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Lanczos Algorithm, the Transfer Matrix, and the Signal-to-Noise Problem

Michael L. Wagman

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Phys. Rev. Lett. 134, 241901 – Published 17 June, 2025

DOI: https://doi.org/10.1103/pcvc-734h

Abstract

This Letter introduces a method for determining the energy spectrum of lattice quantum chromodynamics by applying the Lanczos algorithm to the transfer matrix and using a bootstrap generalization of the Cullum-Willoughby method to filter out spurious eigenvalues. Proof-of-principle analyses of the simple harmonic oscillator and the lattice quantum chromodynamics proton mass demonstrate that this method provides faster ground-state convergence than the “effective mass,” which is related to the power-iteration algorithm. Lanczos provides more accurate energy estimates than multistate fits to correlation functions with small imaginary times while achieving comparable statistical precision. Two-sided error bounds are computed for Lanczos results and guarantee that excited-state effects cannot shift Lanczos results far outside their statistical uncertainties.

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References (108)

  1. Y.-C. Jang, R. Gupta, B. Yoon, and T. Bhattacharya, Axial vector form factors from lattice QCD that satisfy the PCAC relation, Phys. Rev. Lett. 124, 072002 (2020).
  2. G. S. Bali, L. Barca, S. Collins, M. Gruber, M. Löffler, A. Schäfer, W. Söldner, P. Wein, S. Weishäupl, and T. Wurm (RQCD Collaboration), Nucleon axial structure from lattice QCD, J. High Energy Phys. 05 (2020) 126.
  3. C. Alexandrou et al., Nucleon axial and pseudoscalar form factors from lattice QCD at the physical point, Phys. Rev. D 103, 034509 (2021).
  4. S. Park, R. Gupta, B. Yoon, S. Mondal, T. Bhattacharya, Y.-C. Jang, B. Joó, and F. Winter (Nucleon Matrix Elements (NME) Collaboration), Precision nucleon charges and form factors using (2+1)-flavor lattice QCD, Phys. Rev. D 105, 054505 (2022).
  5. D. Djukanovic, G. von Hippel, J. Koponen, H. B. Meyer, K. Ottnad, T. Schulz, and H. Wittig, Isovector axial form factor of the nucleon from lattice QCD, Phys. Rev. D 106, 074503 (2022).
  6. Y.-C. Jang, R. Gupta, T. Bhattacharya, B. Yoon, and H.-W. Lin (Precision Neutron Decay Matrix Elements (PNDME) Collaboration), Nucleon isovector axial form factors, Phys. Rev. D 109, 014503 (2024).
  7. C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, R. Frezzotti, B. Kostrzewa, G. Koutsou, G. Spanoudes, and C. Urbach (Extended Twisted Mass), Nucleon axial and pseudoscalar form factors using twisted-mass fermion ensembles at the physical point, Phys. Rev. D 109, 034503 (2024).
  8. R. Gupta, Isovector axial charge and form factors of nucleons from lattice QCD, Proc. Sci., LATTICE2023 (2024) 124.
  9. A. Francis, J. R. Green, P. M. Junnarkar, C. Miao, T. D. Rae, and H. Wittig, Lattice QCD study of the H dibaryon using hexaquark and two-baryon interpolators, Phys. Rev. D 99, 074505 (2019).
  10. B. Hörz et al., Two-nucleon S-wave interactions at the SU(3) flavor-symmetric point with mud≃msphys: A first lattice QCD calculation with the stochastic Laplacian Heaviside method, Phys. Rev. C 103, 014003 (2021).
  11. S. Amarasinghe, R. Baghdadi, Z. Davoudi, W. Detmold, M. Illa, A. Parreño, A. V. Pochinsky, P. E. Shanahan, and M. L. Wagman, Variational study of two-nucleon systems with lattice QCD, Phys. Rev. D 107, 094508 (2023).
  12. J. R. Green, A. D. Hanlon, P. M. Junnarkar, and H. Wittig, Weakly bound H dibaryon from SU(3)-flavor-symmetric QCD, Phys. Rev. Lett. 127, 242003 (2021).
  13. W. Detmold, M. Illa, W. I. Jay, A. Parreño, R. J. Perry, P. E. Shanahan, and M. L. Wagman, Constraints on the finite volume two-nucleon spectrum at mπ≈806  MeV, arXiv:2404.12039.
  14. G. Parisi, The strategy for computing the hadronic mass spectrum, Phys. Rep. 103, 203 (1984).
  15. G. P. Lepage, The analysis of algorithms for lattice field theory, in Theoretical Advanced Study Institute in Elementary Particle Physics (1989).
