Advances in quantum computing for fluid dynamics

MENG Zhaoyuan; LU Zhen; XIONG Shiying; ZHAO Yaomin; YANG Yue

doi:10.6052/1000-0992-24-041

Article Contents

Article Navigation > Advances in Mechanics > 2025 > Accepted Manuscript

Meng Z Y, Lu Z, Xiong S Y, Zhao Y M, Yang Y. Advances in quantum computing for fluid dynamics. Advances in Mechanics, in press doi: 10.6052/1000-0992-24-041

Citation:

Meng Z Y, Lu Z, Xiong S Y, Zhao Y M, Yang Y. Advances in quantum computing for fluid dynamics. Advances in Mechanics, in press doi: 10.6052/1000-0992-24-041

Meng Z Y, Lu Z, Xiong S Y, Zhao Y M, Yang Y. Advances in quantum computing for fluid dynamics. Advances in Mechanics, in press doi: 10.6052/1000-0992-24-041

Citation:

Meng Z Y, Lu Z, Xiong S Y, Zhao Y M, Yang Y. Advances in quantum computing for fluid dynamics. Advances in Mechanics, in press doi: 10.6052/1000-0992-24-041

PDF( 6989 KB)

Advances in quantum computing for fluid dynamics

doi: 10.6052/1000-0992-24-041 cstr: 32046.14.1000-0992-24-041

MENG Zhaoyuan¹,
LU Zhen¹,
XIONG Shiying²,
ZHAO Yaomin^{1, 3},
YANG Yue^{1, 3
,
,}

1.
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, PR China
2.
Department of Engineering Mechanics, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, 31002, PR China
3.
HEDPS-CAPT, Peking University, Beijing 100871, PR China

More Information

Corresponding author: yyg@pku.edu.cn
Available Online: 2025-03-11

Abstract

Abstract

We review progress and challenges in the emerging field of quantum computing for fluid dynamics (QCFD). Quantum computing, a potentially disruptive technology, is expected to tackle pressing problems in the real world. Fluid dynamics, a complex problem in classical physics and engineering, can serve as an example to demonstrate quantum utility and advantage. Conversely, quantum computing can introduce new paradigms in fluid dynamics research. In this review, we first introduce quantum computing features, such as superposition and entanglement, and highlight the challenges of QCFD in initial state preparation, quantum state evolution, and measurement. We then focus on hybrid quantum-classical algorithms and Hamiltonian simulation for fluid dynamics, reviewing their hardware implementation on current quantum computers. In conclusion, QCFD is in its infancy, facing both challenges in quantum devices and algorithms. Although quantum computing has not yet shown an advantage in simulating strongly nonlinear fluid dynamics over classical methods, recent progress suggests its potential in enhancing simulations of complex flows, including turbulence.
- fluid mechanics,
- quantum computing,
- turbulence,
- vortex dynamics,
- computational fluid dynamics

FullText(HTML)

References(166)

