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 > Corrected proof

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( 8260 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, China
2.
Department of Engineering Mechanics, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China
3.
HEDPS-CAPT, Peking University, Beijing 100871, 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(165)

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 Q, Yuan X X, Tu G H, Guo Q L. 2023. Computational fluid dynamics 2035 vision in China. Science Press). Chen J Q, Yuan X X, Tu G H, Guo Q L. 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]	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.
[11]	Babukhin D V. 2023. Harrow−Hassidim−Lloyd algorithm without ancilla postselection. Phys. Rev. A, 107: 042408. doi: 10.1103/PhysRevA.107.042408
[12]	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
[13]	Basheer A, Afham A, Goyal S K. 2020. Quantum k-nearest neighbors algorithm. arXiv preprint arXiv: 2003.09187,
[14]	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
[15]	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
[16]	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
[17]	Bernstein D, Lange T. 2017. Post-quantum cryptography. Nature, 549: 188-194. doi: 10.1038/nature23461
[18]	Bharadwaj S S, Sreenivasan K R. 2020. Quantum computation of fluid dynamics. Indian Acad. Sci. Conf. Ser., 3: 77.
[19]	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
[20]	Bharadwaj S S, Sreenivasan K R. 2024a. Compact quantum algorithms that can potentially maintain quantum advantage for solving time-dependent differential equations. arXiv: 2405.09767.
[21]	Bharadwaj S S, Sreenivasan K R. 2024b. Simulating fluid flows with quantum computing. arXiv: 2409.09736.
[22]	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
[23]	Biamonte J, Wittek P, Pancotti N, Rebentrost P, Wiebe N, Lloyd S. 2017. Quantum machine learning. Nature, 549: 195-202. doi: 10.1038/nature23474
[24]	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
[25]	Borle A, Lomonaco S J. 2022. How viable is quantum annealing for solving linear algebra problems? arXiv: 2206.10576.
[26]	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
[27]	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
[28]	Brüstle N, Wiebe N. 2024. Quantum and classical algorithms for nonlinear unitary dynamics. arXiv: 2407.07685.
[29]	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
[30]	Budinski L. 2022. Quantum algorithm for the Navier−Stokes equations by using the stream function vorticity formulation and the lattice Boltzmann method. Int. J. Quantum Inf., 20: 2150039. doi: 10.1142/S0219749921500398
[31]	Buluta I, Nori F. 2009. Quantum simulators. Science, 326: 108-111. doi: 10.1126/science.1177838
[32]	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
[33]	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
[34]	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
[35]	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
[36]	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
[37]	Chern A, Knöppel F, Pinkall U, Schröder P. 2017. Inside fluids: Clebsch maps for visualization and processing. ACM Trans. Graphics, 36: 1-11.
[38]	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
[39]	Chiribella G, D’Ariano G M, Perinotti P. 2008. Quantum circuit architecture. Phys. Rev. Lett., 101: 060401. doi: 10.1103/PhysRevLett.101.060401
[40]	Cirac J I, Zoller P. 2012. Goals and opportunities in quantum simulation. Nat. Phys., 8: 264-266. doi: 10.1038/nphys2275
[41]	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
[42]	Cottet G H, Koumoutsakos P. 2000. Vortex methods: Theory and practice. Cambridge University Press.
[43]	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
[44]	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
[45]	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.
[46]	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
[47]	Ferreira G J, Loss D. 2013. Magnetically defined qubits on 3D topological insulators. Phys. Rev. Lett., 111: 106802. doi: 10.1103/PhysRevLett.111.106802
[48]	Feynman R P. 1982. Simulating physics with computers. Int. J. Theor. Phys., 21: 467-488. doi: 10.1007/BF02650179
[49]	Feynman R, Leighton R, Sands M. 2015. The Feynman lectures on physics, Vol. II: The new millennium edition: Mainly electromagnetism and matter. Basic Books.
[50]	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
[51]	Fukagata K. 2022. Towards quantum computing of turbulence. Nat. Comput. Sci., 2: 68-69. doi: 10.1038/s43588-022-00189-1
[52]	Georgescu I M, Ashhab S, Nori F. 2014. Quantum simulation. Rev. Mod. Phys., 86: 153-185. doi: 10.1103/RevModPhys.86.153
[53]	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.
