Quantum dynamics simulation of the advection-diffusion equation

Alipanah, Hirad; Zhang, Feng; Yao, Yong-Xin; Thompson, Richard; Nguyen, Nam; Liu, Junyu; Givi, Peyman; McDermott, Brian J.; Mendoza-Arenas, Juan José

doi:10.1103/ndc3-bdwt

Quantum dynamics simulation of the advection-diffusion equation

Journal Article · Fri Dec 19 00:00:00 EST 2025 · Physical Review Research

DOI:https://doi.org/10.1103/ndc3-bdwt· OSTI ID:3012376

^[1]; ^[2]; ^[2]; ^[3]; Nguyen, Nam ^[4]; Liu, Junyu ^[1]; ^[1]; ^[5]; ^[1]

Univ. of Pittsburgh, PA (United States)
Ames Laboratory, and Iowa State Univ., Ames, IA (United States)
Boeing Research and Technology (BR&T), Huntsville, AL (United States)
Boeing Research and Technology (BR&T), Huntington Beach, CA (United States)
Naval Nuclear Laboratory, Schenectady, NY (United States)

The advection-diffusion equation is simulated via several quantum algorithms. Three formulations are considered: (1) Trotterization, (2) variational quantum time evolution (VarQTE), and (3) adaptive variational quantum dynamics simulation (AVQDS). These schemes were originally developed for the Hamiltonian simulation of many-body quantum systems. The finite-difference discretized operator of the transport equation is formulated as a Hamiltonian and solved without the need for ancillary qubits. Computations are conducted on a quantum simulator (IBM Qiskit Aer) and a superconducting quantum hardware (IBM Fez). The former emulates the latter without the noise. The actual hardware implementation experiences significant noise. The results of the quantum simulator are compared with data from direct numerical simulation (DNS) with infidelities of the order 10⁻⁵. In the quantum simulator, Trotterization is observed to have the lowest infidelity and is suitable for fault-tolerant computation. The AVQDS algorithm requires the lowest gate count and circuit depth. The VarQTE algorithm is the next best in terms of gate counts, but the number of its optimization variables is directly proportional to the number of qubits. Due to current hardware limitations, Trotterization cannot be implemented, as it has an overwhelmingly large number of operations. Meanwhile, AVQDS and VarQTE can be executed at the hardware level. These algorithms present a new paradigm for computational transport phenomena on quantum computers.

View Accepted Manuscript (DOE)

Research Organization:: Ames Laboratory (AMES), Ames, IA (United States)

Sponsoring Organization:: National Science Foundation (NSF); US Air Force Office of Scientific Research (AFOSR); USDOE Office of Science (SC), Basic Energy Sciences (BES). Materials Sciences & Engineering Division (MSE)

Grant/Contract Number:: 89233018CNR000004; AC02-07CH11358

OSTI ID:: 3012376

Journal Information:: Physical Review Research, Journal Name: Physical Review Research Journal Issue: 4 Vol. 7; ISSN 2643-1564

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (85)