References

[1]	陈坚强, 袁先旭, 涂国华, 郭启龙. 2023. 计算流体力学 2035 愿景. 科学出版社. (Chen J, Yuan X, Tu G, Guo Q. 2023. Computational Fluid Dynamics 2035 Vision in China. Science Press). Chen J, Yuan X, Tu G, Guo Q. 2023. Computational Fluid Dynamics 2035 Vision in China. Science Press.
[2]	Aaronson S. 2015. Read the fine print. Nat. Phys., 11: 291-293 doi: 10.1038/nphys3272
[3]	Aharonov Y, Davidovich L, Zagury N. 1993. Quantum random walks. Phys. Rev. A, 48: 1687-1690 doi: 10.1103/PhysRevA.48.1687
[4]	Albash T, Lidar D A. 2018. Adiabatic quantum computation. Rev. Mod. Phys., 90: 015002 doi: 10.1103/RevModPhys.90.015002
[5]	An D, Lin L. 2022. Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm. ACM Transactions on Quantum Computing, 3 .
[6]	An D, Liu J P, Lin L. 2023. Linear combination of Hamiltonian simulation for nonunitary dynamics with optimal state preparation cost. Phys. Rev. Lett., 131: 150603 doi: 10.1103/PhysRevLett.131.150603
[7]	Arute F, Arya K, Babbush R, Bacon D, Bardin J C, Barends R, Biswas R, Boixo S, Brandao F, Buell D A, et al. 2019. Quantum supremacy using a programmable superconducting processor. Nature, 574: 505-510 doi: 10.1038/s41586-019-1666-5
[8]	Ashida Y, Gong Z, Ueda M. 2021. Non-Hermitian physics. Adv. Phys., 69: 249-435
[9]	Asztalos K, Steijl R, Maulik R. 2024. Reduced-order modeling on a near-term quantum computer. J. Comput. Phys., 510: 113070 doi: 10.1016/j.jcp.2024.113070
[10]	Asztalos K, Steijl R, Maulik R. 2024. Reduced-order modeling on a near-term quantum computer. J. Comput. Phys., 510: 113070 doi: 10.1016/j.jcp.2024.113070
[11]	Babbush R, Berry D W, Kothari R, Somma R D, Wiebe N. 2023. Exponential quantum speedup in simulating coupled classical oscillators. Phys. Rev. X, 13: 041041
[12]	Babukhin D V. 2023. Harrow-Hassidim-Lloyd algorithm without ancilla postselection. Phys. Rev. A, 107: 042408 doi: 10.1103/PhysRevA.107.042408
[13]	Barends R, Shabani A, Lamata L, Kelly J, Mezzacapo A, Las Heras U, Babbush R, Fowler A G, Campbell B, Chen Y, et al. 2016. Digitized adiabatic quantum computing with a superconducting circuit. Nature, 534: 222-226 doi: 10.1038/nature17658
[14]	Basheer A, Afham A, Goyal S K. 2020. Quantum k-nearest neighbors algorithm. arXiv preprint arXiv: 2003.09187,
[15]	Begušić T, Gray J, Chan G K L. 2024. Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance. Sci. Adv., 10: eadk4321 doi: 10.1126/sciadv.adk4321
[16]	Benedetti M, Lloyd E, Sack S, Fiorentini M. 2019. Parameterized quantum circuits as machine learning models. Quantum Sci. Technol., 4: 043001 doi: 10.1088/2058-9565/ab4eb5
[17]	Bennett C H, Bernstein E, Brassard G, Vazirani U. 1997. Strengths and weaknesses of quantum computing. SIAM J. Comput., 26: 1510-1523 doi: 10.1137/S0097539796300933
[18]	Bernstein D, Lange T. 2017. Post-quantum cryptography. Nature, 549: 188-194 doi: 10.1038/nature23461
[19]	Bharadwaj S S, Sreenivasan K R. 2020. Quantum computation of fluid dynamics. Indian Acad. Sci. Conf. Ser., 3: 77
[20]	Bharadwaj S S, Sreenivasan K R. 2023. Hybrid quantum algorithms for flow problems. Proc. Natl. Acad. Sci. U.S.A., 120: e2311014120 doi: 10.1073/pnas.2311014120
[21]	Bharadwaj S S, Sreenivasan K R. 2024. Compact quantum algorithms that can potentially maintain quantum advantage for solving time-dependent differential equations, arXiv: 2405.09767.
[22]	Bharadwaj S S, Sreenivasan K R. 2024. Simulating fluid flows with quantum computing, arXiv: 2409.09736.
[23]	Bharti K, Cervera-Lierta A, Kyaw T H, Haug T, Alperin-Lea S, Anand A, Degroote M, Heimonen H, Kottmann J S, Menke T, et al. 2022. Noisy intermediate-scale quantum algorithms. Rev. Mod. Phys., 94: 015004 doi: 10.1103/RevModPhys.94.015004
[24]	Biamonte J, Wittek P, Pancotti N, Rebentrost P, Wiebe N, Lloyd S. 2017. Quantum machine learning. Nature, 549: 195-202 doi: 10.1038/nature23474
[25]	Bishwas A K, Mani A, Palade V. 2020. An investigation on support vector clustering for big data in quantum paradigm. Quantum Inf. Process., 19: 108 doi: 10.1007/s11128-020-2606-x