[54]	Giovannetti V, Lloyd S, Maccone L. 2008. Quantum random access memory. Phys. Rev. Lett., 100: 160501. doi: 10.1103/PhysRevLett.100.160501
[55]	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
[56]	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
[57]	Großardt A. 2024. Nonlinear-ancilla aided quantum algorithm for nonlinear Schrödinger equations. arXiv: 2403.10102.
[58]	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.
[59]	Gupta S, Zia R. 2001. Quantum neural networks. J. Comput. System Sci., 63: 355-383. doi: 10.1006/jcss.2001.1769
[60]	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
[61]	Hangleiter D, Eisert J. 2023. Computational advantage of quantum random sampling. Rev. Mod. Phys., 95: 035001. doi: 10.1103/RevModPhys.95.035001
[62]	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
[63]	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
[64]	Hu J, Jin S, Liu N, Zhang L. 2024. Quantum circuits for partial differential equations via Schrödingerisation. arXiv: 2403.10032.
[65]	Ishida S, Wojtan C, Chern A. 2022. Hidden degrees of freedom in implicit vortex filaments. ACM Trans. Graph., 41: 241.
[66]	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
[67]	Itani W, Succi S. 2022. Analysis of carleman linearization of lattice Boltzmann. Fluids, 7: 24. doi: 10.3390/fluids7010024
[68]	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
[69]	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
[70]	Jin S, Liu N, Ma C. 2023a. Quantum simulation of Maxwell’s equations via Schrödingerisation. arXiv: 2308.08408.
[71]	Jin S, Liu N, Yu Y. 2023b. Quantum simulation of partial differential equations: Applications and detailed analysis. Phys. Rev. A, 108: 032603. doi: 10.1103/PhysRevA.108.032603
[72]	Jin S, Liu N, Yu Y. 2023c. 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
[73]	Jin S, Liu N, Yu Y. 2024a. Quantum simulation of partial differential equations via Schrödingerization. Phys. Rev. Lett., 133: 230602. doi: 10.1103/PhysRevLett.133.230602
[74]	Jin S, Liu N, Yu Y. 2024b. Quantum simulation of the Fokker−Planck equation via Schrödingerization. arXiv: 2404.13585.
[75]	Jin S, Liu N. 2024. Quantum algorithms for nonlinear partial differential equations. Bull. Sci. Math., 194: 103457. doi: 10.1016/j.bulsci.2024.103457
[76]	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
[77]	Kempe J. 2003. Quantum random walks: An introductory overview. Contemp. Phys., 44: 307-327. doi: 10.1080/00107151031000110776
[78]	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
[79]	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
[80]	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
[81]	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
[82]	Kumar E D, Frankel S H. 2024. Decomposition of nonlinear collision operator in quantum lattice Boltzmann algorithm. arXiv: 2408.00387.
[83]	Kumar E D, Frankel S H. 2025. Quantum unitary matrix representation of the lattice Boltzmann model for low Reynolds fluid flow simulation. AVS Quantum Sci., 7: 013802. doi: 10.1116/5.0245082
[84]	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
[85]	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
[86]	Lapworth L. 2022. A hybrid quantum-classical CFD methodology with benchmark HHL solutions. arXiv: 2206.00419.
[87]	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
[88]	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
[89]	Liao S. 2024. A general frame of quantum simulation for nonlinear partial differential equations. arXiv: 2406.15821.
[90]	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
[91]	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
[92]	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.
[93]	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
[94]	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
[95]	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.
[96]	Loss D, DiVincenzo D P. 1998. Quantum computation with quantum dots. Phys. Rev. A, 57: 120-126. doi: 10.1103/PhysRevA.57.120
[97]	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
[98]	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
[99]	Low G H, Chuang I L. 2019. Hamiltonian simulation by qubitization. Quantum, 3: 163. doi: 10.22331/q-2019-07-12-163
[100]	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
[101]	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
[102]	Markidis S. 2022. On physics-informed neural networks for quantum computers. Front. Appl. Math. Stat., 8: 1036711. doi: 10.3389/fams.2022.1036711
[103]	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
[104]	Meng Z, Song C, Yang Y. 2025. Challenges of simulating fluid flows on near-term quantum computer. Sci. China-Phys. Mech. Astron. (in press
[105]	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
[106]	Meng Z, Yang Y. 2024a. Lagrangian dynamics and regularity of the spin Euler equation. J. Fluid Mech., 985: A34. doi: 10.1017/jfm.2024.319
[107]	Meng Z, Yang Y. 2024b. Quantum spin representation for the Navier−Stokes equation. Phys. Rev. Res., 6: 043130. doi: 10.1103/PhysRevResearch.6.043130
[108]	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
[109]	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
[110]	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
[111]	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
[112]	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
[113]	Over P, Bengoechea S, Brearley P, Laizet S, Rung T. 2024. Quantum algorithm for the advectiondiffusion equation with optimal success probability. arXiv: 2410.07909.
