Variational Quantum Linear Solver

Carlos Bravo-Prieto; Ryan LaRose; M. Cerezo; Yigit Subasi; Lukasz Cincio; Patrick J. Coles

doi:10.22331/q-2023-11-22-1188

Variational Quantum Linear Solver

Carlos Bravo-Prieto^1,2,3, Ryan LaRose⁴, M. Cerezo^1,5, Yigit Subasi⁶, Lukasz Cincio¹, and Patrick J. Coles¹

¹Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
²Barcelona Supercomputing Center, Barcelona, Spain.
³Institut de Ciències del Cosmos, Universitat de Barcelona, Barcelona, Spain.
⁴Department of Computational Mathematics, Science, and Engineering & Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48823, USA.
⁵Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA
⁶Computer, Computational and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Published:	2023-11-22, volume 7, page 1188
Eprint:	arXiv:1909.05820v4
Doi:	https://doi.org/10.22331/q-2023-11-22-1188
Citation:	Quantum 7, 1188 (2023).

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Abstract

Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare $|x\rangle$ such that $A|x\rangle\propto|b\rangle$. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision $\epsilon$ is achieved. Specifically, we prove that $C \geqslant \epsilon^2 / \kappa^2$, where $C$ is the VQLS cost function and $\kappa$ is the condition number of $A$. We present efficient quantum circuits to estimate $C$, while providing evidence for the classical hardness of its estimation. Using Rigetti's quantum computer, we successfully implement VQLS up to a problem size of $1024\times1024$. Finally, we numerically solve non-trivial problems of size up to $2^{50}\times2^{50}$. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in $\epsilon$, $\kappa$, and the system size $N$.

Featured image: Schematic diagram for the Variational Quantum Linear Solver (VQLS) algorithm. The input to VQLS is a matrix $A$ written as a linear combination of unitaries $A_l$ and a short-depth quantum circuit $U$ which prepares the state $|b\rangle$. The output of VQLS is a quantum state $|x\rangle$ that is approximately proportional to the solution of the linear system $A \vec{x} = \vec{b}$. Parameters $\vec{\alpha}$ in the ansatz $V(\vec{\alpha})$ are adjusted in a hybrid quantum-classical optimization loop until the cost $C(\vec{\alpha})$ (local or global) is below a user-specified threshold. When this loop terminates, the resulting gate sequence $V(\vec{\alpha}_\text{opt})$ prepares the state $|x\rangle = \vec{x} / ||\vec{x}||_2$, from which observable quantities can be computed. Furthermore, the final value of the cost $C(\vec{\alpha}_\text{opt})$ provides an upper bound on the deviation of observables measured on $|x\rangle$ from observables measured on the exact solution.

► BibTeX data

@article{BravoPrieto2023variationalquantum,
  doi = {10.22331/q-2023-11-22-1188},
  url = {https://doi.org/10.22331/q-2023-11-22-1188},
  title = {Variational {Q}uantum {L}inear {S}olver},
  author = {Bravo-Prieto, Carlos and LaRose, Ryan and Cerezo, M. and Subasi, Yigit and Cincio, Lukasz and Coles, Patrick J.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1188},
  month = nov,
  year = {2023}
}

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[27] Guanghui Li, Shasha Wang, Xiumei Zhao, Fei Gao, Sujuan Qin, Fenzhuo Guo, and Zhengping Jin, "Quantum alternating operator ansatz for solving the minimum dominating set problem on sparse graphs with a specific structure", Quantum Information Processing 24 6, 166 (2025).

[28] Francesco Turro, Alessandra Lignarolo, and Daniele Dragoni, "Toward Practical Application of the Quantum Carleman Lattice Boltzmann Method in Industrial CFD Simulations", IEEE Transactions on Quantum Engineering 6, 1 (2025).

[29] Parikshit Pareek, Abhijith Jayakumar, Carleton Coffrin, and Sidhant Misra, "Limitations of Fault-Tolerant Quantum Linear System Solvers for Quantum Power Flow", IEEE Transactions on Power Systems 41 2, 811 (2026).

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[31] Fanfan Lin, Zeheng Wang, Chao Ren, Xinze Li, Juan José Rodríguez-Andina, Sergio Vazquez, H. Alan Mantooth, Mikael Skoglund, Timothy van der Laan, and Muhammad Usman, "Quantum computing for smart grid", Nature Reviews Electrical Engineering 3 7, 406 (2026).