Adaptive Variational Quantum Imaginary Time Evolution Approach for Ground State Preparation Gomes, Niladri; Mukherjee, Anirban; Zhang, Feng Advanced Quantum Technologies, Vol. 4, Issue 12 https://doi.org/10.1002/qute.202100114	journal	October 2021
Quantum algorithm for the advection–diffusion equation simulated with the lattice Boltzmann method Budinski, Ljubomir Quantum Information Processing, Vol. 20, Issue 2 https://doi.org/10.1007/s11128-021-02996-3	journal	February 2021
Investigating and mitigating barren plateaus in variational quantum circuits: a survey Cunningham, Jack; Zhuang, Jun Quantum Information Processing, Vol. 24, Issue 2 https://doi.org/10.1007/s11128-025-04665-1	journal	January 2025
High-entanglement capabilities for variational quantum algorithms: the Poisson equation case Ayoub, Fouad; Baeder, James D. Quantum Information Processing, Vol. 24, Issue 8 https://doi.org/10.1007/s11128-025-04846-y	journal	July 2025
Time-evolution methods for matrix-product states Paeckel, Sebastian; Köhler, Thomas; Swoboda, Andreas Annals of Physics, Vol. 411 https://doi.org/10.1016/j.aop.2019.167998	journal	December 2019
Quantum algorithms for nonlinear partial differential equations Jin, Shi; Liu, Nana Bulletin des Sciences Mathématiques, Vol. 194 https://doi.org/10.1016/j.bulsci.2024.103457	journal	September 2024
Two quantum algorithms for solving the one-dimensional advection–diffusion equation Ingelmann, Julia; Bharadwaj, Sachin S.; Pfeffer, Philipp Computers & Fluids, Vol. 281 https://doi.org/10.1016/j.compfluid.2024.106369	journal	August 2024
Incompressible Navier–Stokes solve on noisy quantum hardware via a hybrid quantum–classical scheme Song, Zhixin; Deaton, Robert; Gard, Bryan Computers & Fluids, Vol. 288 https://doi.org/10.1016/j.compfluid.2024.106507	journal	February 2025
Boundary treatment for variational quantum simulations of partial differential equations on quantum computers Over, Paul; Bengoechea, Sergio; Rung, Thomas Computers & Fluids, Vol. 288 https://doi.org/10.1016/j.compfluid.2024.106508	journal	February 2025
A quantum algorithm for the lattice-Boltzmann method advection-diffusion equation Wawrzyniak, David; Winter, Josef; Schmidt, Steffen Computer Physics Communications, Vol. 306 https://doi.org/10.1016/j.cpc.2024.109373	journal	January 2025
A variational quantum algorithm-based numerical method for solving potential and Stokes flows Liu, Y. Y.; Chen, Z.; Shu, C. Ocean Engineering, Vol. 292 https://doi.org/10.1016/j.oceaneng.2023.116494	journal	January 2024
Quantum computing with trapped ions Haffner, H.; Roos, C.; Blatt, R. Physics Reports, Vol. 469, Issue 4 https://doi.org/10.1016/j.physrep.2008.09.003	journal	December 2008
A review on Quantum Approximate Optimization Algorithm and its variants Blekos, Kostas; Brand, Dean; Ceschini, Andrea Physics Reports, Vol. 1068 https://doi.org/10.1016/j.physrep.2024.03.002	journal	June 2024
Variational quantum simulation of the Fokker–Planck equation applied to quantum radiation reaction Amaro, Óscar; Iñigo Gamiz, Lucas Ivan; Vranic, Marija Journal of Plasma Physics, Vol. 91, Issue 4 https://doi.org/10.1017/S0022377825100652	journal	August 2025
Efficient Step-Merged Quantum Imaginary Time Evolution Algorithm for Quantum Chemistry Gomes, Niladri; Zhang, Feng; Berthusen, Noah F. Journal of Chemical Theory and Computation, Vol. 16, Issue 10 https://doi.org/10.1021/acs.jctc.0c00666	journal	September 2020
Barren plateaus in quantum neural network training landscapes McClean, Jarrod R.; Boixo, Sergio; Smelyanskiy, Vadim N. Nature Communications, Vol. 9, Issue 1 https://doi.org/10.1038/s41467-018-07090-4	journal	November 2018