[26]	Borle A, Lomonaco S J. 2022. How viable is quantum annealing for solving linear algebra problems?, arXiv: 2206.10576.
[27]	Brearley P, Laizet S. 2024. Quantum algorithm for solving the advection equation using Hamiltonian simulation. Phys. Rev. A, 110: 012430 doi: 10.1103/PhysRevA.110.012430
[28]	Brunton S L, Noack B R, Koumoutsakos P. 2020. Machine learning for fluid mechanics. Annu. Rev. Fluid Mech., 52: 477-508 doi: 10.1146/annurev-fluid-010719-060214
[29]	Brüstle N, Wiebe N. 2024. Quantum and classical algorithms for nonlinear unitary dynamics, arXiv: 2407.07685.
[30]	Budinski L. 2021. Quantum algorithm for the advection-diffusion equation simulated with the lattice Boltzmann method. Quantum Inf. Process., 20: 57 doi: 10.1007/s11128-021-02996-3
[31]	Budinski L. 2022. Quantum algorithm for the Navier-Stokes equations by using the streamfunctionvorticity formulation and the lattice Boltzmann method. Int. J. Quantum Inf., 20: 2150039 doi: 10.1142/S0219749921500398
[32]	Buluta I, Nori F. 2009. Quantum simulators. Science, 326: 108-111 doi: 10.1126/science.1177838
[33]	Cerezo M, Arrasmith A, Babbush R, Benjamin S C, Endo S, Fujii K, McClean J R, Mitarai K, Yuan X, Cincio L, et al. 2021. Variational quantum algorithms. Nat. Rev. Phys., 3: 625-644 doi: 10.1038/s42254-021-00348-9
[34]	Cerezo M, Verdon G, Huang H Y, Cincio L, Coles P J. 2022. Challenges and opportunities in quantum machine learning. Nat. Comput. Sci., 2: 567-576 doi: 10.1038/s43588-022-00311-3
[35]	Chen Z Y, Ma T Y, Ye C C, Xu L, Bai W, Zhou L, Tan M Y, Zhuang X N, Xu X F, Wang Y J, et al. 2024. Enabling large-scale and high-precision fluid simulations on near-term quantum computers. Comput. Methods Appl. Mech. Eng., 432: 117428 doi: 10.1016/j.cma.2024.117428
[36]	Chen Z Y, Xue C, Chen S M, Lu B H, Wu Y C, Ding J C, Huang S H, Guo G P. 2022. Quantum approach to accelerate finite volume method on steady computational fluid dynamics problems. Quantum Inf. Process., 21: 137 doi: 10.1007/s11128-022-03478-w
[37]	Cheng B, Deng X H, Gu X, He Y, Hu G, Huang P, Li J, Lin B C, Lu D, Lu Y, et al. 2023. Noisy intermediate-scale quantum computers. Front. Phys., 18: 21308 doi: 10.1007/s11467-022-1249-z
[38]	Chern A, Knöppel F, Pinkall U, Schröder P. 2017. Inside fluids: Clebsch maps for visualization and processing. ACM Trans. Graphics, 36: 1-11
[39]	Childs A M, Maslov D, Nam Y, Ross N J, Su Y. 2018. Toward the first quantum simulation with quantum speedup. Proc. Natl. Acad. Sci. U.S.A., 115: 9456-9461 doi: 10.1073/pnas.1801723115
[40]	Chiribella G, D’Ariano G M, Perinotti P. 2008. Quantum circuit architecture. Phys. Rev. Lett., 101: 060401 doi: 10.1103/PhysRevLett.101.060401
[41]	Cirac J I, Zoller P. 2012. Goals and opportunities in quantum simulation. Nat. Phys., 8: 264-266 doi: 10.1038/nphys2275
[42]	Costa P C S, An D, Sanders Y R, Su Y, Babbush R, Berry D W. 2022. Optimal scaling quantum linear-systems solver via discrete adiabatic theorem. PRX Quantum, 3: 040303 doi: 10.1103/PRXQuantum.3.040303
[43]	Cottet G H, Koumoutsakos P. 2000. Vortex Methods: Theory and Practice. Cambridge University Press.
[44]	Daley A J, Bloch I, Kokail C, Flannigan S, Pearson N, Troyer M, Zoller P. 2022. Practical quantum advantage in quantum simulation. Nature, 607: 667-676 doi: 10.1038/s41586-022-04940-6
[45]	Das A, Chakrabarti B K. 2008. Colloquium: Quantum annealing and analog quantum computation. Rev. Mod. Phys., 80: 1061-1081 doi: 10.1103/RevModPhys.80.1061
[46]	Deutsch D. 1985. Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci., 400: 97-117
[47]	Esmaeilifar E, Ahn D, Myong R S. 2024. Quantum algorithm for nonlinear Burgers’ equation for high-speed compressible flows. Phys. Fluids, 36: 106110 doi: 10.1063/5.0231994
[48]	Ferreira G J, Loss D. 2013. Magnetically defined qubits on 3D topological insulators. Phys. Rev. Lett., 111: 106802 doi: 10.1103/PhysRevLett.111.106802
[49]	Feynman R P. 1982. Simulating physics with computers. Int. J. Theor. Phys., 21: 467-488 doi: 10.1007/BF02650179
