4] Michael Lubasch, Yuta Kikuchi, Lewis Wright, and Conor Mc Keever, "Quantum circuits for partial differential equations in Fourier space", Physical Review Research 7 4, 043326 (2025).

[5] Kazuki Sakamoto and Keisuke Fujii, "On the quantum computational complexity of classical linear dynamics with geometrically local interactions: Dequantization and universality", Quantum 10, 2182 (2026).

[6] Dekuan Dong, Yingzhou Li, and Jungong Xue, "A quantum algorithm for linear autonomous differential equations via Padé approximation", Quantum 9, 1770 (2025).

[7] Chelsea A. Williams, Annie E. Paine, Antonio A. Gentile, Daniel Berger, and Oleksandr Kyriienko, "Vortex detection from quantum data", Physical Review A 112 6, 062409 (2025).

[8] Junpeng Hu, Shi Jin, Nana Liu, and Lei Zhang, "Quantum random feature method for solving partial differential equations", Discrete and Continuous Dynamical Systems 55, 497 (2026).

[9] Shi Jin, Nana Liu, and Wei Wei, "Quantum Algorithms for Stochastic Differential Equations: A Schrödingerisation Approach", Journal of Scientific Computing 104 2, 56 (2025).

[10] Nikita Guseynov, Xiajie Huang, and Nana Liu, "Gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations", Physical Review Research 7 3, 033100 (2025).

[11] Hayato Higuchi, Juan William Pedersen, Kiichiro Toyoizumi, Kohji Yoshikawa, Chusei Kiumi, and Akimasa Yoshikawa, "Quantum calculation for two-stream instability and advection test of Vlasov–Maxwell equations: numerical evaluation of Hamiltonian simulation", Journal of Plasma Physics 91 4, E116 (2025).

[12] Sachin S Bharadwaj and Katepalli R Sreenivasan, "Towards simulating fluid flows with quantum computing", Sādhanā 50 2, 57 (2025).

[13] Chuwen Ma, Shi Jin, Nana Liu, Kezhen Wang, and Lei Zhang, "Schrödingerization-Based Quantum Circuits For Maxwell’s Equations With Time-Dependent Source Terms", Journal of Scientific Computing 108 2, 43 (2026).

[14] Tamiya Onodera, Yuki Sato, Toshinari Itoko, and Naoki Yamamoto, "Multilevel Gate Set Optimization of Quantum Circuits for Partial Differential Equations", IEEE Transactions on Quantum Engineering 7, 1 (2026).

[15] Jiahua Yang, Zhen Lu, and Yue Yang, "Modeling noise in quantum computing of scalar convection", Acta Mechanica Sinica 42 6, 351281 (2026).

[16] Julien Zylberman, Thibault Fredon, Nuno F Loureiro, and Fabrice Debbasch, "Trotter-based quantum algorithm for solving transport equations with exponentially fewer time-steps", Quantum Science and Technology 11 2, 025015 (2026).

[17] Dingjie Lu, Zhao Wang, Jun Liu, Yangfan Li, Wei-Bin Ewe, and Zhuangjian Liu, "Quantum finite element algorithm for solving Euler–Bernoulli and heat transfer PDEs with Dirichlet, Neumann, and Robin boundary conditions", Quantum Science and Technology 11 1, 015051 (2026).

[18] Yunah Choi, Jeihee Cho, and Shiho Kim, Advances in Computers 140, 1 (2026) ISBN:9780443223822.

[19] Kevin Lively, Vittorio Pagni, and Gonzalo Camacho, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 1926 (2025) ISBN:979-8-3315-5736-2.

[20] Junpeng Hu, Shi Jin, Nana Liu, and Lei Zhang, "Dilation Theorem Via Schrödingerization, With Applications to the Quantum Simulation of Differential Equations", Studies in Applied Mathematics 154 4, e70047 (2025).

[21] Nikita Guseynov, Xiajie Huang, and Nana Liu, "Quantum framework for simulating linear PDEs with Robin boundary conditions", Quantum Science and Technology 11 1, 015057 (2026).

[22] Niladri Gomes, Gautam Sharma, and Jay Pathak, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 1849 (2025) ISBN:979-8-3315-5736-2.

[23] I. Novikau and I. Joseph, "An efficient explicit implementation of a near-optimal quantum algorithm for simulating linear dissipative differential equations", Journal of Computational Physics 564, 115132 (2026).