Noise-induced barren plateaus in variational quantum algorithms Wang, Samson; Fontana, Enrico; Cerezo, M. Nature Communications, Vol. 12, Issue 1 https://doi.org/10.1038/s41467-021-27045-6	journal	November 2021
Variational ansatz-based quantum simulation of imaginary time evolution McArdle, Sam; Jones, Tyson; Endo, Suguru npj Quantum Information, Vol. 5, Issue 1 https://doi.org/10.1038/s41534-019-0187-2	journal	September 2019
Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution Motta, Mario; Sun, Chong; Tan, Adrian T. K. Nature Physics, Vol. 16, Issue 2 https://doi.org/10.1038/s41567-019-0704-4	journal	November 2019
Drug design on quantum computers Santagati, Raffaele; Aspuru-Guzik, Alan; Babbush, Ryan Nature Physics, Vol. 20, Issue 4 https://doi.org/10.1038/s41567-024-02411-5	journal	March 2024
Practical quantum advantage in quantum simulation Daley, Andrew J.; Bloch, Immanuel; Kokail, Christian Nature, Vol. 607, Issue 7920 https://doi.org/10.1038/s41586-022-04940-6	journal	July 2022
Variational quantum evolution equation solver Leong, Fong Yew; Ewe, Wei-Bin; Koh, Dax Enshan Scientific Reports, Vol. 12, Issue 1 https://doi.org/10.1038/s41598-022-14906-3	journal	June 2022
Generalising quantum imaginary time evolution to solve linear partial differential equations Kumar, Swagat; Wilmott, Colin Michael Scientific Reports, Vol. 14, Issue 1 https://doi.org/10.1038/s41598-024-70423-5	journal	August 2024
Numerical solution of nonlinear Schrödinger equation by a hybrid pseudospectral-variational quantum algorithm Köcher, Nikolas; Rose, Hendrik; Bharadwaj, Sachin S. Scientific Reports, Vol. 15, Issue 1 https://doi.org/10.1038/s41598-025-05660-3	journal	July 2025
Variational quantum algorithms Cerezo, M.; Arrasmith, Andrew; Babbush, Ryan Nature Reviews Physics, Vol. 3, Issue 9 https://doi.org/10.1038/s42254-021-00348-9	journal	August 2021
Quantum computing for nonlinear differential equations and turbulence Tennie, Felix; Laizet, Sylvain; Lloyd, Seth Nature Reviews Physics, Vol. 7, Issue 4 https://doi.org/10.1038/s42254-024-00799-w	journal	January 2025
Barren plateaus in variational quantum computing Larocca, Martín; Thanasilp, Supanut; Wang, Samson Nature Reviews Physics, Vol. 7, Issue 4 https://doi.org/10.1038/s42254-025-00813-9	journal	March 2025
A quantum-inspired approach to exploit turbulence structures Gourianov, Nikita; Lubasch, Michael; Dolgov, Sergey Nature Computational Science, Vol. 2, Issue 1 https://doi.org/10.1038/s43588-021-00181-1	journal	January 2022
Trapped-ion quantum computing: Progress and challenges Bruzewicz, Colin D.; Chiaverini, John; McConnell, Robert Applied Physics Reviews, Vol. 6, Issue 2 https://doi.org/10.1063/1.5088164	journal	June 2019
Elucidating reaction mechanisms on quantum computers Reiher, Markus; Wiebe, Nathan; Svore, Krysta M. Proceedings of the National Academy of Sciences, Vol. 114, Issue 29 https://doi.org/10.1073/pnas.1619152114	journal	July 2017
Efficient quantum algorithm for dissipative nonlinear differential equations Liu, Jin-Peng; Kolden, Herman Øie; Krovi, Hari K. Proceedings of the National Academy of Sciences, Vol. 118, Issue 35 https://doi.org/10.1073/pnas.2026805118	journal	August 2021
Hybrid quantum algorithms for flow problems Bharadwaj, Sachin S.; Sreenivasan, Katepalli R. Proceedings of the National Academy of Sciences, Vol. 120, Issue 49 https://doi.org/10.1073/pnas.2311014120	journal	December 2023
A variational solution of the time-dependent Schrodinger equation McLachlan, A. D. Molecular Physics, Vol. 8, Issue 1 https://doi.org/10.1080/00268976400100041	journal	January 1964
Quantum algorithms for scientific computing Au-Yeung, R.; Camino, B.; Rathore, O. Reports on Progress in Physics, Vol. 87, Issue 11 https://doi.org/10.1088/1361-6633/ad85f0	journal	October 2024