[50]	Feynman R, Leighton R, Sands M. 2015. The Feynman Lectures on Physics, Vol. II: The New Millennium Edition: Mainly Electromagnetism and Matter. Basic Books.
[51]	Frisch U, Hasslacher B, Pomeau Y. 1986. Lattice-gas automata for the Navier-Stokes equation. Phys. Rev. Lett., 56: 1505-1508 doi: 10.1103/PhysRevLett.56.1505
[52]	Fukagata K. 2022. Towards quantum computing of turbulence. Nat. Comput. Sci., 2: 68-69 doi: 10.1038/s43588-022-00189-1
[53]	Georgescu I M, Ashhab S, Nori F. 2014. Quantum simulation. Rev. Mod. Phys., 86: 153-185 doi: 10.1103/RevModPhys.86.153
[54]	Gilyén A, Su Y, Low G H, Wiebe N. 2019. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. 193–204.
[55]	Giovannetti V, Lloyd S, Maccone L. 2008. Quantum random access memory. Phys. Rev. Lett., 100: 160501 doi: 10.1103/PhysRevLett.100.160501
[56]	Gourianov N, Lubasch M, Dolgov S, van den Berg Q Y, Babaee H, Givi P, Kiffner M, Jaksch D. 2022. A quantum-inspired approach to exploit turbulence structures. Nat. Comput. Sci., 2: 30-37 doi: 10.1038/s43588-021-00181-1
[57]	Grant E, Wossnig L, Ostaszewski M, Benedetti M. 2019. An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum, 3: 214 doi: 10.22331/q-2019-12-09-214
[58]	Großardt A. 2024. Nonlinear-ancilla aided quantum algorithm for nonlinear Schrödinger equations, arXiv: 2403.10102.
[59]	Grover L K. 1996. A fast quantum mechanical algorithm for database search. Proceedings of the 28th Annual ACM Symposium on Theory of Computing. 212–219.
[60]	Gupta S, Zia R. 2001. Quantum neural networks. J. Comput. System Sci., 63: 355-383 doi: 10.1006/jcss.2001.1769
[61]	Han Z, Yang Y. 2025. Quantum-enriched large-eddy simulation with the incompressible Schrödinger flow. Acta Sin. Mech., 41: 324054 doi: 10.1007/s10409-024-24054-x
[62]	Hangleiter D, Eisert J. 2023. Computational advantage of quantum random sampling. Rev. Mod. Phys., 95: 035001 doi: 10.1103/RevModPhys.95.035001
[63]	Harrow A W, Hassidim A, Lloyd S. 2009. Quantum algorithm for linear systems of equations. Phys. Rev. Lett., 103: 150502 doi: 10.1103/PhysRevLett.103.150502
[64]	Hibat-Allah M, Mauri M, Carrasquilla J, Perdomo-Ortiz A. 2024. A framework for demonstrating practical quantum advantage: comparing quantum against classical generative models. Commun. Phys., 7: 68 doi: 10.1038/s42005-024-01552-6
[65]	Hu J, Jin S, Liu N, Zhang L. 2024. Quantum circuits for partial differential equations via Schrödingerisation, arXiv: 2403.10032.
[66]	Ishida S, Wojtan C, Chern A. 2022. Hidden Degrees of Freedom in Implicit Vortex Filaments. ACM Trans. Graph., 41: 241
[67]	Itani W, Sreenivasan K R, Succi S. 2024. Quantum algorithm for lattice Boltzmann (QALB) simulation of incompressible fluids with a nonlinear collision term. Phys. Fluids, 36: 017112 doi: 10.1063/5.0176569
[68]	Itani W, Succi S. 2022. Analysis of Carleman linearization of lattice Boltzmann. Fluids, 7: 24 doi: 10.3390/fluids7010024
[69]	Jain N, Landman J, Mathur N, Kerenidis I. 2024. Quantum Fourier Networks for Solving Parametric PDEs. Quantum Sci. Technol., 9: 35026 doi: 10.1088/2058-9565/ad42ce
[70]	Jaksch D, Givi P, Daley A J, Rung T. 2023. Variational quantum algorithms for computational fluid dynamics. AIAA J., 61: 1885-1894 doi: 10.2514/1.J062426
[71]	Jin S, Liu N, Ma C. 2023. Quantum simulation of Maxwell’s equations via Schrödingerisation, arXiv: 2308.08408.
[72]	Jin S, Liu N, Yu Y. 2023. Quantum simulation of partial differential equations: Applications and detailed analysis. Phys. Rev. A, 108: 032603 doi: 10.1103/PhysRevA.108.032603
[73]	Jin S, Liu N, Yu Y. 2023. Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations. J. Comput. Phys., 487: 112149 doi: 10.1016/j.jcp.2023.112149
[74]	Jin S, Liu N, Yu Y. 2024. Quantum simulation of partial differential equations via Schrödingerization. Phys. Rev. Lett., 133: 230602 doi: 10.1103/PhysRevLett.133.230602
[75]	Jin S, Liu N, Yu Y. 2024. Quantum simulation of the Fokker-Planck equation via Schrödingerization, arXiv: 2404.13585.
