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Lanczos algorithm for lattice QCD matrix elements

Daniel C. Hackett and Michael L. Wagman

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Phys. Rev. D 112, 054506 – Published 17 September, 2025

DOI: https://doi.org/10.1103/zjzt-rv86

Abstract

Recent work [M. L. Wagman, Lanczos, the transfer matrix, and the signal-to-noise problem, Phys. Rev. Lett. 134, 241901 (2025).] found that an analysis formalism based on the Lanczos algorithm allows energy levels to be extracted from Euclidean correlation functions with faster ground-state convergence than effective masses, convergent estimators for multiple states from a single correlator, and two-sided error bounds. After filtering out spurious eigenvalues and using outlier-robust estimators within a nested bootstrap framework, Lanczos estimators behave more like multistate fit results than effective masses—but without involving statistical fitting. We extend this formalism to the determination of matrix elements from three-point correlation functions and provide a physical picture of “spurious-state filtering” involving restriction to a Hermitian subspace. We demonstrate similar advantages for matrix elements as for spectroscopy through example applications to noiseless mock-data and (bare) forward matrix elements of the strange scalar current between both ground and excited states with the quantum numbers of the nucleon.

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References (128)

  1. W. Detmold, R. G. Edwards, J. J. Dudek, M. Engelhardt, H.-W. Lin, S. Meinel, K. Orginos, and P. Shanahan (USQCD Collaboration), Hadrons and nuclei, Eur. Phys. J. A 55, 193 (2019).
  2. A. S. Kronfeld et al. (USQCD Collaboration), Lattice QCD and particle physics, arXiv:2207.07641.
  3. P. Boyle et al., Lattice QCD and the computational frontier, in Snowmass 2021 (2022).
  4. P. Achenbach et al., The present and future of QCD, Nucl. Phys. A1047, 122874 (2024).
  5. R. A. Brice no, J. J. Dudek, and R. D. Young, Scattering processes and resonances from lattice QCD, Rev. Mod. Phys. 90, 025001 (2018).
  6. J. Bulava et al., Hadron spectroscopy with Lattice QCD, in Snowmass 2021 (White Paper for Snowmas, Washington, 2022).
  7. A. D. Hanlon, Hadron spectroscopy and few-body dynamics from Lattice QCD, Proc. Sci. LATTICE2023 (2024) 106 [arXiv:2402.05185].
  8. K. Cichy and M. Constantinou, A guide to light-cone PDFs from Lattice QCD: An overview of approaches, techniques and results, Adv. High Energy Phys. 2019, 3036904 (2019).
  9. X. Ji, Y.-S. Liu, Y. Liu, J.-H. Zhang, and Y. Zhao, Large-momentum effective theory, Rev. Mod. Phys. 93, 035005 (2021).
  10. M. Constantinou et al., Lattice QCD calculations of parton physics, arXiv:2202.07193.
  11. X. Gao, Overview of hadron structure form lattice QCD, Proc. Sci. LATTICE2023 (2024) 128.
  12. G. T. Fleming, What can lattice QCD theorists learn from NMR spectroscopists?, in Proceedings of the 3rd International Workshop on Numerical Analysis and Lattice QCD (2004), pp. 143–152.
  13. S. R. Beane, W. Detmold, T. C. Luu, K. Orginos, A. Parreño, M. J. Savage, A. Torok, and A. Walker-Loud (NPLQCD Collaboration), High statistics analysis using anisotropic clover lattices: (I) Single hadron correlation functions, Phys. Rev. D 79, 114502 (2009).
  14. W. Detmold and M. G. Endres, Signal/noise enhancement strategies for stochastically estimated correlation functions, Phys. Rev. D 90, 034503 (2014).
  15. M. L. Wagman and M. J. Savage, Statistics of baryon correlation functions in lattice QCD, Phys. Rev. D 96, 114508 (2017).
  16. W. Detmold, G. Kanwar, and M. L. Wagman, Phase unwrapping and one-dimensional sign problems, Phys. Rev. D 98, 074511 (2018).
  17. 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).
  18. C. Yunus and W. Detmold, Large-time correlation functions in bosonic lattice field theories, Phys. Lett. B 840, 137890 (2023).
  19. 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).
  20. R. Gupta, S. Park, M. Hoferichter, E. Mereghetti, B. Yoon, and T. Bhattacharya, Pion–nucleon sigma term from Lattice QCD, Phys. Rev. Lett. 127, 242002 (2021).
