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  • Open Access

Resource-efficient quantum algorithm for linear systems of equations

Francesco Ghisoni1,*, Francesco Scala1, Daniele Bajoni2, and Dario Gerace1

  • *Contact author: francesco.ghisoni01@universitadipavia.it
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Phys. Rev. A 113, 032405 – Published 3 March, 2026

DOI: https://doi.org/10.1103/gwzg-4pls

Abstract

Finding the solution to linear systems is at the heart of many applications in science and technology. Over the years, a number of algorithms have been proposed to solve this problem on a digital quantum device, yet most of these are too demanding to be applied to the current noisy hardware. In this work, an original algorithmic procedure to solve the quantum linear system problem (QLSP) is presented, which combines ideas from variational quantum algorithms and the framework of classical shadows. The result is the shadow quantum linear solver (SQLS), a quantum algorithm solving the QLSP while avoiding the need for large controlled unitaries and requiring a number of qubits that is logarithmic in the system size. In particular, our heuristics show an exponential advantage of the SQLS in circuit execution per cost function evaluation when compared with other notorious variational approaches to solving linear systems of equations. We test the convergence of the SQLS on a number of linear systems, and the results highlight how the theoretical bounds on the number of resources used by the SQLS are conservative. Finally, we apply this algorithm to a physical problem of practical relevance by leveraging decomposition theorems from linear algebra to solve the discretized Laplace equation in a two-dimensional grid.

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References (84)