[76]	Jin S, Liu N. 2024. Quantum algorithms for nonlinear partial differential equations. Bull. Sci. Math., 194: 103457 doi: 10.1016/j.bulsci.2024.103457
[77]	Joseph I. 2020. Koopman-von Neumann approach to quantum simulation of nonlinear classical dynamics. Phys. Rev. Res., 2: 043102 doi: 10.1103/PhysRevResearch.2.043102
[78]	Kempe J. 2003. Quantum random walks: An introductory overview. Contemp. Phys., 44: 307-327 doi: 10.1080/00107151031000110776
[79]	Kim Y, Eddins A, Anand S, Wei K X, van den Berg E, Rosenblatt S, Nayfeh H, Wu Y, Zaletel M, Temme K, et al. 2023. Evidence for the utility of quantum computing before fault tolerance. Nature, 618: 500-505 doi: 10.1038/s41586-023-06096-3
[80]	Kjaergaard M, Schwartz M E, Braumüller J, Krantz P, Wang J I J, Gustavsson S, Oliver W D. 2020. Superconducting qubits: Current state of play. Annu. Rev. Condens. Matter Phys., 11: 369-395 doi: 10.1146/annurev-conmatphys-031119-050605
[81]	Kocherla S, Song Z, Chrit F E, Gard B, Dumitrescu E F, Alexeev A, Bryngelson S H. 2024. Fully quantum algorithm for mesoscale fluid simulations with application to partial differential equations. AVS Quantum Sci., 6: 033806 doi: 10.1116/5.0217675
[82]	Kuang Z, Xu Y, Huang Q, Yang J, Kihal C E, Hu H. 2025. Quantum computing with error mitigation for data-driven computational homogenization. Compos. Struct., 351: 118625 doi: 10.1016/j.compstruct.2024.118625
[83]	Kumar E D, Frankel S H. 2024. Decomposition of nonlinear collision operator in quantum lattice Boltzmann algorithm, arXiv: 2408.00387.
[84]	Kumar E D, Frankel S H. 2024. Quantum circuit model for lattice Boltzmann fluid flow simulations, arXiv: 2405.08669.
[85]	Kuya Y, Komatsu K, Yonaga K, Kobayashi H. 2024. Quantum annealing-based algorithm for lattice gas automata. Comput. Fluids, 274: 106238 doi: 10.1016/j.compfluid.2024.106238
[86]	Ladd T D, Jelezko F, Laflamme R, Nakamura Y, Monroe C, O’Brien J L. 2010. Quantum computers. Nature, 464: 45-53 doi: 10.1038/nature08812
[87]	Lapworth L. 2022. A hybrid quantum-classical CFD methodology with benchmark HHL solutions, arXiv: 2206.00419.
[88]	Leibfried D, Blatt R, Monroe C, Wineland D. 2003. Quantum dynamics of single trapped ions. Rev. Mod. Phys., 75: 281-324 doi: 10.1103/RevModPhys.75.281
[89]	Lewis D, Eidenbenz S, Nadiga B, Subaşı Y. 2024. Limitations for quantum algorithms to solve turbulent and chaotic systems. Quantum, 8: 1509 doi: 10.22331/q-2024-10-24-1509
[90]	Liao S. 2024. A general frame of quantum simulation for nonlinear partial differential equations, arXiv: 2406.15821.
[91]	Lin L, Tong Y. 2020. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum, 4: 361 doi: 10.22331/q-2020-11-11-361
[92]	Liu B, Zhu L, Yang Z, He G. 2023. Quantum implementation of numerical methods for convectiondiffusion equations: toward computational fluid dynamics. Commun. Comput. Phys., 33: 425-451 doi: 10.4208/cicp.OA-2022-0081
[93]	Liu H Y, Lin X, Chen Z Y, Xue C, Sun T P, Li Q S, Zhuang X N, Wang Y J, Wu Y C, Gong M, et al. 2024. Simulation of open quantum systems on universal quantum computers, arXiv: 2405.20712.
[94]	Liu J P, Kolden H O, Krovi H K, Loureiro N F, Trivisa K, Childs A M. 2021. Effcient quantum algorithm for dissipative nonlinear differential equations. Proc. Natl. Acad. Sci. U.S.A., 118: e2026805118 doi: 10.1073/pnas.2026805118
[95]	Liu Y, Ke Y, Zhou J, Liu Y, Luo H, Wen S, Fan D. 2017. Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements. Sci. Rep., 7: 44096 doi: 10.1038/srep44096
[96]	Lloyd S, Palma G D, Gokler C, Kiani B, Liu Z W, Marvian M, Tennie F, Palmer T. 2020. Quantum algorithm for nonlinear differential equations, arXiv: 2011.06571.
[97]	Loss D, DiVincenzo D P. 1998. Quantum computation with quantum dots. Phys. Rev. A, 57: 120-126 doi: 10.1103/PhysRevA.57.120
[98]	Louie S G, Chan Y H, da Jornada F H, Li Z, Qiu D Y. 2021. Discovering and understanding materials through computation. Nat. Mater., 20: 728-735 doi: 10.1038/s41563-021-01015-1
