Quantum Information & Computation
Volume 25, Issue 4
RESEARCH ARTICLE Open access

Matrix Encoding Method in Variational Quantum Singular Value Decomposition

Alexander I. Zenchuk, Wentao Qi, Junde Wu

Author affiliations
  • Alexander I. Zenchuk1
  • Wentao Qi2
  • Junde Wu3
  1. Federal Research Center of Problems of Chemical Physics and Medicinal Chemistry RAS, Chernogolovka, Moscow reg., Russia
  2. Institute of Quantum Computing and Computer Theory, School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou, China
  3. School of Mathematical Sciences, Zhejiang University, Hangzhou, China

pp. 356-368 · 10.2478/qic-2025-0020

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Abstract

We propose the variational quantum singular value decomposition based on encoding the elements of the considered N × N matrix into the state of a quantum system of appropriate dimension. This method doesn’t use the expansion of this matrix in terms of the unitary matrices. Controlled measurement is involved to avoid small success probability in ancilla measurement. The objective function for maximization algorithm can be obtained probabilistically via measurement of the states of two one-qubit subsystems. The circuit requires O(log N) qubits for realization of this algorithm whose depths is proportional to log N / ε, where ε is the precision required for calculation of singular values.
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Keywords

  • quantum singular value decomposition
  • matrix encoding
  • controlled measurement
  • probabilistic method
  • gradient method

Suggested citation

Alexander I. Zenchuk, Wentao Qi, Junde Wu (2025). Matrix Encoding Method in Variational Quantum Singular Value Decomposition. Quantum Information & Computation, 25(4), 356-368. https://doi.org/10.2478/qic-2025-0020

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References (39)