[114]	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
[115]	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
[116]	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
[117]	Pope S B. 2011. Simple models of turbulent flows. Phys. Fluids, 23: 011301. doi: 10.1063/1.3531744
[118]	Prawer S, Greentree A D. 2008. Diamond for quantum computing. Science, 320: 1601-1602. doi: 10.1126/science.1158340
[119]	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).
[120]	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
[121]	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
[122]	Salasnich L, Succi S, Tiribocchi A. 2024. Quantum wave representation of dissipative fluids. Int. J. Mod. Phys. C, 35: 2450100. doi: 10.1142/S0129183124501006
[123]	Sanavio C, Mauri E, Succi S. 2024a. Explicit quantum circuit for simulating the advection-diffusionreaction dynamics. arXiv: 2410.05876.
[124]	Sanavio C, Scatamacchia R, de Falco C, Succi S. 2024b. Three Carleman routes to the quantum simulation of classical fluids. Phys. Fluids, 36: 057143. doi: 10.1063/5.0204955
[125]	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
[126]	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
[127]	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
[128]	Schmid P J. 2010. Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech., 656: 5-28. doi: 10.1017/S0022112010001217
[129]	Shao C. 2018. Quantum algorithms to matrix multiplication. arXiv: 1803.01601.
[130]	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
[131]	Shor P W. 1994. Algorithms for quantum computation: Discrete logarithms and factoring. Proceedings 35th Annual Symposium on Foundations of Computer Science. 124–134.
[132]	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
[133]	Singh H, McCulloch E, Gopalakrishnan S, Vasseur R. 2024. Emergence of Navier−Stokes hydrodynamics in chaotic quantum circuits. arXiv: 2405.13892.
[134]	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
[135]	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
[136]	Su H, Xiong S, Yang Y. 2024. Quantum state preparation for a velocity field based on the spherical Clebsch wave function. arXiv: 2406.04652.
[137]	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
[138]	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
[139]	Succi S, Itani W, Sreenivasan K, Steijl R. 2023. Quantum computing for fluids: Where do we stand? Europhys. Lett., 144: 10001.
[140]	Tennie F, Laizet S, Lloyd S, Magri L. 2025. Quantum computing for nonlinear differential equations and turbulence. Nat. Rev. Phys., arXiv: 2406.04826.
[141]	Tennie F, Magri L. 2024. Solving nonlinear differential equations on quantum computers: A FokkerPlanck approach. arXiv: 2401.13500.
[142]	Terhal B M. 2015. Quantum error correction for quantum memories. Rev. Mod. Phys., 87: 307-346. doi: 10.1103/RevModPhys.87.307
[143]	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
[144]	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
[145]	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.
[146]	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
[147]	Wang B, Meng Z, Zhao Y, Yang Y. 2025. Quantum lattice Boltzmann method for simulating nonlinear fluid dynamics. arXiv: 2502.16568.
[148]	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
[149]	Weißmann S, Pinkall U, Schröder P. 2014. Smoke rings from smoke. ACM Trans. Graph., 33: 4.
[150]	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.
[151]	Wootters W K, Zurek W H. 1982. A single quantum cannot be cloned. Nature, 299: 802-803. doi: 10.1038/299802a0
[152]	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.
[153]	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
[154]	Xiao P, Zheng M, Jiao A, Yang X, Lu L. 2024a. Quantum DeepONet: Neural operators accelerated by quantum computing. arXiv: 2409.15683.
[155]	Xiao Y, Yang L M, Shu C, Chew S C, Khoo B C, Cui Y D, Liu Y Y. 2024b. Physics-informed quantum neural network for solving forward and inverse problems of partial differential equations. Phys. Fluids, 36 .
[156]	Xiong S, Tong Y, He X, Yang S, Yang C, Zhu B. 2021. Nonseparable symplectic neural networks. International Conference on Learning Representations. arXiv: 2010.12636.
[157]	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
[158]	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
[159]	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
[160]	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
[161]	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
[162]	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
[163]	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
[164]	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
[165]	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