[99]	Low G H, Chuang I L. 2017. Optimal Hamiltonian Simulation by Quantum Signal Processing. Phys. Rev. Lett., 118: 010501 doi: 10.1103/PhysRevLett.118.010501
[100]	Low G H, Chuang I L. 2019. Hamiltonian simulation by qubitization. Quantum, 3: 163 doi: 10.22331/q-2019-07-12-163
[101]	Lu Z, Yang Y. 2024. Quantum computing of reacting flows via Hamiltonian simulation. Proc. Combust. Inst., 40: 105440 doi: 10.1016/j.proci.2024.105440
[102]	Lubasch M, Joo J, Moinier P, Kiffner M, Jaksch D. 2020. Variational quantum algorithms for nonlinear problems. Phys. Rev. A, 101: 010301 doi: 10.1103/PhysRevA.101.010301
[103]	Markidis S. 2022. On physics-informed neural networks for quantum computers. Front. Appl. Math. Stat., 8: 1036711 doi: 10.3389/fams.2022.1036711
[104]	McArdle S, Endo S, Aspuru-Guzik A, Benjamin S C, Yuan X. 2020. Quantum computational chemistry. Rev. Mod. Phys., 92: 015003 doi: 10.1103/RevModPhys.92.015003
[105]	Meng Z, Song C, Yang Y. 2025. Challenges of simulating fluid flows on near-term quantum computer. Sci. China-Phys. Mech. Astron.,
[106]	Meng Z, Yang Y. 2023. Quantum computing of fluid dynamics using the hydrodynamic Schrödinger equation. Phys. Rev. Res., 5: 033182 doi: 10.1103/PhysRevResearch.5.033182
[107]	Meng Z, Yang Y. 2024. Lagrangian dynamics and regularity of the spin Euler equation. J. Fluid Mech., 985: A34 doi: 10.1017/jfm.2024.319
[108]	Meng Z, Yang Y. 2024. Quantum spin representation for the Navier-Stokes equation. Phys. Rev. Res., 6: 043130 doi: 10.1103/PhysRevResearch.6.043130
[109]	Meng Z, Zhong J, Xu S, Wang K, Chen J, Jin F, Zhu X, Gao Y, Wu Y, Zhang C, et al. 2024. Simulating unsteady flows on a superconducting quantum processor. Commun. Phys., 7: 349 doi: 10.1038/s42005-024-01845-w
[110]	Mujal P, Martínez-Peña R, Nokkala J, García-Beni J, Giorgi G L, Soriano M C, Zambrini R. 2021. Opportunities in quantum reservoir computing and extreme learning machines. Adv. Quantum Technol., 4: 2100027 doi: 10.1002/qute.202100027
[111]	Nayak C, Simon S H, Stern A, Freedman M, Das Sarma S. 2008. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys., 80: 1083-1159 doi: 10.1103/RevModPhys.80.1083
[112]	Nemkov N A, Kiktenko E O, Luchnikov I A, Fedorov A K. 2023. Effcient variational synthesis of quantum circuits with coherent multi-start optimization. Quantum, 7: 993 doi: 10.22331/q-2023-05-04-993
[113]	Novikau I, Joseph I. 2025. Quantum algorithm for the advection-diffusion equation and the Koopmanvon Neumann approach to nonlinear dynamical systems. Comput. Phys. Commun., 309: 109498 doi: 10.1016/j.cpc.2025.109498
[114]	Over P, Bengoechea S, Brearley P, Laizet S, Rung T. 2024. Quantum algorithm for the advectiondiffusion equation with optimal success probability, arXiv: 2410.07909.
[115]	Peddinti R D, Pisoni S, Marini A, Lott P, Argentieri H, Tiunov E, Aolita L. 2024. Quantum-inspired framework for computational fluid dynamics. Commun. Phys., 7: 135 doi: 10.1038/s42005-024-01623-8
[116]	Pfeffer P, Heyder F, Schumacher J. 2022. Hybrid quantum-classical reservoir computing of thermal convection flow. Phys. Rev. Res., 4: 033176 doi: 10.1103/PhysRevResearch.4.033176
[117]	Pfeffer P, Heyder F, Schumacher J. 2023. Reduced-order modeling of two-dimensional turbulent Rayleigh-Bénard flow by hybrid quantum-classical reservoir computing. Phys. Rev. Res., 5: 043242 doi: 10.1103/PhysRevResearch.5.043242
[118]	Pope S B. 2011. Simple models of turbulent flows. Phys. Fluids, 23: 011301 doi: 10.1063/1.3531744
[119]	Prawer S, Greentree A D. 2008. Diamond for quantum computing. Science, 320: 1601-1602 doi: 10.1126/science.1158340
[120]	Rodríguez J D D, Pérez A P, Fernández E I, Valera A J J. 2024. A quantum annealing approach to fluid dynamics problems solving Navier-Stokes equations. 9th International Conference on Smart and Sustainable Technologies (SpliTech).
[121]	Saffman M, Walker T G, Mølmer K. 2010. Quantum information with Rydberg atoms. Rev. Mod. Phys., 82: 2313-2363 doi: 10.1103/RevModPhys.82.2313