  1. D. Coppersmith (2002). An Approximate Fourier Transform Useful in Quantum Factoring, IBM Research Report, RC 19642. https://api.semanticscholar.org/CorpusID:17450629.Google Scholar
  2. Y.S. Weinstein, M.A. Pravia, E.M. Fortunato, S. Lloyd and D.G. Cory (2001). “Implementation of the quantum fourier transform”. Physical Review Letters, 86, 1889. https://doi.org/10.1103/PhysRevLett.86.1889.DOI: 10.1103/PhysRevLett.86.1889Google Scholar
  3. M.A. Nielsen and I.L. Chuang (2000). Quantum Computation and Quantum Information, Cambridge, England: Cambridge University Press.Google Scholar
  4. H. Wang, L. Wu, Y. Liu and F. Nori (2010). “Measurement-based quantum phase estimation algorithm for finding eigenvalues of non-unitary matrices”. Physical Review A, 82, 062303. https://doi.org/10.1103/PhysRevA.82.062303DOI: 10.1103/PhysRevA.82.062303Google Scholar
  5. H.F. Trotter (1959). “On the product of semi-groups of operators.” Proceedings of the American Mathematical Society, 10, 545.DOI: 10.1090/S0002-9939-1959-0108732-6Google Scholar
  6. M. Suzuki (1976). “Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems”. Communications in Mathematics Physics, 51, 183.DOI: 10.1007/BF01609348Google Scholar
  7. A. T.-Shma (2013). Inverting Well Conditioned Matrices in Quantum Logspace, in Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC ’13, pp. 881–890, New York.DOI: 10.1145/2488608.2488720Google Scholar
  8. D. Aharonov and A. T.-Shma (2007). “Adiabatic Quantum State Generation”. SIAM Journal on Computing, 37: 1, 47. https://doi.org/10.1137/060648829.DOI: 10.1137/060648829Google Scholar
  9. S. Blanes and F. Casas (2004). “On the convergence and optimization of the Baker-Campbell-Hausdorff formula”. Linear Algebra and its Applications, 378, 135.DOI: 10.1016/j.laa.2003.09.010Google Scholar
  10. L. Zhao, Z. Zhao, P. Rebentrost and J. Fitzsimons (2021). “Compiling basic linear algebra subroutines for quantum computers”. Quantum Machine Intelligence, 3, 21.DOI: 10.1007/s42484-021-00048-8Google Scholar
  11. W. Qi, A.I. Zenchuk, A. Kumar and J. Wu (2024). “Quantum algorithms for matrix operations and linear systems of equations”. Communications in Theoretical Physics, 76, 035103. https://doi.org/10.1088/1572-9494/ad2366DOI: 10.1088/1572-9494/ad2366Google Scholar
  12. A.I. Zenchuk, W. Qi, A. Kumar and J. Wu (2024). “Matrix manipulations via unitary transformations and ancilla-state measurements”. Quantum Information & Computing, 24: 13,14, 1099–1109. https://www.rintonpress.com/xxqic24/qic-24-1314/1099-1109.pdfDOI: 10.26421/QIC24.13-14-2Google Scholar
  13. A.I. Zenchuk, G.A. Bochkin, W. Qi, A. Kumar and J. Wu (2025). “Quantum algorithms for calculating determinant and inverse of matrix and solving linear algebraic systems”. Quantum Information & Computing, 25: 2, 195–215. https://doi.org/10.2478/qic-2025-0010.pdfDOI: 10.2478/qic-2025-0010Google Scholar
  14. A.W. Harrow, A. Hassidim and S. Lloyd (2009), “Quantum algorithm for linear systems of equations”. Physical Reviews Letters, 103, 150502. https://doi.org/10.1103/PhysRevLett.103.150502DOI: 10.1103/PhysRevLett.103.150502Google Scholar
  15. B.D. Clader, B.C. Jacobs and C.R. Sprouse (2013). “Preconditioned quantum linear system algorithm”. Physical Review Letters, 110, 250504. https://doi.org/10.1103/PhysRevLett.110.250504DOI: 10.1103/PhysRevLett.110.250504Google Scholar
  16. J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe and S. Lloyd (2017). “Quantum machine learning”. Nature. 549, 195. https://doi.org/10.1038/nature23474DOI: 10.1038/nature23474Google Scholar
  17. L. Wossnig, Z. Zhao and A. Prakash (2018). “Quantum linear system algorithm for dense matrices”. Physical Reviews Letters, 120, 050502. https://doi.org/10.1103/PhysRevLett.120.050502DOI: 10.1103/PhysRevLett.120.050502Google Scholar
  18. X. D. Cai, C. Weedbrook, Z.E. Su, M.C. Chen, M. Gu, M.J. Zhu, L. Li, N.L. Liu, C.Y. Lu and J.W. Pan (2013). “Experimental quantum computing to solve systems of linear equations”. Physical Reviews Letters, 110, 230501. https://doi.org/10.1103/PhysRevLett.110.230501DOI: 10.1103/PhysRevLett.110.230501Google Scholar