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[33] Hirad Alipanah, Feng Zhang, Yong-Xin Yao, Richard Thompson, Nam Nguyen, Junyu Liu, Peyman Givi, Brian J. McDermott, and Juan José Mendoza-Arenas, "Quantum dynamics simulation of the advection-diffusion equation", Physical Review Research 7 4, 043318 (2025).

[34] Arnaud Rémi, François Damanet, and Christophe Geuzaine, 2026 International Conference on Quantum Control, Computing and Learning (qCCL) 1 (2026) ISBN:979-8-3195-1865-1.

[35] Abhishek Arora, Benjamin M. Ward, and Caglar Oskay, "An implementation of the finite element method in hybrid classical/quantum computers", Finite Elements in Analysis and Design 248, 104354 (2025).

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[37] Sajad Fathi Hafshejani, Daya Gaur, Arundhati Dasgupta, Robert Benkoczi, Narasimha Reddy Gosala, and Alfredo Iorio, "A Hybrid Quantum Solver for the Lorenz System", Entropy 26 12, 1009 (2024).

[38] Xiaolong Li, Jize Han, Feng Feng, Jinyi Liu, Jiali Zhang, Ke Liu, Shaochang Liu, Wei Liu, Zhiguo Huang, and Qi-Jun Zhang, "Block-Encoding-Based Hybrid Quantum–Classical Algorithm for S-Parameters With Magnitude and Phase in Finite Element Method", IEEE Transactions on Microwave Theory and Techniques 74 6, 5053 (2026).

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[41] Qimao Yang and Jing Guo, "Variational Quantum Algorithm for Solving Quantum Transport Equation in Semiconductor Device", IEEE Transactions on Electron Devices 72 6, 3265 (2025).

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[44] 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).

[45] Y. Xiao, L. M. Yang, C. Shu, S. C. Chew, B. C. Khoo, Y. D. Cui, and Y. Y. Liu, "Physics-informed quantum neural network for solving forward and inverse problems of partial differential equations", Physics of Fluids 36 9, 097145 (2024).

[46] Yigal Ilin and Itai Arad, "Dissipative Variational Quantum Algorithms for Gibbs State Preparation", IEEE Transactions on Quantum Engineering 6, 1 (2025).

[47] Abeynaya Gnanasekaran, Amit Surana, and Hongyu Zhu, "Variational Quantum Framework for Nonlinear PDE Constrained Optimization Using Carleman Linearization", Quantum Information & Computation 25 3, 260 (2025).

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[49] Rio Honda, Katsuhiro Endo, Yudai Suzuki, Yoshiki Matsuda, Shu Tanaka, and Mayu Muramatsu, "Development of Hamiltonian for Structural Applications by Quantum Annealing Assuming Finite Element Method", International Journal for Numerical Methods in Engineering 127 17, e70414 (2026).

[50] Yusheng Xu, Xiaojun Wang, and Zhenghuan Wang, "Quantum mapping algorithm for structural non-probabilistic reliability optimization", Structural and Multidisciplinary Optimization 68 7, 138 (2025).

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[54] Ji-Chong Yang, Shuai Zhang, and Chong-Xing Yue, "A quantum machine learning classifier to search for new physics", Journal of High Energy Physics 2026 1, 23 (2026).

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[56] Zhao-Yun Chen, Teng-Yang Ma, Chuang-Chao Ye, Liang Xu, Wen Bai, Lei Zhou, Ming-Yang Tan, Xi-Ning Zhuang, Xiao-Fan Xu, Yun-Jie Wang, Tai-Ping Su, Yong Chen, Lei Du, Liang-Liang Guo, Hai-Feng Zhang, Hao-Ran Tao, Tian-Le Wang, Xiao-Yan Yang, Ze-An Zhao, Peng Wang, Sheng Zhang, Ren-Ze Zhao, Chi Zhang, Zhi-Long Jia, Wei-Cheng Kong, Meng-Han Dou, Jun-Chao Wang, Huan-Yu Liu, Cheng Xue, Peng-Jun-Yi Zhang, Shenghong Huang, Peng Duan, Yu-Chun Wu, and Ping Guo, "Enabling Large-Scale and High-Precision Fluid Simulations on Near-Term Quantum Computers", (2024).

[57] Sowndarya Krishnan Navaneetha Kannan and Serveh Kamrava, "Quantum computing for modeling fluid flow in subsurface environments", Advances in Water Resources 215, 105383 (2026).