  16. S. R. Beane, W. Detmold, T. C. Luu, K. Orginos, A. Parreńo, M. J. Savage, A. Torok, and A. Walker-Loud (NPLQCD Collaboration), High statistics analysis using anisotropic clover lattices: (II) Three-baryon systems, Phys. Rev. D 80, 074501 (2009).
  17. S. R. Beane, W. Detmold, H.-W. Lin, T. C. Luu, K. Orginos, M. J. Savage, A. Torok, and A. Walker-Loud (NPLQCD Collaboration), High statistics analysis using anisotropic clover lattices: (III) baryon-baryon interactions, Phys. Rev. D 81, 054505 (2010).
  18. Z. Davoudi, W. Detmold, K. Orginos, A. Parreño, M. J. Savage, P. Shanahan, and M. L. Wagman, Nuclear matrix elements from lattice QCD for electroweak and beyond-standard-model processes, Phys. Rep. 900, 1 (2021).
  19. G. Fox, R. Gupta, O. Martin, and S. Otto, Monte Carlo estimates of the mass gap of the O(2) and O(3) spin models in (1+1)-dimensions, Nucl. Phys. B205, 188 (1982).
  20. C. Michael and I. Teasdale, Extracting glueball masses from lattice QCD, Nucl. Phys. B215, 433 (1983).
  21. M. Lüscher and U. Wolff, How to calculate the elastic scattering matrix in two-dimensional quantum field theories by numerical simulation, Nucl. Phys. B339, 222 (1990).
  22. B. Blossier, M. Della Morte, G. von Hippel, T. Mendes, and R. Sommer, On the generalized eigenvalue method for energies and matrix elements in lattice field theory, J. High Energy Phys. 04 (2009) 094.
  23. G. T. Fleming, Beyond generalized eigenvalues in lattice quantum field theory, in 40th International Symposium on Lattice Field Theory (2023), arXiv:2309.05111.
  24. R. A. Briceno, J. J. Dudek, and R. D. Young, Scattering processes and resonances from lattice QCD, Rev. Mod. Phys. 90, 025001 (2018).
  25. J. Bulava et al., Hadron spectroscopy with lattice QCD, in Snowmass 2021 (2022), arXiv:2203.03230.
  26. A. D. Hanlon, Hadron spectroscopy and few-body dynamics from lattice QCD, Proc. Sci., LATTICE2023 (2024) 106 [arXiv:2402.05185].
  27. G. T. Fleming, What can lattice QCD theorists learn from NMR spectroscopists?, in 3rd International Workshop on Numerical Analysis and Lattice QCD (2004), pp. 143–152, arXiv:hep-lat/0403023.
  28. H.-W. Lin and S. D. Cohen, Lattice QCD beyond ground states, in 4th International Workshop on Numerical Analysis and Lattice QCD (2007).
  29. G. T. Fleming, S. D. Cohen, H.-W. Lin, and V. Pereyra, Excited-state effective masses in lattice QCD, Phys. Rev. D 80, 074506 (2009).
  30. S. R. Beane, W. Detmold, T. C. Luu, K. Orginos, A. Parreno, M. J. Savage, A. Torok, and A. Walker-Loud, High statistics analysis using anisotropic clover lattices: (I) Single hadron correlation functions, Phys. Rev. D 79, 114502 (2009).
  31. M. Fischer, B. Kostrzewa, J. Ostmeyer, K. Ottnad, M. Ueding, and C. Urbach, On the generalised eigenvalue method and its relation to Prony and generalised pencil of function methods, Eur. Phys. J. A 56, 206 (2020).
  32. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Natl. Bur. Stand. B 45, 255 (1950).
  33. B. Parlett, The Symmetric Eigenvalue Problem, Classics in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, 1980).
  34. G. H. Golub and D. P. O’Leary, Some history of the conjugate gradient and Lanczos algorithms: 1948–1976, SIAM Rev. 31, 50 (1989).
  35. B. N. Parlett, Do we fully understand the symmetric Lanczos algorithm yet (1995), https://apps.dtic.mil/sti/citations/tr/ADA289614.
  36. G. Meurant and Z. Strakoš, The Lanczos and conjugate gradient algorithms in finite precision arithmetic, Acta Numer., 15, 471 (2006).
  37. Y. Saad, Numerical Methods for Large Eigenvalue Problems: Revised Edition, Classics in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, 2011).
  38. G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University Press, Philadelphia, PA, 2013).
  39. V. Hernandez, J. E. Roman, A. Tomas, and V. Vidal, A survey of software for sparse eigenvalue problems, Technical Report STR-5, Universitat Politècnica de València, 2006, available at https://slepc.upv.es.
  40. M. Caffarel, F. X. Gadea, and D. M. Ceperley, Lanczós-type algorithm for quantum Monte Carlo data, Europhys. Lett. 16, 249 (1991).
  41. S. Sorella, Generalized Lanczos algorithm for variational quantum Monte Carlo, Phys. Rev. B 64, 024512 (2001).
