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}
}

► References

[1] E. Alpaydin, Introduction to Machine Learning, 4th ed. (The MIT Press, 2020).
https://mitpress.mit.edu/9780262043793/introduction-to-machine-learning/

[2] C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).
https://link.springer.com/book/9780387310732

[3] L. C. Evans, Partial differential equations (American Mathematical Society, 2010).
https://bookstore.ams.org/gsm-19-r

[4] O. Bretscher, Linear Algebra With Applications, 5th ed. (Pearson, 2013).
https://www.pearson.de/linear-algebra-with-applications-pearson-new-international-edition-pdf-ebook-9781292035345

[5] D. A. Spielman and N. Srivastava, ``Graph sparsification by effective resistances,'' SIAM J. Comput. 40, 1913–1926 (2011).
https://doi.org/10.1137/080734029

[6] A. W. Harrow, A. Hassidim, and S. Lloyd, ``Quantum algorithm for linear systems of equations,'' Phys. Rev. Lett. 103, 150502 (2009).
https://doi.org/10.1103/PhysRevLett.103.150502

[7] A. Ambainis, ``Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations,'' arXiv:1010.4458 [quant-ph].
arXiv:1010.4458

[8] Y. Subaşı, R. D. Somma, and D. Orsucci, ``Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing,'' Phys. Rev. Lett. 122, 060504 (2019).
https://doi.org/10.1103/PhysRevLett.122.060504

[9] A. Childs, R. Kothari, and R. Somma, ``Quantum algorithm for systems of linear equations with exponentially improved dependence on precision,'' SIAM J. Computing 46, 1920–1950 (2017).
https://doi.org/10.1137/16M1087072

[10] S. Chakraborty, A. Gilyén, and S. Jeffery, ``The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation,'' in 46th International Colloquium on Automata, Languages, and Programming (Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019) pp. 33:1-33:14.
https://doi.org/10.4230/LIPIcs.ICALP.2019.33

[11] L. Wossnig, Z. Zhao, and A. Prakash, ``Quantum linear system algorithm for dense matrices,'' Phys. Rev. Lett. 120, 050502 (2018).
https://doi.org/10.1103/PhysRevLett.120.050502

[12] J. Preskill, ``Quantum computing in the NISQ era and beyond,'' Quantum 2, 79 (2018).
https://doi.org/10.22331/q-2018-08-06-79

[13] Y. Zheng, C. Song, M.-C. Chen, B. Xia, W. Liu, et al., ``Solving systems of linear equations with a superconducting quantum processor,'' Phys. Rev. Lett. 118, 210504 (2017).
https://doi.org/10.1103/PhysRevLett.118.210504

[14] Y. Lee, J. Joo, and S. Lee, ``Hybrid quantum linear equation algorithm and its experimental test on IBM quantum experience,'' Scientific Reports 9, 4778 (2019).
https://doi.org/10.1038/s41598-019-41324-9

[15] J. Pan, Y. Cao, X. Yao, Z. Li, C. Ju, et al., ``Experimental realization of quantum algorithm for solving linear systems of equations,'' Phys. Rev. A 89, 022313 (2014).
https://doi.org/10.1103/PhysRevA.89.022313

[16] X.-D. Cai, C. Weedbrook, Z.-E. Su, M.-C. Chen, Mile Gu, et al., ``Experimental quantum computing to solve systems of linear equations,'' Phys. Rev. Lett. 110, 230501 (2013).
https://doi.org/10.1103/PhysRevLett.110.230501

[17] S. Barz, I. Kassal, M. Ringbauer, Y. O. Lipp, B. Dakić, et al., ``A two-qubit photonic quantum processor and its application to solving systems of linear equations,'' Scientific Reports 4, 6115 (2014).
https://doi.org/10.1038/srep06115

[18] J. Wen, X. Kong, S. Wei, B. Wang, T. Xin, and G. Long, ``Experimental realization of quantum algorithms for a linear system inspired by adiabatic quantum computing,'' Phys. Rev. A 99, 012320 (2019).
https://doi.org/10.1103/PhysRevA.99.012320

[19] E. Anschuetz, J. Olson, A. Aspuru-Guzik, and Y. Cao, ``Variational quantum factoring,'' in International Workshop on Quantum Technology and Optimization Problems (Springer, 2019) pp. 74–85.
https://doi.org/10.1007/978-3-030-14082-3_7

[20] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'Brien, ``A variational eigenvalue solver on a photonic quantum processor,'' Nature Communications 5, 4213 (2014).
https://doi.org/10.1038/ncomms5213

[21] Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, et al., ``Quantum chemistry in the age of quantum computing,'' Chemical Reviews 119, 10856–10915 (2019).
https://doi.org/10.1021/acs.chemrev.8b00803

[22] O. Higgott, D. Wang, and S. Brierley, ``Variational Quantum Computation of Excited States,'' Quantum 3, 156 (2019).
https://doi.org/10.22331/q-2019-07-01-156

[23] T. Jones, S. Endo, S. McArdle, X. Yuan, and S. C. Benjamin, ``Variational quantum algorithms for discovering Hamiltonian spectra,'' Phys. Rev. A 99, 062304 (2019).
https://doi.org/10.1103/PhysRevA.99.062304

[24] Y. Li and S. C. Benjamin, ``Efficient variational quantum simulator incorporating active error minimization,'' Phys. Rev. X 7, 021050 (2017).
https://doi.org/10.1103/PhysRevX.7.021050

[25] C. Kokail, C. Maier, R. van Bijnen, T. Brydges, M. K. Joshi, P. Jurcevic, C. A. Muschik, P. Silvi, R. Blatt, C. F. Roos, and P. Zoller, ``Self-verifying variational quantum simulation of lattice models,'' Nature 569, 355–360 (2019).
https://doi.org/10.1038/s41586-019-1177-4

[26] K. Heya, K. M. Nakanishi, K. Mitarai, and K. Fujii, ``Subspace variational quantum simulator,'' Phys. Rev. Research 5, 023078 (2023).
https://doi.org/10.1103/PhysRevResearch.5.023078

[27] Cristina Cirstoiu, Zoe Holmes, Joseph Iosue, Lukasz Cincio, Patrick J Coles, and Andrew Sornborger, ``Variational fast forwarding for quantum simulation beyond the coherence time,'' npj Quantum Information 6, 82 (2020).
https://doi.org/10.1038/s41534-020-00302-0

[28] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C Benjamin, ``Theory of variational quantum simulation,'' Quantum 3, 191 (2019).
https://doi.org/10.22331/q-2019-10-07-191

[29] J. Romero, J. P. Olson, and A. Aspuru-Guzik, ``Quantum autoencoders for efficient compression of quantum data,'' Quantum Science and Technology 2, 045001 (2017).
https://doi.org/10.1088/2058-9565/aa8072

[30] R. LaRose, A. Tikku, É. O'Neel-Judy, L. Cincio, and P. J. Coles, ``Variational quantum state diagonalization,'' npj Quantum Information 5, 57 (2018).
https://doi.org/10.1038/s41534-019-0167-6

[31] C. Bravo-Prieto, D. García-Martín, and J. I. Latorre, ``Quantum Singular Value Decomposer,'' Phys. Rev. A 101, 062310 (2020).
https://doi.org/10.1103/PhysRevA.101.062310

[32] M. Cerezo, Kunal Sharma, Andrew Arrasmith, and Patrick J Coles, ``Variational quantum state eigensolver,'' npj Quantum Information 8, 113 (2022).
https://doi.org/10.1038/s41534-022-00611-6

[33] S. Khatri, R. LaRose, A. Poremba, L. Cincio, A. T. Sornborger, and P. J. Coles, ``Quantum-assisted quantum compiling,'' Quantum 3, 140 (2019).
https://doi.org/10.22331/q-2019-05-13-140

[34] T. Jones and S. C Benjamin, ``Robust quantum compilation and circuit optimisation via energy minimisation,'' Quantum 6, 628 (2022).
https://doi.org/10.22331/q-2022-01-24-628

[35] A. Arrasmith, L. Cincio, A. T. Sornborger, W. H. Zurek, and P. J. Coles, ``Variational consistent histories as a hybrid algorithm for quantum foundations,'' Nature communications 10, 3438 (2019).
https://doi.org/10.1038/s41467-019-11417-0

[36] Marco Cerezo, Alexander Poremba, Lukasz Cincio, and Patrick J Coles, ``Variational quantum fidelity estimation,'' Quantum 4, 248 (2020b).
https://doi.org/10.22331/q-2020-03-26-248

[37] Bálint Koczor, Suguru Endo, Tyson Jones, Yuichiro Matsuzaki, and Simon C Benjamin, ``Variational-state quantum metrology,'' New Journal of Physics 22, 083038 (2020b).
https://doi.org/10.1088/1367-2630/ab965e