[122]	Sajjan M, Li J, Selvarajan R, Sureshbabu S H, Kale S S, Gupta R, Singh V, Kais S. 2022. Quantum machine learning for chemistry and physics. Chem. Soc. Rev., 51: 6475-6573 doi: 10.1039/D2CS00203E
[123]	Salasnich L, Succi S, Tiribocchi A. 2024. Quantum wave representation of dissipative fluids. Int. J. Mod. Phys. C, 35: 2450100 doi: 10.1142/S0129183124501006
[124]	Sanavio C, Mauri E, Succi S. 2024. Explicit quantum circuit for simulating the advection-diffusionreaction dynamics, arXiv: 2410.05876.
[125]	Sanavio C, Scatamacchia R, de Falco C, Succi S. 2024. Three Carleman routes to the quantum simulation of classical fluids. Phys. Fluids, 36: 057143 doi: 10.1063/5.0204955
[126]	Santagati R, Aspuru-Guzik A, Babbush R, Degroote M, González L, Kyoseva E, Moll N, Oppel M, Parrish R M, Rubin N C, et al. 2024. Drug design on quantum computers. Nat. Phys., 20: 549-557 doi: 10.1038/s41567-024-02411-5
[127]	Sato Y, Kondo R, Hamamura I, Onodera T, Yamamoto N. 2024. Hamiltonian simulation for hyperbolic partial differential equations by scalable quantum circuits. Phys. Rev. Res., 6: 033246 doi: 10.1103/PhysRevResearch.6.033246
[128]	Schlimgen A W, Head-Marsden K, Sager L M, Narang P, Mazziotti D A. 2021. Quantum simulation of open quantum systems using a unitary decomposition of operators. Phys. Rev. Lett., 127: 270503 doi: 10.1103/PhysRevLett.127.270503
[129]	Schmid P J. 2010. Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech., 656: 5-28 doi: 10.1017/S0022112010001217
[130]	Shao C. 2018. Quantum algorithms to matrix multiplication, arXiv: 1803.01601.
[131]	Shao H J, Wang Y X, Zhu D Z, Zhu Y S, Sun H N, Chen S Y, Zhang C, Fan Z J, Deng Y, Yao X C, et al. 2024. Antiferromagnetic phase transition in a 3D fermionic Hubbard model. Nature, 632: 267-272 doi: 10.1038/s41586-024-07689-2
[132]	Shor P W. 1994. Algorithms for quantum computation: discrete logarithms and factoring. Proceedings 35th Annual Symposium on Foundations of Computer Science. 124–134.
[133]	Shor P W. 1997. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput., 26: 1484 doi: 10.1137/S0097539795293172
[134]	Singh H, McCulloch E, Gopalakrishnan S, Vasseur R. 2024. Emergence of Navier-Stokes hydrodynamics in chaotic quantum circuits, arXiv: 2405.13892.
[135]	Song Z, Deaton R, Gard B, Bryngelson S H. 2025. Incompressible Navier-Stokes solve on noisy quantum hardware via a hybrid quantum-classical scheme. Comput. Fluids, 288: 106507 doi: 10.1016/j.compfluid.2024.106507
[136]	Steijl R, Barakos G N. 2018. Parallel evaluation of quantum algorithms for computational fluid dynamics. Comput. Fluids, 173: 22-28 doi: 10.1016/j.compfluid.2018.03.080
[137]	Su H, Xiong S, Yang Y. 2024. Quantum state preparation for a velocity field based on the spherical Clebsch wave function, arXiv: 2406.04652.
[138]	Subaşı Y, Somma R D, Orsucci D. 2019. Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing. Phys. Rev. Lett., 122: 060504 doi: 10.1103/PhysRevLett.122.060504
[139]	Succi S, Itani W, Sanavio C, Sreenivasan K R, Steijl R. 2024. Ensemble fluid simulations on quantum computers. Comput. Fluids, 270: 106148 doi: 10.1016/j.compfluid.2023.106148
[140]	Succi S, Itani W, Sreenivasan K, Steijl R. 2023. Quantum computing for fluids: Where do we stand? Europhys. Lett., 144: 10001
[141]	Tennie F, Laizet S, Lloyd S, Magri L. 2025. Quantum computing for nonlinear differential equations and turbulence. Nat. Rev. Phys.,
[142]	Tennie F, Magri L. 2024. Solving nonlinear differential equations on quantum computers: A FokkerPlanck approach, arXiv: 2401.13500.
[143]	Terhal B M. 2015. Quantum error correction for quantum memories. Rev. Mod. Phys., 87: 307-346 doi: 10.1103/RevModPhys.87.307
[144]	Thouless D J, Ao P, Niu Q. 1993. Vortex dynamics in superfluids and the Berry phase. Physica A: Stat. Mech. Appl., 200: 42-49 doi: 10.1016/0378-4371(93)90503-V
[145]	Tilly J, Chen H, Cao S, Picozzi D, Setia K, Li Y, Grant E, Wossnig L, Rungger I, Booth G H, et al. 2022. The variational quantum eigensolver: a review of methods and best practices. Phys. Rep., 986: 1-128 doi: 10.1016/j.physrep.2022.08.003