[24] Sachin S. Bharadwaj and Katepalli R. Sreenivasan, "Compact quantum algorithms for time-dependent differential equations", Physical Review Research 7 2, 023262 (2025).

[25] M. Deiml and D. Peterseim, "Quantum realization of the finite element method", Mathematics of Computation 95 362, 2637 (2025).

[26] Thanh Nguyen, Lecture Notes in Networks and Systems 1732, 18 (2025) ISBN:978-3-032-11523-2.

[27] Shi Jin, Zihao Tang, Xu Yin, and Lei Zhang, "Quantum Circuits for Black-Scholes Equations via Schrödingerisation", Communications in Computational Physics 40 5, 1508 (2026).

[28] Shi Jin, Nana Liu, and Yue Yu, "Quantum simulation of the Fokker-Planck equation via Schrodingerization", arXiv:2404.13585, (2024).

[29] Chuwen Ma, Shi Jin, Nana Liu, Kezhen Wang, and Lei Zhang, "Schrödingerization based Quantum Circuits for Maxwell's Equation with time-dependent source terms", arXiv:2411.10999, (2024).

[30] Fong Yew Leong, Dax Enshan Koh, Jian Feng Kong, Siong Thye Goh, Jun Yong Khoo, Wei-Bin Ewe, Hongying Li, Jayne Thompson, and Dario Poletti, "Solving fractional differential equations on a quantum computer: A variational approach", AVS Quantum Science 6 3, 033802 (2024).

[31] Matthias Deiml and Daniel Peterseim, "Quantum Realization of the Finite Element Method", arXiv:2403.19512, (2024).

[32] Sachin S. Bharadwaj and Katepalli R. Sreenivasan, "Simulating fluid flows with quantum computing", arXiv:2409.09736, (2024).

[33] Chelsea A. Williams, Antonio A. Gentile, Vincent E. Elfving, Daniel Berger, and Oleksandr Kyriienko, "Quantum Iterative Methods for Solving Differential Equations with Application to Computational Fluid Dynamics", arXiv:2404.08605, (2024).

[34] Rundi Lu, Hao-En Li, Zhengwei Liu, and Jin-Peng Liu, "Infinite-dimensional Extension of the Linear Combination of Hamiltonian Simulation: Theorems and Applications", arXiv:2502.19688, (2025).

[35] Kevin Lively, Vittorio Pagni, and Gonzalo Camacho, "A Quantum-Inspired Algorithm for Wave Simulation Using Tensor Networks", arXiv:2504.11181, (2025).

[36] Vladyslav Bohun, Maxence Grandadam, and Maciej Koch-Janusz, "Distributed Quantum Dynamics on Near-Term Quantum Processors", arXiv:2502.03542, (2025).

[37] Shijun Liao, "HAM-Schrödingerisation: a generic framework of quantum simulation for any nonlinear PDEs", arXiv:2406.15821, (2024).

[38] Hsin-Yu Wu, Annie E. Paine, Evan Philip, Antonio A. Gentile, and Oleksandr Kyriienko, "Quantum algorithm for solving nonlinear differential equations based on physics-informed effective Hamiltonians", arXiv:2504.13174, (2025).

[39] I. Novikau and I. Joseph, "Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems", Computer Physics Communications 309, 109498 (2025).

[40] Ivan Novikau and Ilon Joseph, "Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems", arXiv:2410.03985, (2024).

[41] Evandro C. R. Rosa, Jerusa Marchi, Eduardo I. Duzzioni, and Rafael de Santiago, "Automated Auxiliary Qubit Allocation in High-Level Quantum Programming", arXiv:2412.20543, (2024).

[42] Shi Jin, Nana Liu, and Wei Wei, "Quantum Algorithms for Stochastic Differential Equations: A Schrödingerisation Approach", arXiv:2412.14868, (2024).

[43] Joseph Li, Gengzhi Yang, Jiaqi Leng, and Xiaodi Wu, "Resource-efficient quantum simulation of transport phenomena via Hamiltonian embedding", arXiv:2602.03099, (2026).

[44] Sachin S. Bharadwaj, Balasubramanya Nadiga, Stephan Eidenbenz, and Katepalli R. Sreenivasan, "Quantum Homotopy Algorithm for Solving Nonlinear PDEs and Flow Problems", arXiv:2512.21033, (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-12 01:10:39) and SAO/NASA ADS (last updated successfully 2026-08-12 01:10:41). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.