  42. F. Becca and S. Sorella, Quantum Monte Carlo Approaches for Correlated Systems (Cambridge University Press, Cambridge, England, 2017).
  43. I. M. Barbour, P. Gibbs, J. P. Gilchrist, H. Schneider, G. Schierholz, and M. Teper, Strong evidence for spontaneous chiral symmetry breaking in (quenched) QCD, Phys. Lett. B 136, 80 (1984).
  44. I. M. Barbour, N. E. Behilil, P. E. Gibbs, G. Schierholz, and M. Teper, THE Lanczos method in lattice gauge theories (1984), https://inspirehep.net/literature/204843.
  45. I. M. Barbour, N. E. Behilil, P. E. Gibbs, M. Rafique, K. J. M. Moriarty, and G. Schierholz, Updating fermions with the Lanczos method, J. Comput. Phys. 68, 227 (1987).
  46. T. Kalkreuter, Study of Cullum’s and Willoughby’s Lanczos method for Wilson fermions, Comput. Phys. Commun. 95, 1 (1996).
  47. A. D. Kennedy, I. Horvath, and S. Sint, A New exact method for dynamical fermion computations with nonlocal actions, Nucl. Phys. B, Proc. Suppl. 73, 834 (1999).
  48. M. A. Clark, C. Jung, and C. Lehner, Multi-grid Lanczos, EPJ Web Conf. 175, 14023 (2018).
  49. H. Jeong, C. DeTar, and S. Gottlieb, Performance of several Lanczos eigensolvers with HISQ fermions, Proc. Sci., LATTICE2021 (2022) 053 [arXiv:2201.03755].
  50. J. B. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11, 395 (1975).
  51. S. Kaniel, Estimates for some computational techniques in linear algebra, Math. Comput. 20, 369 (1966).
  52. C. C. Paige, The computation of eigenvalues and eigenvectors of very large sparse matrices, Ph.D.thesis, London University, London, UK, 1971.
  53. Y. Saad, On the rates of convergence of the Lanczos and the block-Lanczos methods, SIAM J. Num. Anal. 17, 687 (1980).
  54. M. Della Morte and L. Giusti, Exploiting symmetries for exponential error reduction in path integral Monte Carlo, Comput. Phys. Commun. 180, 813 (2009).
  55. M. Della Morte and L. Giusti, Symmetries and exponential error reduction in Yang-Mills theories on the lattice, Comput. Phys. Commun. 180, 819 (2009).
  56. M. Della Morte and L. Giusti, A novel approach for computing glueball masses and matrix elements in Yang-Mills theories on the lattice, J. High Energy Phys. 05 (2011) 056.
  57. M. Lüscher, Construction of a selfadjoint, strictly positive transfer matrix for euclidean lattice gauge theories, Commun. Math. Phys. 54, 283 (1977).
  58. M. Lüscher and P. Weisz, Definition and general properties of the transfer matrix in continuum limit improved lattice gauge theories, Nucl. Phys. B240, 349 (1984).
  59. See Supplemental Material, which includes Refs. [60,61], at http://link.aps.org/supplemental/10.1103/pcvc-734h for additional details on the implementation of the Lanczos algorithm used to compute the results of the main text, including thermal effects, the Cullum-Willoughby test, oblique Lanczos recursion relations, residual bounds, and correlations.
  60. D. C. Hackett, P. R. Oare, D. A. Pefkou, and P. E. Shanahan, Gravitational form factors of the pion from lattice QCD, Phys. Rev. D 108, 114504 (2023).
  61. R. Abbott, W. Detmold, M. Illa, A. Parreño, R. J. Perry, F. Romero-López, P. E. Shanahan, and M. L. Wagman, QCD constraints on isospin-dense matter and the nuclear equation of state, Phys. Rev. Lett. 134, 011903 (2025).
  62. R. V. Mises and H. Pollaczek-Geiringer, Praktische Verfahren der Gleichungsauflösung, Z. Angew. Math. Mech. 9, 58 (1929).
  63. B. N. Parlett, H. Simon, and L. M. Stringer, On estimating the largest eigenvalue with the Lanczos algorithm, Math. Comput. 38, 153 (1982).
  64. J. Kuczyński and H. Woźniakowski, Estimating the largest eigenvalues by the power and Lanczos algorithms with a random start, SIAM J. Matrix Anal. Appl. 13, 1094 (1992).
  65. A. B. J. Kuijlaars, Which eigenvalues are found by the Lanczos method?, SIAM J. Matrix Anal. Appl. 22, 306 (2000).
  66. J. Garza-Vargas and A. Kulkarni, The Lanczos algorithm under few iterations: Concentration and location of the output, SIAM J. Matrix Anal. Appl. 41, 1312 (2020).
  67. W. DeMeo, A Lanczos procedure for approximating eigenvalues of large stochastic matrices, Ph.D. thesis, 1998.