[58] Leonidas Taliadouros, Ilias G. Mavromatis, and Ioannis A. Kougioumtzoglou, "Eigenvalue analysis of stochastic structural systems: A quantum computing approach", Probabilistic Engineering Mechanics 81, 103814 (2025).

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[61] Jinil Lee, Wooyeong Song, Donghwa Lee, Yosep Kim, Seung-Woo Lee, Hyang-Tag Lim, Hojoong Jung, Sang-Wook Han, and Yong-Su Kim, "Photonic variational quantum eigensolver using entanglement measurements", Quantum Science and Technology 9 4, 045028 (2024).

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[63] Toshiki Matsumine, Hideki Ochiai, and Junji Shikata, "Quantum Algorithms for the Physical Layer: Potential Applications to Physical Layer Security", IEEE Access 13, 13988 (2025).

[64] Zi-Wen Huang, Xiao-Hui Ni, Jia-Cheng Fan, Su-Juan Qin, Wei Huang, Bing-Jie Xu, and Fei Gao, "Iterative partition-search variational quantum algorithm for solving the shortest-vector problem", Physical Review A 113 3, 032601 (2026).

[65] Xiang Rao, Xi Ouyang, and Yina Liu, "Quantum‐Classical Physics‐Informed Neural Networks for Permeability Inversion in Single‐Phase Reservoir Flow", Advanced Quantum Technologies 9 8, e70409 (2026).

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[68] Francesco Hoch, Giovanni Rodari, Eugenio Caruccio, Beatrice Polacchi, Gonzalo Carvacho, Taira Giordani, Mina Doosti, Sebastià Nicolau, Ciro Pentangelo, Simone Piacentini, Andrea Crespi, Francesco Ceccarelli, Roberto Osellame, Ernesto F. Galvão, Nicolò Spagnolo, and Fabio Sciarrino, "Variational quantum cloning machine on an integrated photonic interferometer", Optica Quantum 3 4, 351 (2025).

[69] Linxuan Li, Qianli Zhou, Zhen Li, Yong Deng, and Éloi Bossé, "Towards an efficient implementation of Dempster–Shafer: $$\alpha $$-junction fusion rules on quantum circuits", Quantum Information Processing 23 11, 374 (2024).

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

[75] Abeynaya Gnanasekaran and Amit Surana, "Efficient quantum access model for sparse structured matrices using linear combination of “things”", Physical Review A 113 2, 022437 (2026).

[76] Abdul Basit, Ali Hassan Jamal, Kashif Hameed, Saif Ullah, and Rab Nawaz, 2023 20th International Bhurban Conference on Applied Sciences and Technology (IBCAST) 606 (2023) ISBN:979-8-3503-0825-9.

[77] Chuang-Chao Ye, Ning-Bo An, Teng-Yang Ma, Meng-Han Dou, Wen Bai, De-Jun Sun, Zhao-Yun Chen, and Guo-Ping Guo, "A hybrid quantum-classical framework for computational fluid dynamics", Physics of Fluids 36 12, 127111 (2024).

[78] Yongchun Xu, Zengtao Kuang, Liang Li, Qun Huang, Jie Yang, Kaixuan Huang, Ziting Wang, Heng Fan, and Heng Hu, "Towards quantum computing enhanced general computational homogenization without readout problem", International Journal of Mechanical Sciences 311, 111214 (2026).

[79] Guojian Wu, Fang Gao, Rebing Wu, Peijie Li, Shengchao Jiang, Hongying Zhai, and Qi Liu, "Gridded quantum differentiation algorithm for electromagnetic transients analysis", Physica Scripta 101 32, 325203 (2026).

[80] Wei-Bin Ewe, Dax Enshan Koh, Siong Thye Goh, Hong-Son Chu, and Ching Eng Png, "Variational Quantum-Based Simulation of Waveguide Modes", IEEE Transactions on Microwave Theory and Techniques 70 5, 2517 (2022).

[81] Kajornsak Singhun, Siriwat Ninlawat, Thananan Chooseang, Prakasit Prabpal, and Somchat Sonasang, 2026 23rd International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON) 633 (2026) ISBN:979-8-3315-8102-2.

[82] Jinhwan Sul, Jungin E. Kim, and Yan Wang, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 552 (2024) ISBN:979-8-3315-4137-8.

[83] Minjin Choi and Hoon Ryu, "A variational quantum algorithm for tackling multi-dimensional Poisson equations with inhomogeneous boundary conditions", New Journal of Physics 27 5, 054510 (2025).