Tensor quantum programming Termanova, A.; Melnikov, Ar; Mamenchikov, E. New Journal of Physics, Vol. 26, Issue 12 https://doi.org/10.1088/1367-2630/ad985b	journal	December 2024
A variational quantum algorithm for tackling multi-dimensional Poisson equations with inhomogeneous boundary conditions Choi, Minjin; Ryu, Hoon New Journal of Physics, Vol. 27, Issue 5 https://doi.org/10.1088/1367-2630/add8b4	journal	May 2025
On the product of semi-groups of operators Trotter, H. F. Proceedings of the American Mathematical Society, Vol. 10, Issue 4 https://doi.org/10.1090/S0002-9939-1959-0108732-6	journal	April 1959
Compilation of Trotter-Based Time Evolution for Partially Fault-Tolerant Quantum Computing Architecture Akahoshi, Yutaro; Toshio, Riki; Fujisaki, Jun PRX Quantum, Vol. 6, Issue 4 https://doi.org/10.1103/93zr-1ykb	journal	October 2025
Quantum Computation of Finite-Temperature Static and Dynamical Properties of Spin Systems Using Quantum Imaginary Time Evolution Sun, Shi-Ning; Motta, Mario; Tazhigulov, Ruslan N. PRX Quantum, Vol. 2, Issue 1 https://doi.org/10.1103/PRXQuantum.2.010317	journal	February 2021
Quantum Computer Systems for Scientific Discovery Alexeev, Yuri; Bacon, Dave; Brown, Kenneth R. PRX Quantum, Vol. 2, Issue 1 https://doi.org/10.1103/PRXQuantum.2.017001	journal	February 2021
Adaptive Variational Quantum Dynamics Simulations Yao, Yong-Xin; Gomes, Niladri; Zhang, Feng PRX Quantum, Vol. 2, Issue 3 https://doi.org/10.1103/PRXQuantum.2.030307	journal	July 2021
Digital Quantum Simulation of Open Quantum Systems Using Quantum Imaginary–Time Evolution Kamakari, Hirsh; Sun, Shi-Ning; Motta, Mario PRX Quantum, Vol. 3, Issue 1 https://doi.org/10.1103/PRXQuantum.3.010320	journal	February 2022
Complexity of Implementing Trotter Steps Low, Guang Hao; Su, Yuan; Tong, Yu PRX Quantum, Vol. 4, Issue 2 https://doi.org/10.1103/PRXQuantum.4.020323	journal	May 2023
Variational quantum algorithms for nonlinear problems Lubasch, Michael; Joo, Jaewoo; Moinier, Pierre Physical Review A, Vol. 101, Issue 1 https://doi.org/10.1103/PhysRevA.101.010301	journal	January 2020
Encoding of matrix product states into quantum circuits of one- and two-qubit gates Ran, Shi-Ju Physical Review A, Vol. 101, Issue 3 https://doi.org/10.1103/PhysRevA.101.032310	journal	March 2020
Hybrid quantum-classical algorithms in the noisy intermediate-scale quantum era and beyond Callison, Adam; Chancellor, Nicholas Physical Review A, Vol. 106, Issue 1 https://doi.org/10.1103/PhysRevA.106.010101	journal	July 2022
Improved quantum computing with higher-order Trotter decomposition Yang, Xiaodong; Nie, Xinfang; Ji, Yunlan Physical Review A, Vol. 106, Issue 4 https://doi.org/10.1103/PhysRevA.106.042401	journal	October 2022
Depth analysis of variational quantum algorithms for the heat equation Guseynov, N. M.; Zhukov, A. A.; Pogosov, W. V. Physical Review A, Vol. 107, Issue 5 https://doi.org/10.1103/PhysRevA.107.052422	journal	May 2023
Variational quantum algorithms for Poisson equations based on the decomposition of sparse Hamiltonians Li, Hui-Min; Wang, Zhi-Xi; Fei, Shao-Ming Physical Review A, Vol. 108, Issue 3 https://doi.org/10.1103/PhysRevA.108.032418	journal	September 2023
Quantum variational solving of nonlinear and multidimensional partial differential equations Sarma, Abhijat; Watts, Thomas W.; Moosa, Mudassir Physical Review A, Vol. 109, Issue 6 https://doi.org/10.1103/PhysRevA.109.062616	journal	June 2024
Quantum algorithm for solving the advection equation using Hamiltonian simulation Brearley, Peter; Laizet, Sylvain Physical Review A, Vol. 110, Issue 1 https://doi.org/10.1103/PhysRevA.110.012430	journal	July 2024