  68. Nonspurious eigenvalues with βj2≈0 indicate that a Ritz value has converged to an eigenvalue of T; see Refs. [33, 35, 52, 69]. After convergence has been achieved, statistical fluctuations can easily lead to βj2<0 and therefore complex Tij(m).

  69. B. N. Parlett and D. S. Scott, The Lanczos algorithm with selective orthogonalization, Math. Comput. 33, 217 (1979).
  70. Y. Saad, The Lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems, SIAM J. Num. Anal. 19, 485 (1982).
  71. B. N. Parlett, D. R. Taylor, and Z. A. Liu, A look-ahead Lanczos algorithm for unsymmetric matrices, Math. Comput. 44, 105 (1985).
  72. N. M. Nachtigal, A look-ahead variant of the Lanczos algorithm and its application to the quasi-minimal residual method for non-Hermitian linear systems, Ph.D. thesis, 1993.
  73. D. C. Hackett and M. L. Wagman, Lanczos for lattice QCD matrix elements, arXiv:2407.21777.
  74. J. H. J. H. Wilkinson, The Algebraic Eigenvalue Problem, Monographs on Numerical Analysis (Clarendon Press, Oxford, 1965).
  75. B. Parlett, Misconvergence in the Lanczos algorithm, in Reliable Numerical Commputation (Oxford University Press, New York, 1990).
  76. J. J. Dudek, R. G. Edwards, and C. E. Thomas (Hadron Spectrum Collaboration), Energy dependence of the ρ resonance in ππ elastic scattering from lattice QCD, Phys. Rev. D 87, 034505 (2013); 90, 099902(E) (2014).
  77. C. B. Lang and V. Verduci, Scattering in the πN negative parity channel in lattice QCD, Phys. Rev. D 87, 054502 (2013).
  78. D. J. Wilson, R. A. Briceno, J. J. Dudek, R. G. Edwards, and C. E. Thomas, Coupled ππ,KK¯ scattering in P-wave and the ρ resonance from lattice QCD, Phys. Rev. D 92, 094502 (2015).
  79. J. Ostmeyer, A. Sen, and C. Urbach, On the equivalence of Prony and Lanczos methods for Euclidean correlation functions, Eur. Phys. J. A 61, 26 (2025).
  80. D. Chakraborty, D. Sood, A. Radhakrishnan, and N. Mathur, Estimating energy levels from lattice QCD correlation functions using a transfer matrix formalism, arXiv:2412.01900.
  81. R. Abbott, D. C. Hackett, G. T. Fleming, D. A. Pefkou, and M. L. Wagman, Filtered Rayleigh-Ritz is all you need, arXiv:2503.17357.

  82. LQCD applications of Prony’s method have used fixed m in the range m≤4 [28, 29, 30, 31] and noted that unphysical solutions arising from noise become increasingly common as m is increased [29].

  83. J. Cullum and R. A. Willoughby, Computing eigenvalues of very large symmetric matrices–an implementation of a Lanczos algorithm with no reorthogonalization, J. Comput. Phys. 44, 329 (1981).
  84. J. K. Cullum and R. A. Willoughby, Lanczos procedures, in Lanczos Algorithms for Large Symmetric Eigenvalue Computations Vol. I Theory (Birkhäuser Boston, Boston, MA, 1985), pp. 92–163.
  85. U. Elsner, V. Mehrmann, F. Milde, R. A. Römer, and M. Schreiber, The Anderson model of localization: A challenge for modern eigenvalue methods, SIAM J. Sci. Comput. 20, 2089 (1999).
  86. B. Efron, Nonparametric Estimates of Standard Error: The Jackknife, the Bootstrap and Other Methods (Oxford University Press, New York, 1981), Vol. 68, pp. 589–599.