[38] M Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J Coles, ``Cost function dependent barren plateaus in shallow parametrized quantum circuits,'' Nature Communications 12, 1791 (2020b).
https://doi.org/10.1038/s41467-021-21728-w

[39] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th ed. (Cambridge University Press, New York, NY, USA, 2011).
https://doi.org/10.1017/CBO9780511976667

[40] E. Knill and R. Laflamme, ``Power of one bit of quantum information,'' Phys. Rev. Lett. 81, 5672–5675 (1998).
https://doi.org/10.1103/PhysRevLett.81.5672

[41] K. Fujii, H. Kobayashi, T. Morimae, H. Nishimura, S. Tamate, and S. Tani, ``Impossibility of Classically Simulating One-Clean-Qubit Model with Multiplicative Error,'' Phys. Rev. Lett. 120, 200502 (2018).
https://doi.org/10.1103/PhysRevLett.120.200502

[42] T. Morimae, ``Hardness of classically sampling the one-clean-qubit model with constant total variation distance error,'' Phys. Rev. A 96, 040302 (2017).
https://doi.org/10.1103/PhysRevA.96.040302

[43] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, ``Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets,'' Nature 549, 242 (2017).
https://doi.org/10.1038/nature23879

[44] Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven, ``Barren plateaus in quantum neural network training landscapes,'' Nature communications 9, 4812 (2018).
https://doi.org/10.1038/s41467-018-07090-4

[45] Edward Grant, Leonard Wossnig, Mateusz Ostaszewski, and Marcello Benedetti, ``An initialization strategy for addressing barren plateaus in parametrized quantum circuits,'' Quantum 3, 214 (2019).
https://doi.org/10.22331/q-2019-12-09-214

[46] Tyler Volkoff and Patrick J Coles, ``Large gradients via correlation in random parameterized quantum circuits,'' Quantum Sci. Technol. 6, 025008 (2021).
https://doi.org/10.1088/2058-9565/abd891

[47] L. Cincio, Y. Subaşı, A. T. Sornborger, and P. J. Coles, ``Learning the quantum algorithm for state overlap,'' New Journal of Physics 20, 113022 (2018).
https://doi.org/10.1088/1367-2630/aae94a

[48] E. Farhi, J. Goldstone, and S. Gutmann, ``A quantum approximate optimization algorithm,'' arXiv:1411.4028 [quant-ph].
arXiv:1411.4028

[49] S. Hadfield, Z. Wang, B. O'Gorman, E. G. Rieffel, D. Venturelli, and R. Biswas, ``From the quantum approximate optimization algorithm to a quantum alternating operator ansatz,'' Algorithms 12, 34 (2019).
https://doi.org/10.3390/a12020034

[50] S. Lloyd, ``Quantum approximate optimization is computationally universal,'' arXiv:1812.11075 [quant-ph].
arXiv:1812.11075

[51] Z. Wang, S. Hadfield, Z. Jiang, and E. G. Rieffel, ``Quantum approximate optimization algorithm for MaxCut: A fermionic view,'' Phys. Rev. A 97, 022304 (2018).
https://doi.org/10.1103/PhysRevA.97.022304

[52] L. Zhou, S.-T. Wang, S. Choi, H. Pichler, and M. D. Lukin, ``Quantum approximate optimization algorithm: performance, mechanism, and implementation on near-term devices,'' Phys. Rev. X 10, 021067 (2020).
https://doi.org/10.1103/PhysRevX.10.021067

[53] G. E. Crooks, ``Performance of the quantum approximate optimization algorithm on the maximum cut problem,'' arXiv preprint arXiv:1811.08419 (2018).
arXiv:1811.08419

[54] J. M. Kübler, A. Arrasmith, L. Cincio, and P. J. Coles, ``An adaptive optimizer for measurement-frugal variational algorithms,'' Quantum 4, 263 (2020).
https://doi.org/10.22331/q-2020-05-11-263

[55] Andrew Arrasmith, Lukasz Cincio, Rolando D Somma, and Patrick J Coles, ``Operator sampling for shot-frugal optimization in variational algorithms,'' arXiv preprint arXiv:2004.06252 (2020).
arXiv:2004.06252

[56] Ryan Sweke, Frederik Wilde, Johannes Meyer, Maria Schuld, Paul K Fährmann, Barthélémy Meynard-Piganeau, and Jens Eisert, ``Stochastic gradient descent for hybrid quantum-classical optimization,'' Quantum 4, 314 (2020).
https://doi.org/10.22331/q-2020-08-31-314

[57] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, ``Quantum circuit learning,'' Phys. Rev. A 98, 032309 (2018).
https://doi.org/10.1103/PhysRevA.98.032309

[58] M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, ``Evaluating analytic gradients on quantum hardware,'' Phys. Rev. A 99, 032331 (2019).
https://doi.org/10.1103/PhysRevA.99.032331

[59] A. Harrow and J. Napp, ``Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms,'' Phys. Rev. Lett. 126, 140502 (2021).
https://doi.org/10.1103/PhysRevLett.126.140502

[60] Kunal Sharma, Sumeet Khatri, Marco Cerezo, and Patrick Coles, ``Noise resilience of variational quantum compiling,'' New Journal of Physics 22, 043006 (2020).
https://doi.org/10.1088/1367-2630/ab784c

[61] K. Temme, S. Bravyi, and J. M. Gambetta, ``Error mitigation for short-depth quantum circuits,'' Phys. Rev. Lett. 119, 180509 (2017).
https://doi.org/10.1103/PhysRevLett.119.180509

[62] Y. He and H. Guo, ``The boundary effects of transverse field ising model,'' Journal of Statistical Mechanics: Theory and Experiment 2017, 093101 (2017).
https://doi.org/10.1088/1742-5468/aa85b0

[63] D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders, ``Efficient quantum algorithms for simulating sparse Hamiltonians,'' Communications in Mathematical Physics 270, 359–371 (2007).
https://doi.org/10.1007/s00220-006-0150-x

[64] Y. Atia and D. Aharonov, ``Fast-forwarding of hamiltonians and exponentially precise measurements,'' Nature communications 8, 1572 (2017).
https://doi.org/10.1038/s41467-017-01637-7

[65] X. Xu, J. Sun, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, ``Variational algorithms for linear algebra,'' Science Bulletin 66, 2181–2188 (2021).
https://doi.org/10.1016/j.scib.2021.06.023

[66] H.-Y. Huang, K. Bharti, and P. Rebentrost, ``Near-term quantum algorithms for linear systems of equations with regression loss functions,'' New Journal of Physics 23, 113021 (2021).
https://doi.org/10.1088/1367-2630/ac325f

[67] A. Asfaw, L. Bello, Y. Ben-Haim, S. Bravyi, L. Capelluto, et al., ``Learn quantum computation using qiskit.'' (2019).
http://community.qiskit.org/textbook

[68] A. Mari, ``Variational quantum linear solver.'' (2019).
https://pennylane.ai/qml/app/tutorial_vqls.html

[69] M. Szegedy, ``Quantum speed-up of markov chain based algorithms,'' in Proceedings of the 45th Annual IEEE Symposium on FOCS. (IEEE, 2004) pp. 32–41.
https://doi.org/10.1109/FOCS.2004.53

[70] D. W. Berry, A. M. Childs, and R. Kothari, ``Hamiltonian simulation with nearly optimal dependence on all parameters,'' in Proceedings of the 56th Symposium on Foundations of Computer Science (2015).
https://doi.org/10.1109/FOCS.2015.54

[71] J. C. Garcia-Escartin and P. Chamorro-Posada, ``Swap test and Hong-Ou-Mandel effect are equivalent,'' Phys. Rev. A 87, 052330 (2013).
https://doi.org/10.1103/PhysRevA.87.052330

[72] M. J. D. Powell, ``A fast algorithm for nonlinearly constrained optimization calculations,'' in Numerical analysis (Springer, 1978) pp. 144–157.
https://doi.org/10.1007/BFb0067703

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[26] 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).

[27] 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).

[28] Benjamin Y. L. Tan, Beng Yee Gan, Daniel Leykam, and Dimitris G. Angelakis, "Landscape approximation of low-energy solutions to binary optimization problems", Physical Review A 109 1, 012433 (2024).

[29] 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).

[30] 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).

[31] 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).

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

[33] Prabhat Anand, M Girish Chandra, and Ankit Khandelwal, 2024 16th International Conference on COMmunication Systems & NETworkS (COMSNETS) 994 (2024) ISBN:979-8-3503-8311-9.

[34] 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).

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

[36] 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).

[37] 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).

[38] 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).

[39] 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.

[40] 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).