  19. J.W. Pan, Y. Cao, X. Yao, Z. Li, C. Ju, H. Chen, X. Peng, S. Kais and J. Du (2014). “Experimental realization of quantum algorithm for solving linear systems of equations”. Physical Reviews A, 89, 022313. https://doi.org/10.1103/PhysRevA.89.022313DOI: 10.1103/PhysRevA.89.022313Google Scholar
  20. S. Barz, I. Kassal, M. Ringbauer, Y.O. Lipp, B. Dakic, A. Aspuru-Guzik and P. Walther (2014). “A two-qubit photonic quantum processor and its application to solving systems of linear equations”. Scientific Reports, 4, 6115. https://doi.org/10.1038/srep06115DOI: 10.1038/srep06115Google Scholar
  21. Y. Zheng, C. Song, M.C. Chen, B. Xia, W. Liu, Q. Guo, L. Zhang, D. Xu, H. Deng, K. Huang, Y. Wu, Z. Yan, D. Zheng, L. Lu, J.W. Pan, H. Wang, C.Y. Lu and X. Zhu (2017). “Solving systems of linear equations with a superconducting quantum processor”. Physical Reviews Letters, 118, 210504. https://doi.org/10.1103/PhysRevLett.118.210504DOI: 10.1103/PhysRevLett.118.210504Google Scholar
  22. J.M. Martyn, Z.M. Rossi, A.K. Tan, and I.L. Chuang (2021). “A grand unification of quantum algorithms”. PRX Quantum, 2, 040203.DOI: 10.1103/PRXQuantum.2.040203Google Scholar
  23. I. Novikau and I. Joseph (2024). ‘Estimating QSVT angles for matrix inversion with large condition numbers”. arXiv, 2408.15453v2 [quant-ph].DOI: 10.2139/ssrn.4973413Google Scholar
  24. A. Gilyén, Y. Su, G.H. Low and N. Wiebe (2019). Quantum Singular Value Transformation and beyond: exponential improvements for quantum matrix arithmetics, STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, 193–204. https://doi.org/10.1145/3313276.3316366DOI: 10.1145/3313276.3316366Google Scholar
  25. C. Bravo-Prieto, D. García-Martín and J.I. Latorre (2020). “Quantum singular value decomposer”. Physical Review A, 101, 062310.DOI: 10.1103/PhysRevA.101.062310Google Scholar
  26. X. Wang, Zh. Song and Y. Wang (2021). “Variational quantum singular value decomposition”. Quantum, 5, 483.DOI: 10.22331/q-2021-06-29-483Google Scholar
  27. J. Jojo, A. Khandelwal, M.G. Chandra (2024). “On modifying the variational quantum singular value decomposition algorithm”. arXiv:2310.19504v2 [quant-ph].Google Scholar
  28. Paddle Quantum: a quantum machine learning toolkit (2020). Available at: https://qml.baidu.com (Accessed on 12 June 2025).Google Scholar
  29. PaddlePaddle. Available at: https://github.com/paddlepaddle/paddle (Accessed on 12 June 2025).Google Scholar
  30. Ya. Ma, D. Yu, T. Wu and H. Wang (2019). PaddlePaddle: An open-source deep learning platform from industrial practice. Frontiers of Data and Domputing, 11: 1, 105.Google Scholar
  31. I. Kerenidis and A. Prakash (2016). “Quantum recommendation systems”. arXiv:1603.08675.Google Scholar
  32. P. Rebentrost, A. Steffens, I. Marvian and S. Lloyd (2018). “Quantum singular-value decomposition of nonsparse low-rank matrices”. Physical Review A, 97: 1, 012327.DOI: 10.1103/PhysRevA.97.012327Google Scholar
  33. A. Bellante, A. Luongo, and S. Zanero. (2022). “Quantum algorithms for SVD-based data representation and analysis”. Quantum Machine Intelligence, 4: 2.DOI: 10.1007/s42484-022-00076-yGoogle Scholar
  34. I. Kerenidis, A. Prakash (2020). “Quantum gradient descent for linear systems and least squares”. Physical Review A, 101: 2, 022316.DOI: 10.1103/PhysRevA.101.022316Google Scholar
  35. L. Wossnig, Zh. Zhao, and A. Prakash (2018). “Quantum linear system algorithm for dense matrices”. Physical Review Letters, 120: 5, 050502.DOI: 10.1103/PhysRevLett.120.050502Google Scholar
  36. Sh. Zheng, Ch. Ding and F. Nie (2018). “Regularized singular value decomposition and application to recommender system”. arXiv:1804.05090v1 [cs.LG].Google Scholar
  37. P. Bhavana, V. Kumar and V. Padmanabhan (2019). “Block based Singular Value Decomposition approach to matrix factorization for recommender systems”. arXiv:1907.07410v1 [cs.LG].Google Scholar
  38. E.B. Fel’dman, A.I. Zenchuk, W. Qi and J. Wu (2025). “Remarks on controlled measurement and quantum algorithm for calculating Hermitian conjugate”. arXiv:2501.16028v1.Google Scholar
  39. A.Y. Kitaev, A.H. Shen and M.N. Vyalyi (2002). Classical and Quantum Computation, Graduate Studies in Mathematics, V.47, Rhode Island: American Mathematical Society, Providence.DOI: 10.1090/gsm/047Google Scholar