Variational quantum Crank-Nicolson and method-of-lines schemes for the solution of initial value problems Guzman-Cajica, Francisco; Guzmán, Francisco S. Physical Review A, Vol. 110, Issue 4 https://doi.org/10.1103/PhysRevA.110.042415	journal	October 2024
Variational quantum algorithm based on Lagrange polynomial encoding to solve differential equations Hunout, Josephine; Laizet, Sylvain; Iannucci, Lorenzo Physical Review A, Vol. 111, Issue 6 https://doi.org/10.1103/PhysRevA.111.062404	journal	June 2025
Dynamical suppression of decoherence in two-state quantum systems Viola, Lorenza; Lloyd, Seth Physical Review A, Vol. 58, Issue 4 https://doi.org/10.1103/PhysRevA.58.2733	journal	October 1998
Noise tailoring for scalable quantum computation via randomized compiling Wallman, Joel J.; Emerson, Joseph Physical Review A, Vol. 94, Issue 5 https://doi.org/10.1103/PhysRevA.94.052325	journal	November 2016
Performance Study of Variational Quantum Algorithms for Solving the Poisson Equation on a Quantum Computer Ali, Mazen; Kabel, Matthias Physical Review Applied, Vol. 20, Issue 1 https://doi.org/10.1103/PhysRevApplied.20.014054	journal	July 2023
Demonstration of Fidelity Improvement Using Dynamical Decoupling with Superconducting Qubits Pokharel, Bibek; Anand, Namit; Fortman, Benjamin Physical Review Letters, Vol. 121, Issue 22 https://doi.org/10.1103/PhysRevLett.121.220502	journal	November 2018
Efficient Simulation of One-Dimensional Quantum Many-Body Systems Vidal, Guifré Physical Review Letters, Vol. 93, Issue 4 https://doi.org/10.1103/PhysRevLett.93.040502	journal	July 2004
Mixed-State Dynamics in One-Dimensional Quantum Lattice Systems: A Time-Dependent Superoperator Renormalization Algorithm Zwolak, Michael; Vidal, Guifré Physical Review Letters, Vol. 93, Issue 20 https://doi.org/10.1103/PhysRevLett.93.207205	journal	November 2004
Classical Simulation of Infinite-Size Quantum Lattice Systems in One Spatial Dimension Vidal, G. Physical Review Letters, Vol. 98, Issue 7 https://doi.org/10.1103/PhysRevLett.98.070201	journal	February 2007
Quantum computing enhanced computational catalysis von Burg, Vera; Low, Guang Hao; Häner, Thomas Physical Review Research, Vol. 3, Issue 3 https://doi.org/10.1103/PhysRevResearch.3.033055	journal	July 2021
Quantum computing of fluid dynamics using the hydrodynamic Schrödinger equation Meng, Zhaoyuan; Yang, Yue Physical Review Research, Vol. 5, Issue 3 https://doi.org/10.1103/PhysRevResearch.5.033182	journal	September 2023
Nonlinear dynamics as a ground-state solution on quantum computers Pool, Albert J.; Somoza, Alejandro D.; Mc Keever, Conor Physical Review Research, Vol. 6, Issue 3 https://doi.org/10.1103/PhysRevResearch.6.033257	journal	September 2024
Efficient Variational Quantum Simulator Incorporating Active Error Minimization Li, Ying; Benjamin, Simon C. Physical Review X, Vol. 7, Issue 2 https://doi.org/10.1103/PhysRevX.7.021050	journal	June 2017
Noisy intermediate-scale quantum algorithms Bharti, Kishor; Cervera-Lierta, Alba; Kyaw, Thi Ha Reviews of Modern Physics, Vol. 94, Issue 1 https://doi.org/10.1103/RevModPhys.94.015004	journal	February 2022
Gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations Guseynov, Nikita; Huang, Xiajie; Liu, Nana Physical Review Research, Vol. 7, Issue 3 https://doi.org/10.1103/xlpd-fb1g	journal	July 2025
Variational quantum simulation of partial differential equations: applications in colloidal transport Leong, Fong Yew; Koh, Dax Enshan; Ewe, Wei-Bin International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 33, Issue 11 https://doi.org/10.1108/HFF-05-2023-0265	journal	July 2023