[41] Lorenzo Leone, Salvatore F.E. Oliviero, Lukasz Cincio, and M. Cerezo, "On the practical usefulness of the Hardware Efficient Ansatz", Quantum 8, 1395 (2024).

[42] Hsin‐Yu Wu, Vincent E. Elfving, and Oleksandr Kyriienko, "Multidimensional Quantum Generative Modeling by Quantum Hartley Transform", Advanced Quantum Technologies 2400337 (2024).

[43] 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).

[44] 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.

[45] 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.

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

[47] 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).

[48] Akash Kundu, Ludmila Botelho, and Adam Glos, "Hamiltonian-oriented homotopy quantum approximate optimization algorithm", Physical Review A 109 2, 022611 (2024).

[49] Junxiang Xiao, Jingwei Wen, Zengrong Zhou, Ling Qian, Zhiguo Huang, Shijie Wei, and Guilu Long, "A quantum algorithm for linear differential equations with layerwise parameterized quantum circuits", AAPPS Bulletin 34 1, 12 (2024).

[50] Pablo Bermejo, Borja Aizpurua, and Román Orús, "Improving gradient methods via coordinate transformations: Applications to quantum machine learning", Physical Review Research 6 2, 023069 (2024).

[51] R.H. Drebotiy and H.A. Shynkarenko, "Quantum-assisted hλ-adaptive finite element method", Partial Differential Equations in Applied Mathematics 101120 (2025).

[52] Muhammad AbuGhanem and Hichem Eleuch, "NISQ Computers: A Path to Quantum Supremacy", IEEE Access 12, 102941 (2024).

[53] 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).

[54] Farshad Amani and Amin Kargarian, 2024 IEEE Texas Power and Energy Conference (TPEC) 1 (2024) ISBN:979-8-3503-3120-2.

[55] Mohammadreza Saghafi, Lamine Mili, and Ravi Raghunathan, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 176 (2024) ISBN:979-8-3315-4137-8.

[56] Qingyu Li, Yuhan Huang, Xiaokai Hou, Ying Li, Xiaoting Wang, and Abolfazl Bayat, "Ensemble-learning error mitigation for variational quantum shallow-circuit classifiers", Physical Review Research 6 1, 013027 (2024).

[57] Patrick Rebentrost and Seth Lloyd, "Quantum Computational Finance: Quantum Algorithm for Portfolio Optimization", KI - Künstliche Intelligenz (2024).

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

[59] Océane Koska, Marc Baboulin, and Arnaud Gazda, ISC High Performance 2024 Research Paper Proceedings (39th International Conference) 1 (2024) ISBN:978-3-9826336-0-2.

[60] WenShan Xu, Ri-Gui Zhou, YaoChong Li, and XiaoXue Zhang, "Towards an efficient variational quantum algorithm for solving linear equations", Communications in Theoretical Physics 76 11, 115103 (2024).

[61] Jared D. Weidman, Manas Sajjan, Camille Mikolas, Zachary J. Stewart, Johannes Pollanen, Sabre Kais, and Angela K. Wilson, "Quantum computing and chemistry", Cell Reports Physical Science 5 9, 102105 (2024).

[62] 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).

[63] Hansheng Jiang, Zuo-Jun Max Shen, and Junyu Liu, 2022 IEEE/ACM 7th Symposium on Edge Computing (SEC) 400 (2022) ISBN:978-1-6654-8611-8.

[64] Osama Ahmed, Felix Tennie, and Luca Magri, "Prediction of chaotic dynamics and extreme events: A recurrence-free quantum reservoir computing approach", Physical Review Research 6 4, 043082 (2024).

[65] Samson Wang, Piotr Czarnik, Andrew Arrasmith, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, "Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?", Quantum 8, 1287 (2024).

[66] Christo Meriwether Keller, Stephan Eidenbenz, Andreas Bärtschi, Daniel O'Malley, John Golden, and Satyajayant Misra, ISC High Performance 2024 Research Paper Proceedings (39th International Conference) 1 (2024) ISBN:978-3-9826336-0-2.

[67] Phattharaporn Singkanipa and Daniel A. Lidar, "Beyond unital noise in variational quantum algorithms: noise-induced barren plateaus and limit sets", Quantum 9, 1617 (2025).

[68] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, "Variational quantum algorithms", Nature Reviews Physics 3 9, 625 (2021).

[69] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum algorithms", Reviews of Modern Physics 94 1, 015004 (2022).

[70] M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles, "Cost function dependent barren plateaus in shallow parametrized quantum circuits", Nature Communications 12, 1791 (2021).

[71] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, and Jonathan Tennyson, "The Variational Quantum Eigensolver: A review of methods and best practices", Physics Reports 986, 1 (2022).

[72] Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles, "Noise-induced barren plateaus in variational quantum algorithms", Nature Communications 12, 6961 (2021).

[73] Andrew Arrasmith, M. Cerezo, Piotr Czarnik, Lukasz Cincio, and Patrick J. Coles, "Effect of barren plateaus on gradient-free optimization", Quantum 5, 558 (2021).

[74] Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles, "Connecting Ansatz Expressibility to Gradient Magnitudes and Barren Plateaus", PRX Quantum 3 1, 010313 (2022).

[75] Suguru Endo, Zhenyu Cai, Simon C. Benjamin, and Xiao Yuan, "Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation", Journal of the Physical Society of Japan 90 3, 032001 (2021).

[76] J. Abhijith, Adetokunbo Adedoyin, John Ambrosiano, Petr Anisimov, William Casper, Gopinath Chennupati, Carleton Coffrin, Hristo Djidjev, David Gunter, Satish Karra, Nathan Lemons, Shizeng Lin, Alexander Malyzhenkov, David Mascarenas, Susan Mniszewski, Balu Nadiga, Daniel O'Malley, Diane Oyen, Scott Pakin, Lakshman Prasad, Randy Roberts, Phillip Romero, Nandakishore Santhi, Nikolai Sinitsyn, Pieter J. Swart, James G. Wendelberger, Boram Yoon, Richard Zamora, Wei Zhu, Stephan Eidenbenz, Andreas Bärtschi, Patrick J. Coles, Marc Vuffray, and Andrey Y. Lokhov, "Quantum Algorithm Implementations for Beginners", arXiv:1804.03719, (2018).

[77] Suguru Endo, Jinzhao Sun, Ying Li, Simon C. Benjamin, and Xiao Yuan, "Variational Quantum Simulation of General Processes", Physical Review Letters 125 1, 010501 (2020).

[78] Arthur Pesah, M. Cerezo, Samson Wang, Tyler Volkoff, Andrew T. Sornborger, and Patrick J. Coles, "Absence of Barren Plateaus in Quantum Convolutional Neural Networks", Physical Review X 11 4, 041011 (2021).

[79] Daniel Stilck França and Raul García-Patrón, "Limitations of optimization algorithms on noisy quantum devices", Nature Physics 17 11, 1221 (2021).

[80] Xiaosi Xu, Jinzhao Sun, Suguru Endo, Ying Li, Simon C. Benjamin, and Xiao Yuan, "Variational algorithms for linear algebra", Science Bulletin 66 21, 2181 (2021).

[81] Ryan LaRose and Brian Coyle, "Robust data encodings for quantum classifiers", Physical Review A 102 3, 032420 (2020).

[82] Kunal Sharma, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, "Trainability of Dissipative Perceptron-Based Quantum Neural Networks", Physical Review Letters 128 18, 180505 (2022).

[83] Kunal Sharma, Sumeet Khatri, M. Cerezo, and Patrick J. Coles, "Noise resilience of variational quantum compiling", New Journal of Physics 22 4, 043006 (2020).

[84] Dylan Herman, Cody Googin, Xiaoyuan Liu, Alexey Galda, Ilya Safro, Yue Sun, Marco Pistoia, and Yuri Alexeev, "A Survey of Quantum Computing for Finance", arXiv:2201.02773, (2022).

[85] Bojia Duan, Jiabin Yuan, Chao-Hua Yu, Jianbang Huang, and Chang-Yu Hsieh, "A survey on HHL algorithm: From theory to application in quantum machine learning", Physics Letters A 384, 126595 (2020).

[86] Oleksandr Kyriienko, Annie E. Paine, and Vincent E. Elfving, "Solving nonlinear differential equations with differentiable quantum circuits", Physical Review A 103 5, 052416 (2021).

[87] Martin Larocca, Piotr Czarnik, Kunal Sharma, Gopikrishnan Muraleedharan, Patrick J. Coles, and M. Cerezo, "Diagnosing Barren Plateaus with Tools from Quantum Optimal Control", Quantum 6, 824 (2022).

[88] Jonathan Wei Zhong Lau, Kian Hwee Lim, Harshank Shrotriya, and Leong Chuan Kwek, "NISQ computing: where are we and where do we go?", Association of Asia Pacific Physical Societies Bulletin 32 1, 27 (2022).