[84] Nikita A. Zemlevskiy, "Scalable quantum simulations of scattering in scalar field theory on 120 qubits", Physical Review D 112 3, 034502 (2025).

[85] Thibaut Pellerin, Robin Van Gaalen, and Ronny I. A. Harmanny, 2024 21st European Radar Conference (EuRAD) 15 (2024) ISBN:978-2-87487-079-8.

[86] 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).

[87] Nico Meyer, Jakob Murauer, Alexander Popov, Christian Ufrecht, Axel Plinge, Christopher Mutschler, and Daniel D. Scherer, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 1458 (2024) ISBN:979-8-3315-4137-8.

[88] Anne-Solène Bornens and Michel Nowak, "Variational quantum algorithms on cat-state qubits", Physical Review A 112 2, 022412 (2025).

[89] Sachin S. Bharadwaj, "QFlowS: Quantum simulator for fluid flows", Physics of Fluids 36 10, 107112 (2024).

[90] E. Dinesh Kumar and Steven H. Frankel, "Quantum unitary matrix representation of the lattice Boltzmann model for low Reynolds fluid flow simulation", AVS Quantum Science 7 1, 013802 (2025).

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[92] Li Xu, Jing Wang, Jiang Chen, Jiamin Xu, Ming Li, and Weishan Zhang, "Quantum Support Vector Machines and Quantum Kernel Methods", Software: Practice and Experience 56 6, 786 (2026).

[93] Yulin Chi, Hongyi Shi, Wen Zheng, Haoyang Cai, Yu Zhang, Xinsheng Tan, Shaoxiong Li, Jianwei Wang, Jiangyu Cui, Man-Hong Yung, and Yang Yu, "Variational quantum algorithms with invariant probabilistic error cancellation on noisy quantum processors", Science China Physics, Mechanics & Astronomy 69 1, 210312 (2026).

[94] Zhenghuan Wang and Xiaojun Wang, "Quantum genetic evolutionary algorithm for discrete and continuous optimization on quantum–classical hybrid computing architectures", Computational Optimization and Applications (2026).

[95] Saleh Almutairi, Asem Alenaizan, Abduljabar Al-Sayoud, Joakim Beck, Sheikha Lardhi, Shuroog Al-Ogbi, and Muhamad Felemban, "Quantum Computing for Computational Sciences", IEEE Transactions on Quantum Engineering 7, 3103728 (2026).

[96] 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 3, 033246 (2024).

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[101] Hieu Trung Nguyen and Anh Phuong Ngo, "Exploring Chebyshev Spectral Method With Quantum Annealing via Carleman Linearization", IEEE Control Systems Letters 9, 2801 (2025).

[102] Soham Pawar, Siddharth Ayathu, and Naganand Y, 2026 International Conference on Quantum Communications, Networking, and Computing (QCNC) 1 (2026) ISBN:979-8-3315-6110-9.

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[104] Francesco Preti, Michael Schilling, Sofiene Jerbi, Lea M. Trenkwalder, Hendrik Poulsen Nautrup, Felix Motzoi, and Hans J. Briegel, "Hybrid discrete-continuous compilation of trapped-ion quantum circuits with deep reinforcement learning", Quantum 8, 1343 (2024).

[105] Alistair Letcher, Stefan Woerner, and Christa Zoufal, "Tight and Efficient Gradient Bounds for Parameterized Quantum Circuits", Quantum 8, 1484 (2024).

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[108] Nico Meyer, Martin Röhn, Jakob Murauer, Axel Plinge, Christopher Mutschler, and Daniel D. Scherer, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 1 (2024) ISBN:979-8-3315-4137-8.

[109] N. Renaud, P. Rodríguez-Sánchez, J. Hidding, and P. Chris Broekema, "Quantum radio astronomy: Quantum linear solvers for redundant baseline calibration", Astronomy and Computing 47, 100803 (2024).

[110] Afrah Farea, Saiful Khan, and Mustafa Serdar Celebi, "QCPINN: quantum-classical physics-informed neural networks for solving PDEs", Machine Learning: Science and Technology 6 4, 045053 (2025).

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[112] Paul Over, Sergio Bengoechea, Thomas Rung, Francesco Clerici, Leonardo Scandurra, Eugene de Villiers, and Dieter Jaksch, "Boundary treatment for variational quantum simulations of partial differential equations on quantum computers", Computers & Fluids 288, 106508 (2025).