Explicit Quantum Circuit for Simulating the Advection–Diffusion–Reaction Dynamics Sanavio, Claudio; Mauri, Enea; Succi, Sauro IEEE Transactions on Quantum Engineering, Vol. 6 https://doi.org/10.1109/TQE.2025.3544839	journal	January 2025
Tensor networks enable the calculation of turbulence probability distributions Gourianov, Nikita; Givi, Peyman; Jaksch, Dieter Science Advances, Vol. 11, Issue 5 https://doi.org/10.1126/sciadv.ads5990	journal	January 2025
Neutral atom quantum computing hardware: performance and end-user perspective Wintersperger, Karen; Dommert, Florian; Ehmer, Thomas EPJ Quantum Technology, Vol. 10, Issue 1 https://doi.org/10.1140/epjqt/s40507-023-00190-1	journal	August 2023
Disentangling Hype from Practicality: On Realistically Achieving Quantum Advantage Hoefler, Torsten; Häner, Thomas; Troyer, Matthias Communications of the ACM, Vol. 66, Issue 5 https://doi.org/10.1145/3571725	journal	April 2023
Quantum Computers for Weather and Climate Prediction: The Good, the Bad, and the Noisy Tennie, F.; Palmer, T. N. Bulletin of the American Meteorological Society, Vol. 104, Issue 2 https://doi.org/10.1175/BAMS-D-22-0031.1	journal	February 2023
Quantum computing for fluids: Where do we stand? Succi, Sauro; Itani, W.; Sreenivasan, K. Europhysics Letters, Vol. 144, Issue 1 https://doi.org/10.1209/0295-5075/acfdc7	journal	October 2023
Quantum Computing in the NISQ era and beyond Preskill, John Quantum, Vol. 2 https://doi.org/10.22331/q-2018-08-06-79	journal	August 2018
Theory of variational quantum simulation Yuan, Xiao; Endo, Suguru; Zhao, Qi Quantum, Vol. 3 https://doi.org/10.22331/q-2019-10-07-191	journal	October 2019
Quantum computing with neutral atoms Henriet, Loïc; Beguin, Lucas; Signoles, Adrien Quantum, Vol. 4 https://doi.org/10.22331/q-2020-09-21-327	journal	September 2020
A variational quantum algorithm for the Feynman-Kac formula Alghassi, Hedayat; Deshmukh, Amol; Ibrahim, Noelle Quantum, Vol. 6 https://doi.org/10.22331/q-2022-06-07-730	journal	June 2022
Variational Quantum Linear Solver Bravo-Prieto, Carlos; LaRose, Ryan; Cerezo, M. Quantum, Vol. 7 https://doi.org/10.22331/q-2023-11-22-1188	journal	November 2023
Improved Accuracy for Trotter Simulations Using Chebyshev Interpolation Rendon, Gumaro; Watkins, Jacob; Wiebe, Nathan Quantum, Vol. 8 https://doi.org/10.22331/q-2024-02-26-1266	journal	February 2024
Variational Quantum Framework for Nonlinear PDE Constrained Optimization Using Carleman Linearization Gnanasekaran, Abeynaya; Surana, Amit; Zhu, Hongyu Quantum Information & Computation, Vol. 25, Issue 3 https://doi.org/10.2478/qic-2025-0014	journal	June 2025
Quantum Speedup for Aeroscience and Engineering Givi, Peyman; Daley, Andrew J.; Mavriplis, Dimitri AIAA Journal, Vol. 58, Issue 8 https://doi.org/10.2514/1.J059183	journal	August 2020
Variational Quantum Algorithms for Computational Fluid Dynamics Jaksch, Dieter; Givi, Peyman; Daley, Andrew J. AIAA Journal, Vol. 61, Issue 5 https://doi.org/10.2514/1.J062426	journal	May 2023
Basic Research Perspective on Quantum Information Science for the Aerospace Community Metcalfe, Grace D.; Tabakov, Boyan; Nguyen, Tristan AIAA Journal, Vol. 61, Issue 12 https://doi.org/10.2514/1.J062978	journal	December 2023
Toward Variational Quantum Algorithms for Generalized Linear and Nonlinear Transport Phenomena Bengoechea, Sergio; Over, Paul; Jaksch, Dieter AIAA Journal https://doi.org/10.2514/1.J065582	journal	November 2025
A Hybrid Quantum Solver for the Lorenz System Fathi Hafshejani, Sajad; Gaur, Daya; Dasgupta, Arundhati Entropy, Vol. 26, Issue 12 https://doi.org/10.3390/e26121009	journal	November 2024

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