[89] Tyler Volkoff and Patrick J. Coles, "Large gradients via correlation in random parameterized quantum circuits", Quantum Science and Technology 6 2, 025008 (2021).

[90] Hsin-Yuan Huang, Kishor Bharti, and Patrick Rebentrost, "Near-term quantum algorithms for linear systems of equations", arXiv:1909.07344, (2019).

[91] M. Cerezo and Patrick J. Coles, "Higher order derivatives of quantum neural networks with barren plateaus", Quantum Science and Technology 6 3, 035006 (2021).

[92] Andrew Arrasmith, Lukasz Cincio, Rolando D. Somma, and Patrick J. Coles, "Operator Sampling for Shot-frugal Optimization in Variational Algorithms", arXiv:2004.06252, (2020).

[93] Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles, "An Adaptive Optimizer for Measurement-Frugal Variational Algorithms", Quantum 4, 263 (2020).

[94] Zoë Holmes, Andrew Arrasmith, Bin Yan, Patrick J. Coles, Andreas Albrecht, and Andrew T. Sornborger, "Barren Plateaus Preclude Learning Scramblers", Physical Review Letters 126 19, 190501 (2021).

[95] Aram W. Harrow and John C. Napp, "Low-Depth Gradient Measurements Can Improve Convergence in Variational Hybrid Quantum-Classical Algorithms", Physical Review Letters 126 14, 140502 (2021).

[96] M. Cerezo, Kunal Sharma, Andrew Arrasmith, and Patrick J. Coles, "Variational Quantum State Eigensolver", arXiv:2004.01372, (2020).

[97] Benjamin Commeau, M. Cerezo, Zoë Holmes, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, "Variational Hamiltonian Diagonalization for Dynamical Quantum Simulation", arXiv:2009.02559, (2020).

[98] A. K. Fedorov, N. Gisin, S. M. Beloussov, and A. I. Lvovsky, "Quantum computing at the quantum advantage threshold: a down-to-business review", arXiv:2203.17181, (2022).

[99] Youle Wang, Guangxi Li, and Xin Wang, "Variational Quantum Gibbs State Preparation with a Truncated Taylor Series", Physical Review Applied 16 5, 054035 (2021).

[100] Hsin-Yuan Huang, Kishor Bharti, and Patrick Rebentrost, "Near-term quantum algorithms for linear systems of equations with regression loss functions", New Journal of Physics 23 11, 113021 (2021).

[101] Jacob Biamonte, "Universal variational quantum computation", Physical Review A 103 3, L030401 (2021).

[102] Yu Tong, Dong An, Nathan Wiebe, and Lin Lin, "Fast inversion, preconditioned quantum linear system solvers, fast Green's-function computation, and fast evaluation of matrix functions", Physical Review A 104 3, 032422 (2021).

[103] Alexandre Choquette, Agustin Di Paolo, Panagiotis Kl. Barkoutsos, David Sénéchal, Ivano Tavernelli, and Alexandre Blais, "Quantum-optimal-control-inspired ansatz for variational quantum algorithms", Physical Review Research 3 2, 023092 (2021).

[104] P. Chandarana, N. N. Hegade, K. Paul, F. Albarrán-Arriagada, E. Solano, A. del Campo, and Xi Chen, "Digitized-counterdiabatic quantum approximate optimization algorithm", Physical Review Research 4 1, 013141 (2022).

[105] Peter J. Karalekas, Nikolas A. Tezak, Eric C. Peterson, Colm A. Ryan, Marcus P. da Silva, and Robert S. Smith, "A quantum-classical cloud platform optimized for variational hybrid algorithms", Quantum Science and Technology 5 2, 024003 (2020).

[106] Lin Lin and Yu Tong, "Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems", Quantum 4, 361 (2020).

[107] Carlos Bravo-Prieto, Diego García-Martín, and José I. Latorre, "Quantum singular value decomposer", Physical Review A 101 6, 062310 (2020).

[108] M. Cerezo, Akira Sone, Jacob L. Beckey, and Patrick J. Coles, "Sub-quantum Fisher information", Quantum Science and Technology 6 3, 035008 (2021).

[109] Austin Gilliam, Stefan Woerner, and Constantin Gonciulea, "Grover Adaptive Search for Constrained Polynomial Binary Optimization", Quantum 5, 428 (2021).

[110] Chenfeng Cao and Xin Wang, "Noise-Assisted Quantum Autoencoder", Physical Review Applied 15 5, 054012 (2021).

[111] Ting Zhang, Jinzhao Sun, Xiao-Xu Fang, Xiao-Ming Zhang, Xiao Yuan, and He Lu, "Experimental Quantum State Measurement with Classical Shadows", Physical Review Letters 127 20, 200501 (2021).

[112] Xin Wang, Zhixin Song, and Youle Wang, "Variational Quantum Singular Value Decomposition", Quantum 5, 483 (2021).

[113] Budinski Ljubomir, "Quantum algorithm for the Navier-Stokes equations by using the streamfunction-vorticity formulation and the lattice Boltzmann method", International Journal of Quantum Information 20 2, 2150039-27 (2022).

[114] Antonio A. Mele, Glen B. Mbeng, Giuseppe E. Santoro, Mario Collura, and Pietro Torta, "Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz", Physical Review A 106 6, L060401 (2022).

[115] Carlos Bravo-Prieto, Josep Lumbreras-Zarapico, Luca Tagliacozzo, and José I. Latorre, "Scaling of variational quantum circuit depth for condensed matter systems", Quantum 4, 272 (2020).

[116] Nikolay V. Tkachenko, James Sud, Yu Zhang, Sergei Tretiak, Petr M. Anisimov, Andrew T. Arrasmith, Patrick J. Coles, Lukasz Cincio, and Pavel A. Dub, "Correlation-Informed Permutation of Qubits for Reducing Ansatz Depth in the Variational Quantum Eigensolver", PRX Quantum 2 2, 020337 (2021).

[117] Ranyiliu Chen, Zhixin Song, Xuanqiang Zhao, and Xin Wang, "Variational quantum algorithms for trace distance and fidelity estimation", Quantum Science and Technology 7 1, 015019 (2022).

[118] Adrián Pérez-Salinas, Juan Cruz-Martinez, Abdulla A. Alhajri, and Stefano Carrazza, "Determining the proton content with a quantum computer", Physical Review D 103 3, 034027 (2021).

[119] Shi-Xin Zhang, Chang-Yu Hsieh, Shengyu Zhang, and Hong Yao, "Neural predictor based quantum architecture search", Machine Learning: Science and Technology 2 4, 045027 (2021).

[120] Jacob L. Beckey, M. Cerezo, Akira Sone, and Patrick J. Coles, "Variational quantum algorithm for estimating the quantum Fisher information", Physical Review Research 4 1, 013083 (2022).

[121] Aidan Pellow-Jarman, Ilya Sinayskiy, Anban Pillay, and Francesco Petruccione, "A comparison of various classical optimizers for a variational quantum linear solver", Quantum Information Processing 20 6, 202 (2021).

[122] Juneseo Lee, Alicia B. Magann, Herschel A. Rabitz, and Christian Arenz, "Progress toward favorable landscapes in quantum combinatorial optimization", Physical Review A 104 3, 032401 (2021).

[123] Sergi Ramos-Calderer, Adrián Pérez-Salinas, Diego García-Martín, Carlos Bravo-Prieto, Jorge Cortada, Jordi Planagumà, and José I. Latorre, "Quantum unary approach to option pricing", Physical Review A 103 3, 032414 (2021).

[124] Kunal Sharma, M. Cerezo, Zoë Holmes, Lukasz Cincio, Andrew Sornborger, and Patrick J. Coles, "Reformulation of the No-Free-Lunch Theorem for Entangled Datasets", Physical Review Letters 128 7, 070501 (2022).

[125] Dong An and Lin Lin, "Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm", arXiv:1909.05500, (2019).

[126] Pei Zeng, Jinzhao Sun, and Xiao Yuan, "Universal quantum algorithmic cooling on a quantum computer", arXiv:2109.15304, (2021).

[127] Kosuke Mitarai and Keisuke Fujii, "Overhead for simulating a non-local channel with local channels by quasiprobability sampling", Quantum 5, 388 (2021).

[128] Ruizhe Zhang, Guoming Wang, and Peter Johnson, "Computing Ground State Properties with Early Fault-Tolerant Quantum Computers", Quantum 6, 761 (2022).

[129] Andi Gu, Angus Lowe, Pavel A. Dub, Patrick J. Coles, and Andrew Arrasmith, "Adaptive shot allocation for fast convergence in variational quantum algorithms", arXiv:2108.10434, (2021).