[113] Thinh Viet Le, Md Obaidur Rahman, and Vassilis Kekatos, 2025 IEEE International Conference on Communications, Control, and Computing Technologies for Smart Grids (SmartGridComm) 1 (2025) ISBN:979-8-3315-2084-7.

[114] Yagnik Chatterjee, Eric Bourreau, and Marko J. Rančić, "Solving various NP-hard problems using exponentially fewer qubits on a quantum computer", Physical Review A 109 5, 052441 (2024).

[115] Daniel Mastropietro, Georgios Korpas, Vyacheslav Kungurtsev, and Jakub Marecek, "Parallel Variational Quantum Algorithms With Gradient-Informed Restart to Speed Up Optimization in the Presence of Barren Plateaus", IEEE Transactions on Quantum Engineering 7, 1 (2026).

[116] Shreyan Prakash, Raj Bhattacherjee, Sainath Bitragunta, Ashutosh Bhatia, and Kamlesh Tiwari, "Quantum Computing-Accelerated Kalman Filtering for Satellite Clusters: Algorithms and Comparative Analysis", IEEE Open Journal of the Computer Society 6, 307 (2025).

[117] Kwassi Joseph Dzahini, Jeffrey M. Larson, Matt Menickelly, and Stefan M. Wild, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 20 (2025) ISBN:979-8-3315-5736-2.

[118] Beimbet Daribayev, Aksultan Mukhanbet, Nurtugan Azatbekuly, and Timur Imankulov, "A Quantum Approach for Exploring the Numerical Results of the Heat Equation", Algorithms 17 8, 327 (2024).

[119] Fei Feng, Peng Zhang, Yifan Zhou, and Yacov A. Shamash, "Noisy-intermediate-scale quantum power system state estimation", iEnergy 3 3, 135 (2024).

[120] Pietro Asinari, Matteo Maria Piredda, Giulio Barletta, Matteo Fasano, and Eliodoro Chiavazzo, "Extending the Step-by-Step HHL Algorithm Walkthrough: Advanced Concepts for Practical Quantum Computing Applications", IEEE Access 14, 105941 (2026).

[121] Tian Liu, Bingbing Song, Fanxu Meng, Wu Yang, Jiaqi Li, Jianwei You, and Weibing Lu, "Rapid Estimation Method for Coupling Degree of Airborne Antenna Based on Quantum Neural Network", IEEE Antennas and Wireless Propagation Letters 23 12, 4089 (2024).

[122] Rajesh K. Malla, Hiroki Sukeno, Hongye Yu, Tzu-Chieh Wei, Andreas Weichselbaum, and Robert M. Konik, "Feedback-based quantum algorithm inspired by counterdiabatic driving", Physical Review Research 6 4, 043068 (2024).

[123] Yovav Tene-Cohen, Tomer Kelman, Ohad Lev, and Adi Makmal, "A Variational Qubit-Efficient MaxCut Heuristic Algorithm", npj Quantum Information 12 1, 50 (2026).

[124] Dimitris Badounas, Marvin Bechtold, Martin Beisel, Patricia Bickert, Arcesio Castañeda Medina, Michael Fromm, Alexander Geng, Ilie-Daniel Gheorghe-Pop, Florian Girtler, Cristian Grozea, Dominik Heldwein, Michael Holzki, Matthias Kabel, Colin Kai-Uwe Becker, Sophia Lahs, Fernando Lima, Théo Lisart-Liebermann, Darya Martyniuk, Merlin Mengel, Ali Moghiseh, Andreas Müller, Joe Rixon, Marcel Seelbach Benkner, Sebastian Senge, Nikolay Tcholtchev, Felix Truger, Juris Ulmanis, Sebastian Wagner, Benjamin Weder, and Armin Wolf, Communications in Computer and Information Science 2744, 150 (2026) ISBN:978-3-032-13854-5.

[125] Efekan Kökcü, Roeland Wiersema, Alexander F. Kemper, and Bojko N. Bakalov, "Classification of dynamical Lie algebras generated by spin interactions on undirected graphs", Journal of Mathematical Physics 67 5, 052205 (2026).

[126] Shengzhe Chang, Wenliang Nan, Jiyuan Liu, Yongming Tang, and He Li, "Acc-VQLS: Accelerated Variational Quantum Linear Solver for VSC Simulation", ACM Transactions on Quantum Computing 3821429 (2026).