  21. R. Gupta, Isovector axial charge and form factors of nucleons from Lattice QCD, Proc. Sci. LATTICE2023 (2024) 124.
  22. R. Gupta, Y.-C. Jang, H.-W. Lin, B. Yoon, and T. Bhattacharya, Axial vector form factors of the nucleon from Lattice QCD, Phys. Rev. D 96, 114503 (2017).
  23. 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.
  24. 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).
  25. R. Gupta et al., Nucleon form factors and the pion-nucleon sigma term, Proc. Sci. LATTICE2022 (2023) 427 [arXiv:2301.07885].
  26. L. Barca, G. Bali, and S. Collins, Progress on nucleon transition matrix elements with a lattice QCD variational analysis, in Proceedings of the EuroPLEx Final Conference (2024).
  27. G. Parisi, The strategy for computing the hadronic mass spectrum, Phys. Rep. 103, 203 (1984).
  28. G. P. Lepage, The analysis of algorithms for lattice field theory, in Theoretical Advanced Study Institute in Elementary Particle Physics (1989).
  29. 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).
  30. C. Michael and I. Teasdale, Extracting glueball masses from Lattice QCD, Nucl. Phys. B215, 433 (1983).
  31. M. Luscher and U. Wolff, How to calculate the elastic scattering matrix in two-dimensional quantum field theories by numerical simulation, Nucl. Phys. B339, 222 (1990).
  32. 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.
  33. G. T. Fleming, Beyond generalized eigenvalues in lattice quantum field theory, in Proceedings of the 40th International Symposium on Lattice Field Theory (2023).
  34. C. Aubin and K. Orginos, An improved method for extracting matrix elements from lattice three-point functions, Proc. Sci. LATTICE2011 (2011) 148.
  35. T. Khan et al. (HadStruc Collaboration), Unpolarized gluon distribution in the nucleon from lattice quantum chromodynamics, Phys. Rev. D 104, 094516 (2021).
  36. J. Bulava, M. Donnellan, and R. Sommer, On the computation of hadron-to-hadron transition matrix elements in lattice QCD, J. High Energy Phys. 01 (2012) 140.
  37. L. Barca, G. Bali, and S. Collins, Toward N to Nπ matrix elements from lattice QCD, Phys. Rev. D 107, L051505 (2023).
  38. T. Blum, P. A. Boyle, D. Hoying, T. Izubuchi, L. Jin, C. Jung, C. Kelly, C. Lehner, A. Soni, and M. Tomii (RBC Collaboration and UKQCD Collaboration), ΔI=3/2 and ΔI=1/2 channels of K→ππ decay at the physical point with periodic boundary conditions, Phys. Rev. D 108, 094517 (2023).
  39. J. R. Green, M. Engelhardt, N. Hasan, S. Krieg, S. Meinel, J. W. Negele, A. V. Pochinsky, and S. N. Syritsyn, Excited-state effects in nucleon structure on the lattice using hybrid interpolators, Phys. Rev. D 100, 074510 (2019).
  40. F. M. Stokes, W. Kamleh, and D. B. Leinweber, Opposite-parity contaminations in lattice nucleon form factors, Phys. Rev. D 99, 074506 (2019).
  41. J. Dragos, R. Horsley, W. Kamleh, D. B. Leinweber, Y. Nakamura, P. E. L. Rakow, G. Schierholz, R. D. Young, and J. M. Zanotti, Nucleon matrix elements using the variational method in lattice QCD, Phys. Rev. D 94, 074505 (2016).
  42. C. J. Shultz, J. J. Dudek, and R. G. Edwards, Excited meson radiative transitions from lattice QCD using variationally optimized operators, Phys. Rev. D 91, 114501 (2015).
  43. J. R. Green, J. W. Negele, A. V. Pochinsky, S. N. Syritsyn, M. Engelhardt, and S. Krieg, Nucleon electromagnetic form factors from lattice QCD using a nearly physical pion mass, Phys. Rev. D 90, 074507 (2014).
  44. J. Green, M. Engelhardt, S. Krieg, S. Meinel, J. Negele, A. Pochinsky, and S. Syritsyn, Nucleon form factors with light Wilson quarks, Proc. Sci. LATTICE2013 (2014) 276 [arXiv:1310.7043].