[130] Reuben Demirdjian, Daniel Gunlycke, Carolyn A. Reynolds, James D. Doyle, and Sergio Tafur, "Variational quantum solutions to the advection-diffusion equation for applications in fluid dynamics", Quantum Information Processing 21 9, 322 (2022).

[131] Enrico Fontana, M. Cerezo, Andrew Arrasmith, Ivan Rungger, and Patrick J. Coles, "Non-trivial symmetries in quantum landscapes and their resilience to quantum noise", Quantum 6, 804 (2022).

[132] Pierre-Luc Dallaire-Demers, Michał Stęchły, Jerome F. Gonthier, Ntwali Toussaint Bashige, Jonathan Romero, and Yudong Cao, "An application benchmark for fermionic quantum simulations", arXiv:2003.01862, (2020).

[133] Brian Coyle, Mina Doosti, Elham Kashefi, and Niraj Kumar, "Progress toward practical quantum cryptanalysis by variational quantum cloning", Physical Review A 105 4, 042604 (2022).

[134] Yuhan Huang, Qingyu Li, Xiaokai Hou, Rebing Wu, Man-Hong Yung, Abolfazl Bayat, and Xiaoting Wang, "Robust resource-efficient quantum variational ansatz through an evolutionary algorithm", Physical Review A 105 5, 052414 (2022).

[135] Jin-Min Liang, Shu-Qian Shen, Ming Li, and Lei Li, "Variational quantum algorithms for dimensionality reduction and classification", Physical Review A 101 3, 032323 (2020).

[136] Lorenzo Leone, Salvatore F. E. Oliviero, Stefano Piemontese, Sarah True, and Alioscia Hamma, "Retrieving information from a black hole using quantum machine learning", Physical Review A 106 6, 062434 (2022).

[137] Yohei Ibe, Yuya O. Nakagawa, Nathan Earnest, Takahiro Yamamoto, Kosuke Mitarai, Qi Gao, and Takao Kobayashi, "Calculating transition amplitudes by variational quantum deflation", Physical Review Research 4 1, 013173 (2022).

[138] Fong Yew Leong, Wei-Bin Ewe, and Dax Enshan Koh, "Variational quantum evolution equation solver", Scientific Reports 12, 10817 (2022).

[139] Supanut Thanasilp, Samson Wang, Nhat A. Nghiem, Patrick J. Coles, and M. Cerezo, "Subtleties in the trainability of quantum machine learning models", arXiv:2110.14753, (2021).

[140] Jacob L. Beckey, M. Cerezo, Akira Sone, and Patrick J. Coles, "Variational Quantum Algorithm for Estimating the Quantum Fisher Information", arXiv:2010.10488, (2020).

[141] Xiaoxia Cai, Wei-Hai Fang, Heng Fan, and Zhendong Li, "Quantum computation of molecular response properties", Physical Review Research 2 3, 033324 (2020).

[142] Yohei Ibe, Yuya O. Nakagawa, Nathan Earnest, Takahiro Yamamoto, Kosuke Mitarai, Qi Gao, and Takao Kobayashi, "Calculating transition amplitudes by variational quantum deflation", arXiv:2002.11724, (2020).

[143] Suguru Endo, Jinzhao Sun, Ying Li, Simon Benjamin, and Xiao Yuan, "Variational quantum simulation of general processes", arXiv:1812.08778, (2018).

[144] Yulong Dong and Lin Lin, "Random circuit block-encoded matrix and a proposal of quantum LINPACK benchmark", Physical Review A 103 6, 062412 (2021).

[145] Ranyiliu Chen, Zhixin Song, Xuanqiang Zhao, and Xin Wang, "Variational Quantum Algorithms for Trace Distance and Fidelity Estimation", arXiv:2012.05768, (2020).

[146] Hrushikesh Patil, Yulun Wang, and Predrag S. Krstić, "Variational quantum linear solver with a dynamic ansatz", Physical Review A 105 1, 012423 (2022).

[147] Fan-Xu Meng, Ze-Tong Li, Yu Xu-Tao, and Zai-Chen Zhang, "Quantum algorithm for MUSIC-based DOA estimation in hybrid MIMO systems", Quantum Science and Technology 7 2, 025002 (2022).

[148] Sheng-Jie Li, Jin-Min Liang, Shu-Qian Shen, and Ming Li, "Variational quantum algorithms for trace norms and their applications", Communications in Theoretical Physics 73 10, 105102 (2021).

[149] Bujiao Wu, Maharshi Ray, Liming Zhao, Xiaoming Sun, and Patrick Rebentrost, "Quantum-classical algorithms for skewed linear systems with an optimized Hadamard test", Physical Review A 103 4, 042422 (2021).

[150] Fong Yew Leong, Wei-Bin Ewe, and Dax Enshan Koh, "Variational Quantum Evolution Equation Solver", arXiv:2204.02912, (2022).

[151] M. R. Perelshtein, A. I. Pakhomchik, A. A. Melnikov, A. A. Novikov, A. Glatz, G. S. Paraoanu, V. M. Vinokur, and G. B. Lesovik, "Solving Large-Scale Linear Systems of Equations by a Quantum Hybrid Algorithm", Annalen der Physik 534 7, 2200082 (2022).

[152] Johanna Barzen, "From Digital Humanities to Quantum Humanities: Potentials and Applications", arXiv:2103.11825, (2021).

[153] Kok Chuan Tan and Tyler Volkoff, "Variational quantum algorithms to estimate rank, quantum entropies, fidelity, and Fisher information via purity minimization", Physical Review Research 3 3, 033251 (2021).

[154] Benjamin A. Cordier, Nicolas P. D. Sawaya, Gian G. Guerreschi, and Shannon K. McWeeney, "Biology and medicine in the landscape of quantum advantages", arXiv:2112.00760, (2021).

[155] Davide Orsucci and Vedran Dunjko, "On solving classes of positive-definite quantum linear systems with quadratically improved runtime in the condition number", Quantum 5, 573 (2021).

[156] Ruho Kondo, Yuki Sato, Satoshi Koide, Seiji Kajita, and Hideki Takamatsu, "Computationally Efficient Quantum Expectation with Extended Bell Measurements", Quantum 6, 688 (2022).

[157] Xi He, Li Sun, Chufan Lyu, and Xiaoting Wang, "Quantum locally linear embedding for nonlinear dimensionality reduction", Quantum Information Processing 19 9, 309 (2020).

[158] Michael R. Geller, Zoë Holmes, Patrick J. Coles, and Andrew Sornborger, "Experimental quantum learning of a spectral decomposition", Physical Review Research 3 3, 033200 (2021).

[159] S. Biedron, L. Brouwer, D. L. Bruhwiler, N. M. Cook, A. L. Edelen, D. Filippetto, C. -K. Huang, A. Huebl, T. Katsouleas, N. Kuklev, R. Lehe, S. Lund, C. Messe, W. Mori, C. -K. Ng, D. Perez, P. Piot, J. Qiang, R. Roussel, D. Sagan, A. Sahai, A. Scheinker, M. Thévenet, F. Tsung, J. -L. Vay, D. Winklehner, and H. Zhang, "Snowmass21 Accelerator Modeling Community White Paper", arXiv:2203.08335, (2022).

[160] Rozhin Eskandarpour, Kumar Ghosh, Amin Khodaei, Liuxi Zhang, Aleksi Paaso, and Shay Bahramirad, "Quantum Computing Solution of DC Power Flow", arXiv:2010.02442, (2020).

[161] Hao-Kai Zhang, Chengkai Zhu, Geng Liu, and Xin Wang, "Fundamental limitations on optimization in variational quantum algorithms", arXiv:2205.05056, (2022).

[162] Lukasz Cincio, Kenneth Rudinger, Mohan Sarovar, and Patrick J. Coles, "Machine learning of noise-resilient quantum circuits", arXiv:2007.01210, (2020).

[163] Filippo M. Miatto and Nicolás Quesada, "Fast optimization of parametrized quantum optical circuits", Quantum 4, 366 (2020).

[164] Xi He, "Quantum correlation alignment for unsupervised domain adaptation", Physical Review A 102 3, 032410 (2020).

[165] Sayantan Pramanik, M Girish Chandra, C V Sridhar, Aniket Kulkarni, Prabin Sahoo, Vishwa Chethan D V, Hrishikesh Sharma, Ashutosh Paliwal, Vidyut Navelkar, Sudhakara Poojary, Pranav Shah, and Manoj Nambiar, "A Quantum-Classical Hybrid Method for Image Classification and Segmentation", arXiv:2109.14431, (2021).