  1. P. Wang, S. Mou, J. Lian, and W. Ren, Solving a system of linear equations: From centralized to distributed algorithms, Annu. Rev. Control 47, 306 (2019).
  2. J. Rahola, Solution of dense systems of linear equations in the discrete-dipole approximation, SIAM J. Sci. Comput. 17, 78 (1996).
  3. B. Carpentieri, I. Duff, L. Giraud, and M. Magolu Monga Made, Sparse symmetric preconditioners for dense linear systems in electromagnetism, Numer. Linear Algebra Appl. 11, 753 (2004).
  4. A. Frank, D. Fabregat-Traver, and P. Bientinesi, Large-scale linear regression: Development of high-performance routines, Appl. Math. Comput. 275, 411 (2016).
  5. R. P. Feynman, There's plenty of room at the bottom, J. Sci. Educ. 16, 890 (1959).
  6. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000).
  7. P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput. 26, 1484 (1997).
  8. L. K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing (STOC'96) (Association for Computing Machinery, New York, 1996), pp. 212–219.
  9. A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, Phys. Rev. Lett. 103, 150502 (2009).
  10. A. Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations, in 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012) (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2012).
  11. 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).
  12. S. Chakraborty, A. Gilyén, and S. Jeffery, The Power of Block-Encoded Matrix Powers: Improved Regression Techniques via Faster Hamiltonian Simulation (Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2019).
  13. A. M. Childs, R. Kothari, and R. D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision, SIAM J. Comput. 46, 1920 (2017).
  14. L. Wossnig, Z. Zhao, and A. Prakash, Quantum linear system algorithm for dense matrices, Phys. Rev. Lett. 120, 050502 (2018).
  15. S.-j. Wei, Z.-r. Zhou, D. Ruan, and G.-L. Long, Realization of the algorithm for system of linear equations in duality quantum computing, in 2017 IEEE 85th Vehicular Technology Conference (VTC Spring) (Sydney, Australia, 2017), pp. 1–4.
  16. L. Gui-Lu, General quantum interference principle and duality computer, Commun. Theor. Phys. 45, 825 (2006).
  17. L. Gui-Lu and L. Yang, Duality computing in quantum computers, Commun. Theor. Phys. 50, 1303 (2008).
  18. L. Gui-Lu, L. Yang, and W. Chuan, Allowable generalized quantum gates, Commun. Theor. Phys. 51, 65 (2009).
  19. N. Wiebe, D. Braun, and S. Lloyd, Quantum algorithm for data fitting, Phys. Rev. Lett. 109, 050505 (2012).
  20. B. D. Clader, B. C. Jacobs, and C. R. Sprouse, Preconditioned quantum linear system algorithm, Phys. Rev. Lett. 110, 250504 (2013).
  21. G. Wang, Efficient quantum algorithms for analyzing large sparse electrical networks, Quantum Inf. Comput. 17, 987 (2017).
  22. P. Rebentrost, M. Mohseni, and S. Lloyd, Quantum support vector machine for big data classification, Phys. Rev. Lett. 113, 130503 (2014).
  23. I. Kerenidis and A. Prakash, Quantum recommendation systems, in 8th Innovations in Theoretical Computer Science Conference (ITCS 2017) (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2017).
  24. R. D. Somma, G. H. Low, D. W. Berry, and R. Babbush, Quantum algorithm for linear matrix equations, arXiv:2508.02822.
  25. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  26. 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).
  27. M. AbuGhanem and H. Eleuch, Two-qubit entangling gates for superconducting quantum computers, Results Phys. 56, 107236 (2024).
  28. S. A. Moses, C. H. Baldwin, M. S. Allman, R. Ancona, L. Ascarrunz, C. Barnes, J. Bartolotta, B. Bjork, P. Blanchard, M. Bohn, J. G. Bohnet, N. C. Brown, N. Q. Burdick, W. C. Burton, S. L. Campbell, J. P. Campora, C. Carron, J. Chambers, J. W. Chan, Y. H. Chen, et al., A race-track trapped-ion quantum processor, Phys. Rev. X 13, 041052 (2023).
  29. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, and A. Kandala, Evidence for the utility of quantum computing before fault tolerance, Nature (London) 618, 500 (2023).
  30. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  31. M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini, Parameterized quantum circuits as machine learning models, Quantum Sci. Technol. 4, 043001 (2019).
  32. J. Jäger and R. V. Krems, Universal expressiveness of variational quantum classifiers and quantum kernels for support vector machines, Nat. Commun. 14, 576 (2023).
  33. 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, Nat. Commun. 5, 4213 (2014).
  34. Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, M. Kieferová, I. D. Kivlichan, T. Menke, B. Peropadre, N. P. D. Sawaya, S. Sim, L. Veis, and A. Aspuru-Guzik, Quantum chemistry in the age of quantum computing, Chem. Rev. 119, 10856 (2019).
  35. O. Higgott, D. Wang, and S. Brierley, Variational quantum computation of excited states, Quantum 3, 156 (2019).
  36. 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).
  37. F. Scala, S. Mangini, C. Macchiavello, D. Bajoni, and D. Gerace, Quantum variational learning for entanglement witnessing, in 2022 International Joint Conference on Neural Networks (IJCNN) (IEEE, Padua, Italy, 2022).
  38. M. Lubasch, J. Joo, P. Moinier, M. Kiffner, and D. Jaksch, Variational quantum algorithms for nonlinear problems, Phys. Rev. A 101, 010301(R) (2020).
  39. A. J. Pool, A. D. Somoza, C. M. Keever, M. Lubasch, and B. Horstmann, Nonlinear dynamics as a ground-state solution on quantum computers, Phys. Rev. Res. 6, 033257 (2024).
  40. A. Pellow-Jarman, I. Sinayskiy, A. Pillay, and F. Petruccione, Near term algorithms for linear systems of equations, Quantum Inf. Process 22, 258 (2023).
  41. C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, and P. J. Coles, Variational quantum linear solver, Quantum 7, 1188 (2023).
  42. H.-L. Liu, Y.-S. Wu, L.-C. Wan, S.-J. Pan, S.-J. Qin, F. Gao, and Q.-Y. Wen, Variational quantum algorithm for the Poisson equation, Phys. Rev. A 104, 022418 (2021).
  43. Y. Liu, Z. Chen, C. Shu, P. Rebentrost, Y. Liu, S. Chew, B. Khoo, and Y. Cui, A variational quantum algorithm-based numerical method for solving potential and Stokes flows, Ocean Eng. 292, 116494 (2024).
  44. Y. Y. Liu, Z. Chen, C. Shu, S. C. Chew, B. C. Khoo, X. Zhao, and Y. D. Cui, Application of a variational hybrid quantum-classical algorithm to heat conduction equation and analysis of time complexity, Phys. Fluids 34, 117121 (2022).