[127] Farshad Amani and Amin Kargarian, "Quantum Optimization for Optimal Power Flow: CVQLS-Augmented Interior Point Method", IEEE Transactions on Smart Grid 16 6, 5040 (2025).

[128] Peiyong Wang, Muhammad Usman, Udaya Parampalli, Lloyd C. L. Hollenberg, and Casey R. Myers, "Automated Quantum Circuit Design With Nested Monte Carlo Tree Search", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[129] Elise Fressart, Michel Nowak, and Nicole Spillane, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 312 (2025) ISBN:979-8-3315-5736-2.

[130] Zefree Lazarus Mayaluri and Sasmita Mishra, "Quantum-enhanced multi-echelon inventory control", Journal of Simulation 1 (2026).

[131] Qimao Yang and Jing Guo, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 2031 (2025) ISBN:979-8-3315-5736-2.

[132] Claudio Sanavio, Fabio Mascherpa, Alessia Marruzzo, Alfonso Amendola, and Sauro Succi, "Variational-adiabatic quantum solver for systems of linear equations with warm starts", International Journal of Modern Physics C 2750108 (2026).

[133] Junxiang Xiao, Shen Dong, and Linghao Xia, 2023 8th International Conference on Communication, Image and Signal Processing (CCISP) 546 (2023) ISBN:979-8-3503-0583-8.

[134] Fouad Ayoub and James D. Baeder, "High-entanglement capabilities for variational quantum algorithms: the Poisson equation case", Quantum Information Processing 24 8, 229 (2025).

[135] Youla Yang, "Quantum2Prompt: Representing Quantum Circuits as Language Prompts for Linear System Solving", (2026).

[136] Yongchun Xu and Heng Hu, "Potential energy minimization for structural analysis via decomposition-free quantum computing", Computers & Structures 330, 108309 (2026).

[137] Dylan Herman, Rudy Raymond, Muyuan Li, Nicolas Robles, Antonio Mezzacapo, and Marco Pistoia, "Expressivity of Variational Quantum Machine Learning on the Boolean Cube", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[138] Ivelina Stoyanova, Orkun Şensebat, Yanjun Ji, Priyanka Arkalgud Ganeshamurthy, Sonja Kajganic, Dennis Willsch, and Antonello Monti, 2025 IEEE PES Innovative Smart Grid Technologies Conference Europe (ISGT Europe) 1 (2025) ISBN:979-8-3315-2503-3.

[139] Leonhard Hölscher, Oliver Ahrend, Lukas Karch, Carlotta L’Estocq, Marc Marfany Andreu, Tobias Stollenwerk, Frank K Wilhelm, and Julia Kowalski, "End-to-end quantum algorithm for topology optimization in structural mechanics", Quantum Science and Technology 11 2, 025029 (2026).

[140] Hsin‐Yu Wu, Vincent E. Elfving, and Oleksandr Kyriienko, "Multidimensional Quantum Generative Modeling by Quantum Hartley Transform", Advanced Quantum Technologies 8 3, 2400337 (2025).

[141] Maximilian Zorn, Jonas Stein, Philipp Altmann, Michael Kölle, Claudia Linnhoff-Popien, and Thomas Gabor, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 1721 (2024) ISBN:979-8-3315-4137-8.

[142] Nithin Reddy Govindugari and Hiu Yung Wong, 2024 International Conference on Simulation of Semiconductor Processes and Devices (SISPAD) 01 (2024) ISBN:979-8-3315-1635-2.

[143] Abeynaya Gnanasekaran and Amit Surana, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 199 (2024) ISBN:979-8-3315-4137-8.

[144] Yunfei Wang and Junyu Liu, "A comprehensive review of quantum machine learning: from NISQ to fault tolerance", Reports on Progress in Physics 87 11, 116402 (2024).

[145] Josephine Hunout, Sylvain Laizet, and Lorenzo Iannucci, "Variational quantum algorithm based on Lagrange polynomial encoding to solve differential equations ", Physical Review A 111 6, 062404 (2025).

[146] Francesco Ghisoni, Francesco Scala, Daniele Bajoni, and Dario Gerace, "Resource-efficient quantum algorithm for linear systems of equations", Physical Review A 113 3, 032405 (2026).

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The above citations are from Crossref's cited-by service (last updated successfully 2026-09-08 23:12:34) and SAO/NASA ADS (last updated successfully 2026-09-08 23:12:46). The list may be incomplete as not all publishers provide suitable and complete citation data.

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