[166] Pranav Gokhale, Samantha Koretsky, Shilin Huang, Swarnadeep Majumder, Andrew Drucker, Kenneth R. Brown, and Frederic T. Chong, "Quantum Fan-out: Circuit Optimizations and Technology Modeling", arXiv:2007.04246, (2020).

[167] Alicia B. Magann, Christian Arenz, Matthew D. Grace, Tak-San Ho, Robert L. Kosut, Jarrod R. McClean, Herschel A. Rabitz, and Mohan Sarovar, "From pulses to circuits and back again: A quantum optimal control perspective on variational quantum algorithms", arXiv:2009.06702, (2020).

[168] Ze-Tong Li, Fan-Xu Meng, Han Zeng, Zai-Chen Zhang, and Xu-Tao Yu, "An Efficient Gradient Sensitive Alternate Framework for VQE with Variable Ansatz", arXiv:2205.03031, (2022).

[169] Guoming Wang, Dax Enshan Koh, Peter D. Johnson, and Yudong Cao, "Minimizing estimation runtime on noisy quantum computers", arXiv:2006.09350, (2020).

[170] M. R. Perelshtein, A. I. Pakhomchik, A. A. Melnikov, A. A. Novikov, A. Glatz, G. S. Paraoanu, V. M. Vinokur, and G. B. Lesovik, "Large-scale quantum hybrid solution for linear systems of equations", arXiv:2003.12770, (2020).

[171] Quoc Chuong Nguyen, Le Bin Ho, Lan Nguyen Tran, and Hung Q. Nguyen, "Qsun: an open-source platform towards practical quantum machine learning applications", Machine Learning: Science and Technology 3 1, 015034 (2022).

[172] Carlos Bravo-Prieto, "Quantum autoencoders with enhanced data encoding", arXiv:2010.06599, (2020).

[173] Daniel O'Malley, Jessie M. Henderson, Elijah Pelofske, Sarah Greer, Yigit Subasi, John K. Golden, Robert Lowrie, and Stephan Eidenbenz, "A near-term quantum algorithm for solving linear systems of equations based on the Woodbury identity", arXiv:2205.00645, (2022).

[174] Junxiang Xiao, Jingwei Wen, Shijie Wei, and Guilu Long, "Reconstructing unknown quantum states using variational layerwise method", Frontiers of Physics 17 5, 51501 (2022).

[175] Youle Wang, Guangxi Li, and Xin Wang, "A Hybrid Quantum-Classical Hamiltonian Learning Algorithm", arXiv:2103.01061, (2021).

[176] Peter B. Weichman, "Quantum-enhanced algorithms for classical target detection in complex environments", Physical Review A 103 4, 042424 (2021).

[177] Rishabh Gupta, Manas Sajjan, Raphael D. Levine, and Sabre Kais, "Variational approach to quantum state tomography based on maximal entropy formalism", Physical Chemistry Chemical Physics (Incorporating Faraday Transactions) 24 47, 28870 (2022).

[178] Jinfeng Zeng, Zipeng Wu, Chenfeng Cao, Chao Zhang, Shiyao Hou, Pengxiang Xu, and Bei Zeng, "Simulating noisy variational quantum eigensolver with local noise models", arXiv:2010.14821, (2020).

[179] Manas Sajjan, Junxu Li, Raja Selvarajan, Shree Hari Sureshbabu, Sumit Suresh Kale, Rishabh Gupta, Vinit Singh, and Sabre Kais, "Quantum Machine Learning for Chemistry and Physics", arXiv:2111.00851, (2021).

[180] Shweta Sahoo, Utkarsh Azad, and Harjinder Singh, "Quantum phase recognition using quantum tensor networks", European Physical Journal Plus 137 12, 1373 (2022).

[181] Rolando D. Somma and Yigit Subasi, "Complexity of quantum state verification in the quantum linear systems problem", arXiv:2007.15698, (2020).

[182] James R. Wootton, Francis Harkins, Nicholas T. Bronn, Almudena Carrera Vazquez, Anna Phan, and Abraham T. Asfaw, "Teaching quantum computing with an interactive textbook", arXiv:2012.09629, (2020).

[183] Merey M. Sarsengeldin, "A Hybrid Classical-Quantum framework for solving Free Boundary Value Problems and Applications in Modeling Electric Contact Phenomena", arXiv:2205.02230, (2022).

[184] Stefano Mangini, Alessia Marruzzo, Marco Piantanida, Dario Gerace, Daniele Bajoni, and Chiara Macchiavello, "Quantum neural network autoencoder and classifier applied to an industrial case study", arXiv:2205.04127, (2022).

[185] Kaixuan Huang, Xiaoxia Cai, Hao Li, Zi-Yong Ge, Ruijuan Hou, Hekang Li, Tong Liu, Yunhao Shi, Chitong Chen, Dongning Zheng, Kai Xu, Zhi-Bo Liu, Zhendong Li, Heng Fan, and Wei-Hai Fang, "Variational Quantum Computation of Molecular Linear Response Properties on a Superconducting Quantum Processor", arXiv:2201.02426, (2022).

[186] Fei Feng, Peng Zhang, Yifan Zhou, and Zefan Tang, "Quantum microgrid state estimation", Electric Power Systems Research 212, 108386 (2022).

[187] Yipeng Huang, Steven Holtzen, Todd Millstein, Guy Van den Broeck, and Margaret Martonosi, "Logical Abstractions for Noisy Variational Quantum Algorithm Simulation", arXiv:2103.17226, (2021).

[188] Dylan Herman, Rudy Raymond, Muyuan Li, Nicolas Robles, Antonio Mezzacapo, and Marco Pistoia, "Expressivity of Variational Quantum Machine Learning on the Boolean Cube", arXiv:2204.05286, (2022).

[189] Pedro Rivero, Ian C. Cloët, and Zack Sullivan, "An optimal quantum sampling regression algorithm for variational eigensolving in the low qubit number regime", arXiv:2012.02338, (2020).

[190] Fanxu Meng, "Quantum Algorithm for DOA Estimation in Hybrid Massive MIMO", arXiv:2102.03963, (2021).

[191] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

[192] Carlos A. Riofrío, Johannes Klepsch, Jernej Rudi Finžgar, Florian Kiwit, Leonhard Hölscher, Marvin Erdmann, Lukas Müller, Chandan Kumar, Youssef Achari Berrada, and Andre Luckow, "Quantum Computing for Automotive Applications", arXiv:2409.14183, (2024).

[193] Maxwell Aifer, Kaelan Donatella, Max Hunter Gordon, Samuel Duffield, Thomas Ahle, Daniel Simpson, Gavin E. Crooks, and Patrick J. Coles, "Thermodynamic Linear Algebra", arXiv:2308.05660, (2023).

[194] M. Cerezo, Kunal Sharma, Andrew Arrasmith, and Patrick J. Coles, "Variational quantum state eigensolver", npj Quantum Information 8 1, 113 (2022).

[195] Sitan Chen, Jordan Cotler, Hsin-Yuan Huang, and Jerry Li, "The complexity of NISQ", Nature Communications 14, 6001 (2023).

[196] He-Liang Huang, Xiao-Yue Xu, Chu Guo, Guojing Tian, Shi-Jie Wei, Xiaoming Sun, Wan-Su Bao, and Gui-Lu Long, "Near-term quantum computing techniques: Variational quantum algorithms, error mitigation, circuit compilation, benchmarking and classical simulation", Science China Physics, Mechanics, and Astronomy 66 5, 250302 (2023).

[197] Nikita A. Zemlevskiy, "Scalable Quantum Simulations of Scattering in Scalar Field Theory on 120 Qubits", arXiv:2411.02486, (2024).

[198] Christo Meriwether Keller, Stephan Eidenbenz, Andreas Bärtschi, Daniel O'Malley, John Golden, and Satyajayant Misra, "Hierarchical Multigrid Ansatz for Variational Quantum Algorithms", arXiv:2312.15048, (2023).

[199] Han Qi, Sihui Xiao, Zhuo Liu, Changqing Gong, and Abdullah Gani, "Variational quantum algorithms: fundamental concepts, applications and challenges", Quantum Information Processing 23 6, 224 (2024).

[200] Ar A. Melnikov, A. A. Termanova, S. V. Dolgov, F. Neukart, and M. R. Perelshtein, "Quantum state preparation using tensor networks", Quantum Science and Technology 8 3, 035027 (2023).

[201] Bujiao Wu, Jinzhao Sun, Qi Huang, and Xiao Yuan, "Overlapped grouping measurement: A unified framework for measuring quantum states", Quantum 7, 896 (2023).

[202] Travis L. Scholten, Carl J. Williams, Dustin Moody, Michele Mosca, William Hurley, William J. Zeng, Matthias Troyer, and Jay M. Gambetta, "Assessing the Benefits and Risks of Quantum Computers", arXiv:2401.16317, (2024).