  45. H. Luo, Q. Zhou, Z. Li, and Y. Deng, Variational quantum linear solver-based combination rules in Dempster–Shafer theory, Inf. Fusion 102, 102070 (2024).
  46. X. Xing, A. G. Cadavid, A. F. Izmaylov, and T. V. Tscherbul, A hybrid quantum-classical algorithm for multichannel quantum scattering of atoms and molecules, J. Phys. Chem. Lett. 14 (2023).
  47. C. J. Trahan, M. Loveland, N. Davis, and E. Ellison, A variational quantum linear solver application to discrete finite-element methods, Entropy 25, 580 (2023).
  48. G. M. D'Ariano and P. Perinotti, Optimal data processing for quantum measurements, Phys. Rev. Lett. 98, 020403 (2007).
  49. S. Aaronson, Shadow tomography of quantum states, in Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2018) (Association for Computing Machinery, New York, 2018), pp. 325–338.
  50. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  51. F. Truger, J. Barzen, F. Leymann, and J. Obst, Warm-starting the VQE with approximate complex amplitude encoding, in Proceedings of the 1st International Conference on Quantum Software IQSOFT (SciTePress, 2025), Vol. 1, pp. 15–26.
  52. A. Basheer, Y. Feng, C. Ferrie, and S. Li, Alternating layered variational quantum circuits can be classically optimized efficiently using classical shadows, in Proceedings of the AAAI Conference on Artificial Intelligence (AAAI, 2024), Vol. 37.
  53. S. H. Sack, R. A. Medina, A. A. Michailidis, R. Kueng, and M. Serbyn, Avoiding barren plateaus using classical shadows, PRX Quantum 3, 020365 (2022).
  54. J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 4812 (2018).
  55. M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Biamonte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, A review of barren plateaus in variational quantum computing, Nat. Rev. Phys. 7, 174 (2025).
  56. M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Cost function dependent barren plateaus in shallow parametrized quantum circuits, Nat. Commun. 12, 1791 (2021).
  57. K. Zhang, L. Liu, M.-H. Hsieh, and D. Tao, Escaping from the barren plateau via Gaussian initializations in deep variational quantum circuits, in Advances in Neural Information Processing Systems, edited by S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (Curran Associates, Inc., 2022), Vol. 35, pp. 18612–18627.
  58. C.-Y. Park and N. Killoran, Hamiltonian variational ansatz without barren plateaus, Quantum 8, 1239 (2024).
  59. Y. Wang, B. Qi, C. Ferrie, and D. Dong, Trainability enhancement of parameterized quantum circuits via reduced-domain parameter initialization, Phys. Rev. Appl. 22, 054005 (2023).
  60. K. Sharma, S. Khatri, M. Cerezo, and P. J. Coles, Noise resilience of variational quantum compiling, New J. Phys. 22, 043006 (2020).
  61. R. M. N. Pesce and P. D. Stevenson, H2ZIXY: Pauli spin matrix decomposition of real symmetric matrices, arXiv:2111.00627.
  62. S. Vidal Romero and J. Santos-Suárez, Paulicomposer: Compute tensor products of Pauli matrices efficiently, Quantum Info. Proc. 22, 449 (2023).
  63. V. Bergholm, J. Izaac, M. Schuld, C. Gogolin, S. Ahmed, V. Ajith, M. S. Alam, G. Alonso-Linaje, B. AkashNarayanan, A. Asadi, J. M. Arrazola, U. Azad, S. Banning, C. Blank, T. R. Bromley, B. A. Cordier, J. Ceroni, A. Delgado, O. D. Matteo, A. Dusko, et al., Pennylane: Automatic differentiation of hybrid quantum-classical computations, arXiv:1811.04968.
  64. T. Jones, Decomposing dense matrices into dense Pauli tensors, arXiv:2401.16378.
  65. L. Hantzko, L. Binkowski, and S. Gupta, Tensorized Pauli decomposition algorithm, Phys. Scr. 99, 085128 (2024).
  66. E. Knill and R. Laflamme, Power of one bit of quantum information, Phys. Rev. Lett. 81, 5672 (1998).
  67. T. Morimae, Hardness of classically sampling the one-clean-qubit model with constant total variation distance error, Phys. Rev. A 96, 040302(R) (2017).
  68. 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).
  69. A. Bernasconi, A. Berti, G. M. d. Corso, and A. Poggiali, Quantum subroutine for efficient matrix multiplication, IEEE Access 12, 116274 (2024).
  70. F. Ghisoni, Code and data for: Shadow quantum linear solver (2025), https://github.com/fghisoni/ShadowQuantum LinearSolver.
  71. F. Tacchino, A. Chiesa, S. Carretta, and D. Gerace, Quantum computers as universal quantum simulators: State‐of‐the‐art and perspectives, Adv. Quantum Technol. 3, 1900052 (2019).
  72. A. Pellow-Jarman, I. Sinayskiy, A. Pillay, and F. Petruccione, A comparison of various classical optimizers for a variational quantum linear solver, Quantum Info. Proc. 20, 202 (2021).
  73. M. J. D. Powell, An efficient method for finding the minimum of a function of several variables without calculating derivatives, Comput. J. 7, 155 (1964).
  74. D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv:1412.6980.
  75. P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, et al., SciPy 1.0: Fundamental algorithms for scientific computing in python, Nat. Methods 17, 261 (2020).
  76. F. R. K. Chung, Spectral Graph Theory (American Mathematical Society, Providence, 1997).
  77. P. G. Doyle and J. L. Snell, Random walks and electric networks, arXiv:math/0001057.
  78. G. Pedersen, Analysis Now, Graduate Texts in Mathematics (Springer-Verlag, New York, 1989).
  79. H.-Y. Huang, R. Kueng, and J. Preskill, Efficient estimation of Pauli observables by derandomization, Phys. Rev. Lett. 127, 030503 (2021).
  80. H.-Y. Huang, S. Chen, and J. Preskill, Learning to predict arbitrary quantum processes, PRX Quantum 4, 040337 (2023).
  81. C. Hadfield, S. Bravyi, R. Raymond, and A. Mezzacapo, Measurements of quantum Hamiltonians with locally-biased classical shadows, Commun. Math. Phys. 391, 951 (2022).
  82. D. Grier, H. Pashayan, and L. Schaeffer, Principal eigenstate classical shadows, in Proceedings of Thirty Seventh Conference on Learning Theory (PMLR, 2024), pp. 2122–2165.
  83. L. E. Fischer, T. Dao, I. Tavernelli, and F. Tacchino, Dual frame optimization for informationally complete quantum measurements, Phys. Rev. A 109, 062415 (2024).
  84. H. Patil, Y. Wang, and P. S. Krstić, Variational quantum linear solver with a dynamic ansatz, Phys. Rev. A 105, 012423 (2022).

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Phys. Rev. A 113, 032405– Published 3 March, 2026
  • Received 4 September 2025
  • Accepted 11 February 2026
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