  45. E. Bennett, N. Forzano, D. K. Hong, H. Hsiao, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai, D. Vadacchino, and F. Zierler, On the mixing between flavor singlets in lattice gauge theories coupled to matter fields in multiple representations, Phys. Rev. D 110, 074504 (2024).
  46. V. M. Braun, S. Collins, M. Göckeler, P. Pérez-Rubio, A. Schäfer, R. W. Schiel, and A. Sternbeck, Second moment of the pion light-cone distribution amplitude from lattice QCD, Phys. Rev. D 92, 014504 (2015).
  47. A. Bussone, A. Conigli, J. Frison, G. Herdoíza, C. Pena, D. Preti, A. Sáez, and J. Ugarrio (Alpha Collaboration), Hadronic physics from a Wilson fermion mixed-action approach: Charm quark mass and D(s) meson decay constants, Eur. Phys. J. C 84, 506 (2024).
  48. 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).
  49. 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).
  50. J. K. Cullum and R. A. Willoughby, Lanczos procedures, in Lanczos Algorithms for Large Symmetric Eigenvalue Computations Vol. I Theory (Birkhäuser, Boston, MA, 1985), pp. 92–163.
  51. M. L. Wagman, Lanczos, the transfer matrix, and the signal-to-noise problem, Phys. Rev. Lett. 134, 241901 (2025).
  52. 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).
  53. 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.
  54. D. C. Hackett and M. L. Wagman, Block Lanczos for lattice QCD spectroscopy and matrix elements, Phys. Rev. D 112, 014514 (2025).
  55. Z. Leyk, Breakdowns and stagnation in iterative methods, BIT 37, 377 (1997).
  56. S. W. Gaaf and E. Jarlebring, The infinite bi-lanczos method for nonlinear eigenvalue problems, SIAM J. Sci. Comput. 39, S898 (2017).
  57. Y. Saad, Numerical Methods for Large Eigenvalue Problems: Revised Edition, Classics in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2011).
  58. 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.
  59. R. Horsley, R. Millo, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller, F. Winter, and J. M. Zanotti (QCDSF Collaboration and UKQCD Collaboration), A Lattice study of the glue in the nucleon, Phys. Lett. B 714, 312 (2012).
  60. M. Batelaan, K. U. Can, R. Horsley, Y. Nakamura, P. E. L. Rakow, G. Schierholz, H. Stüben, R. D. Young, and J. M. Zanotti (QCDSF-UKQCD-CSSM Collaborations), Feynman-Hellmann approach to transition matrix elements and quasidegenerate energy states, Phys. Rev. D 108, 034507 (2023).
  61. M. Batelaan et al. (QCDSF/UKQCD/CSSM, CSSM, UKQCD, QCDSF Collaborations), Moments and power corrections of longitudinal and transverse proton structure functions from lattice QCD, Phys. Rev. D 107, 054503 (2023).
  62. A. Hannaford-Gunn, R. Horsley, H. Perlt, P. Rakow, G. Schierholz, H. Stüben, R. Young, J. Zanotti, and K. U. Can (CSSM/QCDSF/UKQCD Collaborations), Generalised parton distributions from Lattice Feynman-Hellmann techniques, Proc. Sci. LATTICE2021 (2022) 088 [arXiv:2202.03662].
  63. J. B. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11, 395 (1975).
  64. M. Lüscher, Construction of a selfadjoint, strictly positive transfer matrix for Euclidean lattice gauge theories, Commun. Math. Phys. 54, 283 (1977).
  65. 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).
  66. R. V. Mises and H. Pollaczek-Geiringer, Praktische Verfahren der Gleichungsauflösung, Z. Angew. Math. Mech. 9, 58 (1929).
  67. L. Maiani, G. Martinelli, M. L. Paciello, and B. Taglienti, Scalar densities and baryon mass differences in Lattice QCD with Wilson fermions, Nucl. Phys. B293, 420 (1987).
  68. S. J. Dong, K. F. Liu, and A. G. Williams, Lattice calculation of the strangeness magnetic moment of the nucleon, Phys. Rev. D 58, 074504 (1998).
  69. S. Capitani, M. Della Morte, G. von Hippel, B. Jäger, A. Juttner, B. Knippschild, H. B. Meyer, and H. Wittig, The nucleon axial charge from lattice QCD with controlled errors, Phys. Rev. D 86, 074502 (2012).