[203] Sriharsha Kocherla, Zhixin Song, Fatima Ezahra Chrit, Bryan Gard, Eugene F. Dumitrescu, Alexander Alexeev, and Spencer H. Bryngelson, "Fully quantum algorithm for mesoscale fluid simulations with application to partial differential equations", AVS Quantum Science 6 3, 033806 (2024).

[204] Nic Ezzell, Elliott M. Ball, Aliza U. Siddiqui, Mark M. Wilde, Andrew T. Sornborger, Patrick J. Coles, and Zoë Holmes, "Quantum mixed state compiling", Quantum Science and Technology 8 3, 035001 (2023).

[205] Romina Yalovetzky, Pierre Minssen, Dylan Herman, and Marco Pistoia, "Solving linear systems on quantum hardware with hybrid HHL<SUP>++</SUP>", Scientific Reports 14 1, 20610 (2024).

[206] Samson Wang, Sam McArdle, and Mario Berta, "Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra", PRX Quantum 5 2, 020324 (2024).

[207] Annie E. Paine, Vincent E. Elfving, and Oleksandr Kyriienko, "Quantum kernel methods for solving regression problems and differential equations", Physical Review A 107 3, 032428 (2023).

[208] Yovav Tene-Cohen, Tomer Kelman, Ohad Lev, and Adi Makmal, "A Variational Qubit-Efficient MaxCut Heuristic Algorithm", arXiv:2308.10383, (2023).

[209] Xiang Rao, "Performance study of variational quantum linear solver with an improved ansatz for reservoir flow equations", Physics of Fluids 36 4, 047104 (2024).

[210] Takanori Nishi and Kaoru Yamanouchi, "Simulation of a spin-boson model by iterative optimization of a parametrized quantum circuit", AVS Quantum Science 6 2, 023801 (2024).

[211] Osama Muhammad Raisuddin and Suvranu De, "A Review of Quantum Scientific Computing Algorithms for Engineering Problems", arXiv:2408.13943, (2024).

[212] Yuki Sato, Hiroshi C. Watanabe, Rudy Raymond, Ruho Kondo, Kaito Wada, Katsuhiro Endo, Michihiko Sugawara, and Naoki Yamamoto, "Variational quantum algorithm for generalized eigenvalue problems and its application to the finite-element method", Physical Review A 108 2, 022429 (2023).

[213] S. Golestan, M. R. Habibi, S. Y. Mousazadeh Mousavi, J. M. Guerrero, and J. C. Vasquez, "Quantum computation in power systems: An overview of recent advances", Energy Reports 9, 584 (2023).

[214] N. M. Guseynov, A. A. Zhukov, W. V. Pogosov, and A. V. Lebedev, "Depth analysis of variational quantum algorithms for the heat equation", Physical Review A 107 5, 052422 (2023).

[215] Maksims Dimitrijevs, Mārtiņš Kālis, and Iļja Repko, "Exploring Hybrid Quantum-Classical Methods for Practical Time-Series Forecasting", arXiv:2412.05615, (2024).

[216] Anna Schroeder, Matthias Heller, and Mariami Gachechiladze, "Deterministic Ansätze for the measurement-based variational quantum eigensolver", New Journal of Physics 26 6, 063019 (2024).

[217] Giorgio Tosti Balducci, Boyang Chen, Matthias Möller, and Roeland De Breuker, "Solving 1D Poisson problem with a Variational Quantum Linear Solver", arXiv:2412.04938, (2024).

[218] Guojian Wu, Fang Gao, Qing Gao, and Yu Pan, "Quantum Discrete Adiabatic Linear Solver based on Block Encoding and Eigenvalue Separator", arXiv:2412.06202, (2024).

[219] Óscar Amaro and Diogo Cruz, "A Living Review of Quantum Computing for Plasma Physics", arXiv:2302.00001, (2023).

[220] 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", arXiv:2402.18619, (2024).

[221] Anton Simen Albino, Lucas Correia Jardim, Diego Campos Knupp, Antonio Jose Silva Neto, Otto Menegasso Pires, and Erick Giovani Sperandio Nascimento, "Solving partial differential equations on near-term quantum computers", arXiv:2208.05805, (2022).

[222] M. Schumann, F. K. Wilhelm, and A. Ciani, "Emergence of noise-induced barren plateaus in arbitrary layered noise models", Quantum Science and Technology 9 4, 045019 (2024).

[223] Junpeng Zhan, "Variational Quantum Search with Shallow Depth for Unstructured Database Search", arXiv:2212.09505, (2022).

[224] Mazen Ali and Matthias Kabel, "Performance Study of Variational Quantum Algorithms for Solving the Poisson Equation on a Quantum Computer", Physical Review Applied 20 1, 014054 (2023).

[225] Kaito Wada, Rudy Raymond, Yuki Sato, and Hiroshi C. Watanabe, "Sequential optimal selections of single-qubit gates in parameterized quantum circuits", Quantum Science and Technology 9 3, 035030 (2024).

[226] Marc Andreu Marfany, Alona Sakhnenko, and Jeanette Miriam Lorenz, "Identifying Bottlenecks of NISQ-friendly HHL algorithms", arXiv:2406.06288, (2024).

[227] Gabriel Matos, Chris N. Self, Zlatko Papić, Konstantinos Meichanetzidis, and Henrik Dreyer, "Characterization of variational quantum algorithms using free fermions", Quantum 7, 966 (2023).

[228] Po-Wei Huang and Patrick Rebentrost, "Post-variational quantum neural networks", arXiv:2307.10560, (2023).

[229] Anne-Solène Bornens and Michel Nowak, "Variational quantum algorithms on cat qubits", arXiv:2305.14143, (2023).

[230] Nishant Saurabh, Pradeep Mantha, Florian J. Kiwit, Shantenu Jha, and Andre Luckow, "Quantum Mini-Apps: A Framework for Developing and Benchmarking Quantum-HPC Applications", arXiv:2405.07333, (2024).

[231] A. Avkhadiev, P. E. Shanahan, and R. D. Young, "Strategies for quantum-optimized construction of interpolating operators in classical simulations of lattice quantum field theories", Physical Review D 107 5, 054507 (2023).

[232] Nishant Saurabh, Shantenu Jha, and Andre Luckow, "A Conceptual Architecture for a Quantum-HPC Middleware", arXiv:2308.06608, (2023).

[233] Alexis Ralli, Tim Weaving, Andrew Tranter, William M. Kirby, Peter J. Love, and Peter V. Coveney, "Unitary partitioning and the contextual subspace variational quantum eigensolver", Physical Review Research 5 1, 013095 (2023).

[234] Niraj Kumar, Jamie Heredge, Changhao Li, Shaltiel Eloul, Shree Hari Sureshbabu, and Marco Pistoia, "Expressive variational quantum circuits provide inherent privacy in federated learning", arXiv:2309.13002, (2023).

[235] Aidan Pellow-Jarman, Ilya Sinayskiy, Anban Pillay, and Francesco Petruccione, "Near term algorithms for linear systems of equations", Quantum Information Processing 22 6, 258 (2023).

[236] Melody Lee, Zhixin Song, Sriharsha Kocherla, Austin Adams, Alexander Alexeev, and Spencer H. Bryngelson, "A multiple-circuit approach to quantum resource reduction with application to the quantum lattice Boltzmann method", arXiv:2401.12248, (2024).

[237] L. Zambrano, A. D. Muñoz-Moller, M. Muñoz, L. Pereira, and A. Delgado, "Avoiding barren plateaus in the variational determination of geometric entanglement", Quantum Science and Technology 9 2, 025016 (2024).

[238] Mohammadreza Soltaninia and Junpeng Zhan, "Quantum Global Minimum Finder based on Variational Quantum Search", arXiv:2405.00450, (2024).

[239] Junyu Liu, Han Zheng, Masanori Hanada, Kanav Setia, and Dan Wu, "Quantum Power Flows: From Theory to Practice", arXiv:2211.05728, (2022).

[240] Luca Arceci, Viacheslav Kuzmin, and Rick Van Bijnen, "Gaussian process model kernels for noisy optimization in variational quantum algorithms", arXiv:2412.13271, (2024).

[241] Stefano Markidis, "On Physics-Informed Neural Networks for Quantum Computers", arXiv:2209.14754, (2022).

[242] Payal Kaushik, Sayantan Pramanik, M Girish Chandra, and C V Sridhar, "One-Step Time Series Forecasting Using Variational Quantum Circuits", arXiv:2207.07982, (2022).

[243] Rishabh Gupta, Raja Selvarajan, Manas Sajjan, Raphael D. Levine, and Sabre Kais, "Hamiltonian Learning from Time Dynamics Using Variational Algorithms", Journal of Physical Chemistry A 127 14, 3246 (2023).