[146]	Tsemo P B, Jayashankar A, Sugisaki K, Baskaran N, Chakraborty S, Prasannaa V S. 2024. Enhancing the Harrow-Hassidim-Lloyd (HHL) algorithm in systems with large condition numbers, arXiv: 2407.21641.
[147]	Vandersypen L M K, Chuang I L. 2005. NMR techniques for quantum control and computation. Rev. Mod. Phys., 76: 1037-1069 doi: 10.1103/RevModPhys.76.1037
[148]	Wang B, Meng Z, Zhao Y, Yang Y. 2025. Quantum lattice Boltzmann method for simulating nonlinear fluid dynamics, arXiv: 2502.16568.
[149]	Wawrzyniak D, Winter J, Schmidt S, Indinger T, Janßen C F, Schramm U, Adams N A. 2025. A quantum algorithm for the lattice-Boltzmann method advection-diffusion equation. Comput. Phys. Commun., 306: 109373 doi: 10.1016/j.cpc.2024.109373
[150]	Weißmann S, Pinkall U, Schröder P. 2014. Smoke rings from smoke. ACM Trans. Graph., 33: 4
[151]	Williams C A, Gentile A A, Elfving V E, Berger D, Kyriienko O. 2024. Quantum iterative methods for solving differential equations with application to computational fluid dynamics, arXiv: 2404.08605.
[152]	Wootters W K, Zurek W H. 1982. A single quantum cannot be cloned. Nature, 299: 802-803 doi: 10.1038/299802a0
[153]	Wright L, Keever C M, First J T, Johnston R, Tillay J, Chaney S, Rosenkranz M, Lubasch M. 2024. Noisy intermediate-scale quantum simulation of the one-dimensional wave equation, arXiv: 2402.19247.
[154]	Wu Y, Bao W S, Cao S, Chen F, Chen M C, Chen X, Chung T H, Deng H, Du Y, Fan D, et al. 2021. Strong quantum computational advantage using a superconducting quantum processor. Phys. Rev. Lett., 127: 180501 doi: 10.1103/PhysRevLett.127.180501
[155]	Xiao P, Zheng M, Jiao A, Yang X, Lu L. 2024. Quantum DeepONet: Neural Operators Accelerated by Quantum Computing, arXiv: 2409.15683.
[156]	Xiao Y, Yang L M, Shu C, Chew S C, Khoo B C, Cui Y D, Liu Y Y. 2024. Physics-informed quantum neural network for solving forward and inverse problems of partial differential equations. Phys. Fluids, 36 .
[157]	Xiong S, Tong Y, He X, Yang S, Yang C, Zhu B. 2021. Nonseparable Symplectic Neural Networks. International Conference on Learning Representations.
[158]	Xu X, Sun J, Endo S, Li Y, Benjamin S C, Yuan X. 2021. Variational algorithms for linear algebra. Sci. Bull., 66: 2181-2188 doi: 10.1016/j.scib.2021.06.023
[159]	Xu Y, Yang J, Kuang Z, Huang Q, Huang W, Hu H. 2024. Quantum computing enhanced distanceminimizing data-driven computational mechanics. Comput. Meth. Appl. Mech. Eng., 419: 116675 doi: 10.1016/j.cma.2023.116675
[160]	Yadav S. 2023. Qpde: Quantum neural network based stabilization parameter prediction for numerical solvers for partial differential equations. AppliedMath, 3: 552-562 doi: 10.3390/appliedmath3030029
[161]	Ye C C, An N B, Ma T Y, Dou M H, Bai W, Sun D J, Chen Z Y, Guo G P. 2024. A hybrid quantum-classical framework for computational fluid dynamics. Phys. Fluids, 36: 127111 doi: 10.1063/5.0238193
[162]	Yuan X, Chen Z, Liu Y, Xie Z, Liu Y, Jin X, Wen X, Tang H. 2023. Quantum support vector machines for aerodynamic classification. Int. Comput., 2: 0057 doi: 10.34133/icomputing.0057
[163]	Zamora A D B, Budinski L, Niemimäki O, Lahtinen V. 2025. Effcient quantum lattice gas automata. Comput. Fluids, 286: 106476 doi: 10.1016/j.compfluid.2024.106476
[164]	Zhong H S, Deng Y H, Qin J, Wang H, Chen M C, Peng L C, Luo Y H, Wu D, Gong S Q, Su H, et al. 2021. Phase-programmable Gaussian Boson sampling using stimulated squeezed light. Phys. Rev. Lett., 127: 180502 doi: 10.1103/PhysRevLett.127.180502
[165]	Zhong H S, Wang H, Deng Y H, Chen M C, Peng L C, Luo Y H, Qin J, Wu D, Ding X, Hu Y, et al. 2020. Quantum computational advantage using photons. Science, 370: 1460-1463 doi: 10.1126/science.abe8770
[166]	Zylberman J, Di Molfetta G, Brachet M, Loureiro N F, Debbasch F. 2022. Quantum simulations of hydrodynamics via the Madelung transformation. Phys. Rev. A, 106: 032408 doi: 10.1103/PhysRevA.106.032408