  70. B. N. Parlett, H. Simon, and L. M. Stringer, On estimating the largest eigenvalue with the Lanczos algorithm, Math. Comput. 38, 153 (1982).
  71. G. H. Golub and D. P. O’Leary, Some history of the conjugate gradient and Lanczos algorithms: 1948–1976, SIAM Rev. 31, 50 (1989).
  72. 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).
  73. B. N. Parlett, Do we fully understand the symmetric Lanczos algorithm yet, in Proceedings of the Cornelius Lanczos International Centenary Conference (1994), https://math.berkeley.edu/~parlett/.
  74. G. Meurant and Z. Strakoš, The Lanczos and conjugate gradient algorithms in finite precision arithmetic, Acta Numer. 15, 471 (2006).
  75. G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University Press, Philadelphia, PA, 2013).
  76. Y. Saad, The Lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems, SIAM J. Numer. Anal. 19, 485 (1982).
  77. B. N. Parlett, D. R. Taylor, and Z. A. Liu, A look-ahead Lanczos algorithm for unsymmetric matrices, Math. Comput. 44, 105 (1985).
  78. 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.
  79. C. C. Paige, The computation of eigenvalues and eigenvectors of very large sparse matrices, Ph.D. thesis, London University, 1971.
  80. B. N. Parlett and D. S. Scott, The Lanczos algorithm with selective orthogonalization, Math. Comput. 33, 217 (1979).
  81. S. Kaniel, Estimates for some computational techniques in linear algebra, Math. Comput. 20, 369 (1966).
  82. Y. Saad, On the rates of convergence of the Lanczos and the block-Lanczos methods, SIAM J. Numer. Anal. 17, 687 (1980).
  83. See Supplemental Material at http://link.aps.org/supplemental/10.1103/zjzt-rv86 for random draw of matrix element values used in noiseless demonstration.
  84. R. Edwards, R. Gupta, N. Joó, K. Orginos, D. Richards, F. Winter, and B. Yoon, U.S. 2+1 flavor clover lattice generation program (to be published).
  85. M. Lüscher and P. Weisz, On-shell improved lattice gauge theories, Commun. Math. Phys. 98, 433 (1985); 98, 433(E) (1985).
  86. B. Sheikholeslami and R. Wohlert, Improved continuum limit lattice action for QCD with Wilson fermions, Nucl. Phys. B259, 572 (1985).
  87. C. Morningstar and M. J. Peardon, Analytic smearing of SU(3) link variables in lattice QCD, Phys. Rev. D 69, 054501 (2004).
  88. 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).
  89. B. Yoon et al., Isovector charges of the nucleon from 2+1-flavor QCD with clover fermions, Phys. Rev. D 95, 074508 (2017).
  90. 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 (2025) 004.
  91. 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).
  92. D. C. Hackett, D. A. Pefkou, and P. E. Shanahan, Gravitational form factors of the proton from Lattice QCD, Phys. Rev. Lett. 132, 251904 (2024).
  93. M. Hutchinson, A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines, Commun. Stat. Simul. Comput. 19, 433 (1990).
  94. A. Stathopoulos, J. Laeuchli, and K. Orginos, Hierarchical probing for estimating the trace of the matrix inverse on toroidal lattices, SIAM J. Sci. Comput. 35, S299 (2013).
  95. A. S. Gambhir, Disconnected diagrams in Lattice QCD, Ph.D. thesis, William-Mary Coll, 2017.
  96. G. P. Lepage, gvar v. 11.9.1, 10.5281/zenodo.4290884(2020), https://github.com/gplepage/gvar.
  97. B. Parlett, Misconvergence in the Lanczos algorithm, in Reliable Numerical Commputation (Oxford University Press, New York, 1990).
  98. A. B. J. Kuijlaars, Which eigenvalues are found by the Lanczos method?, SIAM J. Matrix Anal. Appl. 22, 306 (2000).
  99. O. Bär, Nucleon-pion-state contribution in lattice calculations of the nucleon charges gA, gT and gS, Phys. Rev. D 94, 054505 (2016).
  100. A. V. Grebe and M. Wagman, Nucleon-pion spectroscopy from sparsened correlators, Proc. Sci. LATTICE2023 (2024) 049 [arXiv:2312.00321].
  101. R. Zhang, A. V. Grebe, D. C. Hackett, M. L. Wagman, and Y. Zhao, Kinematically-enhanced interpolating operators for boosted hadrons, arXiv:2501.00729.