[244] Brian Coyle, "Machine learning applications for noisy intermediate-scale quantum computers", arXiv:2205.09414, (2022).

[245] Arun Sehrawat, "Interferometric Neural Networks", arXiv:2310.16742, (2023).

[246] Océane Koska, Marc Baboulin, and Arnaud Gazda, "A tree-approach Pauli decomposition algorithm with application to quantum computing", arXiv:2403.11644, (2024).

[247] Tianxiang Yue, Chenchen Wu, Yi Liu, Zhengping Du, Na Zhao, Yimeng Jiao, Zhe Xu, and Wenjiao Shi, "HASM quantum machine learning", Science China Earth Sciences 66 9, 1937 (2023).

[248] Ajinkya Borle and Samuel J. Lomonaco, "How viable is quantum annealing for solving linear algebra problems?", arXiv:2206.10576, (2022).

[249] Xiaodong Xing, Alejandro Gomez Cadavid, Artur F. Izmaylov, and Timur V. Tscherbul, "A hybrid quantum-classical algorithm for multichannel quantum scattering of atoms and molecules", arXiv:2304.06089, (2023).

[250] Sergio Bengoechea, Paul Over, Dieter Jaksch, and Thomas Rung, "Towards Variational Quantum Algorithms for generalized linear and nonlinear transport phenomena", arXiv:2411.14931, (2024).

[251] Mina Doosti, "Unclonability and Quantum Cryptanalysis: From Foundations to Applications", arXiv:2210.17545, (2022).

[252] Katsuhiro Endo, Yuki Sato, Rudy Raymond, Kaito Wada, Naoki Yamamoto, and Hiroshi C. Watanabe, "Optimal parameter configurations for sequential optimization of the variational quantum eigensolver", Physical Review Research 5 4, 043136 (2023).

[253] Hamza Jaffali, Jonas Bastos de Araujo, Nadia Milazzo, Marta Reina, Henri de Boutray, Karla Baumann, and Frédéric Holweck, "H-DES: a Quantum-Classical Hybrid Differential Equation Solver", arXiv:2410.01130, (2024).

[254] Reza Mahroo and Amin Kargarian, "Trainable Variational Quantum-Multiblock ADMM Algorithm for Generation Scheduling", arXiv:2303.16318, (2023).

[255] Anton Simen Albino, Otto Menegasso Pires, Peterson Nogueira, Renato Ferreira de Souza, and Erick Giovani Sperandio Nascimento, "Quantum computational intelligence for traveltime seismic inversion", arXiv:2208.05794, (2022).

[256] Shao-Hen Chiew and Leong-Chuan Kwek, "Scalable Quantum Computation of Highly Excited Eigenstates with Spectral Transforms", arXiv:2302.06638, (2023).

[257] Daniel Mastropietro, Georgios Korpas, Vyacheslav Kungurtsev, and Jakub Marecek, "Parallel variational quantum algorithms with gradient-informed restart to speed up optimisation in the presence of barren plateaus", arXiv:2311.18090, (2023).

[258] Yulun Wang and Predrag S. Krstić, "Multistate transition dynamics by strong time-dependent perturbation in NISQ era", Journal of Physics Communications 7 7, 075004 (2023).

[259] Jessie M. Henderson, Marianna Podzorova, M. Cerezo, John K. Golden, Leonard Gleyzer, Hari S. Viswanathan, and Daniel O'Malley, "Quantum algorithms for geologic fracture networks", Scientific Reports 13, 2906 (2023).

[260] Dekel Meirom, Ittay Alfassi, and Tal Mor, "Discretized Quantum Exhaustive Search for Variational Quantum Algorithms", arXiv:2407.17659, (2024).

[261] Abeynaya Gnanasekaran and Amit Surana, "Efficient Variational Quantum Linear Solver for Structured Sparse Matrices", arXiv:2404.16991, (2024).

[262] Hyeong-Gyu Kim, Siheon Park, and June-Koo Kevin Rhee, "Variational Quantum Approximate Spectral Clustering for Binary Clustering Problems", arXiv:2309.04465, (2023).

[263] Hansheng Jiang, Zuo-Jun Max Shen, and Junyu Liu, "Quantum Computing Methods for Supply Chain Management", arXiv:2209.08246, (2022).

[264] Jessie M. Henderson, Marianna Podzorova, M. Cerezo, John K. Golden, Leonard Gleyzer, Hari S. Viswanathan, and Daniel O'Malley, "Quantum Algorithms for Geologic Fracture Networks", arXiv:2210.11685, (2022).

[265] Oliver Knitter, James Stokes, and Shravan Veerapaneni, "Toward Neural Network Simulation of Variational Quantum Algorithms", arXiv:2211.02929, (2022).

[266] Benjamin Wu, Hrushikesh Patil, and Predrag Krstic, "Effect of matrix sparsity and quantum noise on quantum random walk linear solvers", arXiv:2205.14180, (2022).

[267] Xi He, Feiyu Du, Mingyuan Xue, Xiaogang Du, Tao Lei, and A. K. Nandi, "Quantum classifiers for domain adaptation", Quantum Information Processing 22 2, 105 (2023).

[268] Sunheang Ty, Renaud Vilmart, Axel TahmasebiMoradi, and Chetra Mang, "Double-Logarithmic Depth Block-Encodings of Simple Finite Difference Method's Matrices", arXiv:2410.05241, (2024).

[269] Shuai Zhang, Ke-Xin Chen, and Ji-Chong Yang, "Detect anomalous quartic gauge couplings at muon colliders with quantum kernel k-means", arXiv:2409.07010, (2024).

[270] Dirk Oliver Theis, ""Proper" Shift Rules for Derivatives of Perturbed-Parametric Quantum Evolutions", Quantum 7, 1052 (2023).

[271] Mazen Ali and Matthias Kabel, "Piecewise Polynomial Tensor Network Quantum Feature Encoding", arXiv:2402.07671, (2024).

[272] N. M. Guseynov, W. V. Pogosov, and A. V. Lebedev, "Dynamical quantum Ansatz tree approach for the heat equation", arXiv:2404.14102, (2024).

[273] Ruimin Shang, Zhimin Wang, Shangshang Shi, Jiaxin Li, Yanan Li, and Yongjian Gu, "Algorithm for simulating ocean circulation on a quantum computer", Science China Earth Sciences 66 10, 2254 (2023).

[274] Oxana Shaya, "When could NISQ algorithms start to create value in discrete manufacturing ?", arXiv:2209.09650, (2022).

[275] Amit Surana and Abeynaya Gnanasekaran, "Variational Quantum Framework for Partial Differential Equation Constrained Optimization", arXiv:2405.16651, (2024).

[276] Yoshiyuki Saito, Xinwei Lee, Dongsheng Cai, and Nobuyoshi Asai, "Quantum Multi-Resolution Measurement with application to Quantum Linear Solver", arXiv:2304.05960, (2023).

[277] Dingjie Lu, Zhao Wang, Jun Liu, Yangfan Li, Wei-Bin Ewe, and Zhuangjian Liu, "From Ad-Hoc to Systematic: A Strategy for Imposing General Boundary Conditions in Discretized PDEs in variational quantum algorithm", arXiv:2310.11764, (2023).

[278] Felicia Barbato, Marc Barthelemy, Christian Beck, Jacob D. Biamonte, J. Ignacio Cirac, Daniel Malz, Antigone Marino, Zeki Can Seskir, and Javier Ventura-Traveset, "Physics for secure and efficient societies", EPS Grand Challenges 7 (2024).

[279] Nicolas PD Sawaya and Joonsuk Huh, "Improved resource-tunable near-term quantum algorithms for transition probabilities, with applications in physics and variational quantum linear algebra", arXiv:2206.14213, (2022).

[280] Abeynaya Gnanasekaran, Amit Surana, and Hongyu Zhu, "Variational Quantum Framework for Nonlinear PDE Constrained Optimization Using Carleman Linearization", arXiv:2410.13688, (2024).

[281] Po-Wei Huang, Xiufan Li, Kelvin Koor, and Patrick Rebentrost, "Hybrid quantum-classical and quantum-inspired classical algorithms for solving banded circulant linear systems", arXiv:2309.11451, (2023).

[282] Willie Aboumrad and Dominic Widdows, "Mod2VQLS: a Variational Quantum Algorithm for Solving Systems of Linear Equations Modulo 2", arXiv:2311.12771, (2023).

[283] Sanjay Suresh and Krishnan Suresh, "Computing a Sparse Approximate Inverse on Quantum Annealing Machines", arXiv:2310.02388, (2023).

[284] Kazue Kudo, "Annealing-based approach to solving partial differential equations", arXiv:2406.17364, (2024).

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