  87. A precise recipe for defining ϵCW in terms of three hyperparameters is given in Supplemental Material; examples here use Δ=4, KCW=3, and FCW=10.

  88. A. C. Davison and D. V. Hinkley, The basic bootstraps, in Bootstrap Methods and their Application, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, Cambridge, England, 1997), pp. 11–69.
  89. P. Young, Everything you wanted to know about data analysis and fitting but were afraid to ask, arXiv:1210.3781.
  90. P. Lepage, C. Gohlke, and D. Hackett, https://github.com/gplepage/gvar (2024).
  91. M. L. Wagman and M. J. Savage, Statistics of baryon correlation functions in lattice QCD, Phys. Rev. D 96, 114508 (2017).
  92. W. Detmold, G. Kanwar, and M. L. Wagman, Phase unwrapping and one-dimensional sign problems, Phys. Rev. D 98, 074511 (2018).
  93. S. R. Beane et al. (NPLQCD, QCDSF Collaborations), Charged multihadron systems in lattice QCD+QED, Phys. Rev. D 103, 054504 (2021).
  94. O. Ledoit and M. Wolf, A well-conditioned estimator for large-dimensional covariance matrices, J. Multivariate Anal. 88, 365 (2004).
  95. E. Rinaldi, S. Syritsyn, M. L. Wagman, M. I. Buchoff, C. Schroeder, and J. Wasem, Lattice QCD determination of neutron-antineutron matrix elements with physical quark masses, Phys. Rev. D 99, 074510 (2019).
  96. H. Akaike, A new look at the statistical model identification, IEEE Trans. Autom. Control 19, 716 (1974).
  97. B. Yoon et al., Isovector charges of the nucleon from 2+1-flavor QCD with clover fermions, Phys. Rev. D 95, 074508 (2017).
  98. S. Mondal, R. Gupta, S. Park, B. Yoon, T. Bhattacharya, B. Joó, and F. Winter (Nucleon Matrix Elements (NME) Collaboration), Nucleon momentum fraction, helicity and transversity from 2+1-flavor lattice QCD, J. High Energy Phys. 04 (2020) 004.
  99. M. Lüscher and P. Weisz, On-shell improved lattice gauge theories, Commun. Math. Phys. 98, 433 (1985); 98, 433(E) (1985).
  100. B. Sheikholeslami and R. Wohlert, Improved continuum limit lattice action for QCD with Wilson fermions, Nucl. Phys. B259, 572 (1985).
  101. C. Morningstar and M. J. Peardon, Analytic smearing of SU(3) link variables in lattice QCD, Phys. Rev. D 69, 054501 (2004).
  102. N. I. Fisher, Statistical Analysis of Circular Data (Cambridge University Press, Cambridge, England, 1993).
  103. R. G. Edwards and B. Joó (SciDAC, LHPC, UKQCD Collaborations), The Chroma software system for lattice QCD, Nucl. Phys. B, Proc. Suppl. 140, 832 (2005).
  104. M. A. Clark, R. Babich, K. Barros, R. C. Brower, and C. Rebbi (QUDA Collaboration), Solving Lattice QCD systems of equations using mixed precision solvers on GPUs, Comput. Phys. Commun. 181, 1517 (2010).
  105. R. Babich, M. A. Clark, B. Joó, G. Shi, R. C. Brower, and S. Gottlieb (QUDA Collaboration), Scaling lattice QCD beyond 100 GPUs, in International Conference for High Performance Computing, Networking, Storage and Analysis (2011), arXiv:1109.2935.
  106. M. A. Clark, B. Joó, A. Strelchenko, M. Cheng, A. Gambhir, and R. C. Brower (QUDA), Accelerating lattice QCD multigrid on GPUs using fine-grained parallelization, in International Conference for High Performance Computing, Networking, Storage and Analysis (2016), arXiv:1612.07873.
  107. F. Winter and M.Clark, R. Edwards, and B. Joó, A framework for lattice QCD calculations on GPUs, in 2014 IEEE 28th International Parallel and Distributed Processing Symposium (2014), pp. 1073–1082.
  108. Wolfram Research Inc., mathematica, Version 14.0, https://www.wolfram.com/mathematica.

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Phys. Rev. Lett. 134, 241901– Published 17 June, 2025
  • Received 3 September 2024
  • Revised 23 April 2025
  • Accepted 7 May 2025
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