  102. H. Akaike, A new look at the statistical model identification, IEEE Trans. Autom. Control 19, 716 (1974).
  103. W. I. Jay and E. T. Neil, Bayesian model averaging for analysis of lattice field theory results, Phys. Rev. D 103, 114502 (2021).
  104. S. Borsanyi et al. (BMW Collaboration), Ab initio calculation of the neutron-proton mass difference, Science 347, 1452 (2015).
  105. 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).
  106. E. T. Neil and J. W. Sitison, Improved information criteria for Bayesian model averaging in lattice field theory, Phys. Rev. D 109, 014510 (2024).
  107. P. Shanahan, M. Wagman, and Y. Zhao, Collins-Soper kernel for TMD evolution from lattice QCD, Phys. Rev. D 102, 014511 (2020).
  108. O. Ledoit and M. Wolf, A well-conditioned estimator for large-dimensional covariance matrices, J. Multivariate Anal. 88, 365 (2004).
  109. G. C. F. M. Prony, Essai experimental et analytique, J. l’école Polytechnique Paris 1, 24 (1795).
  110. H.-W. Lin and S. D. Cohen, Lattice QCD beyond ground states, in Proceedings of the 4th International Workshop on Numerical Analysis and Lattice QCD (2007).
  111. 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).
  112. V. Druskin and L. Knizhnerman, Error bounds in the simple lanczos procedure for computing functions of symmetric matrices and eigenvalues, Zh. Vychislitel’noi Matematiki Matematicheskoi Fiz. 31, 970 (1991).
  113. C. R. Harris et al., Array programming with numpy, Nature (London) 585, 357 (2020).
  114. P. Virtanen et al. (SciPy 1.0 Contributors), scipy 1.0: Fundamental algorithms for acientific computing in python, Nat. Methods 17, 261 (2020).
  115. J. Reback et al., pandas-dev/pandas: pandas 1.0.3 (2020), 10.5281/zenodo.3715232.
  116. Wes McKinney, Data structures for statistical computing in python, in Proceedings of the 9th python in Science Conference, edited by Stéfan van der Walt and Jarrod Millman (2010), pp. 56–61.
  117. G. P. Lepage, lsqfit v. 11.7, 10.5281/zenodo.4037174(2020), https://github.com/gplepage/lsqfit.
  118. T. mpmath development team, mpmath: A python Library for Arbitrary-Precision Floating-Point Arithmetic (version 1.3.0) (2023), http://mpmath.org/.
  119. J. D. Hunter, matplotlib: A 2d graphics environment, Comput. Sci. Eng. 9, 90 (2007).
  120. M. L. Waskom, seaborn: sAtatistical data visualization, J. Open Source Softwaare 6, 3021 (2021).
  121. R. G. Edwards and B. Joó (SciDAC Collaboration, LHPC Collaboration, and UKQCD Collaboration), The chroma software system for lattice QCD, Nucl. Phys. B, Proc. Suppl. 140, 832 (2005).
  122. 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).
  123. R. Babich, M. A. Clark, B. Joó, G. Shi, R. C. Brower, and S. Gottlieb (QUDA Collaboration), Scaling lattice QCD beyond 100 GPUs, in Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (2011).
  124. M. A. Clark, B. Joó, A. Strelchenko, M. Cheng, A. Gambhir, and R. C. Brower (QUDA Collaboration), Accelerating lattice QCD multigrid on GPUs using fine-grained parallelization, in Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (2016).
  125. F. Winter, M. Clark, R. Edwards, and B. Joó, A framework for lattice QCD calculations on gpus, in Proceedings of the 2014 IEEE 28th International Parallel and Distributed Processing Symposium (2014), pp. 1073–1082.
  126. E. Romero and C. Kallidonis, chromaform, https://github.com/eromero-vlc/chromaform.
  127. A. Gambhir, D. Brantley, J. Chang, B. Hörz, H. Monge-Camacho, P. Vranas, and A. Walker-Loud, lalibe, https://github.com/callat-qcd/lalibe (2018).
  128. C. Gattringer and C. B. Lang, Quantum Chromodynamics on the Lattice (Springer, Berlin, 2010), Vol. 788.

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Phys. Rev. D 112, 054506– Published 17 September, 2025
  • Received 12 January 2025
  • Accepted 12 August 2025
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