Circuit complexity for free fermions

  • Lucas Hackl
  • Robert C. Myers
Open Access
Regular Article - Theoretical Physics


We study circuit complexity for free fermionic field theories and Gaussian states. Our definition of circuit complexity is based on the notion of geodesic distance on the Lie group of special orthogonal transformations equipped with a right-invariant metric. After analyzing the differences and similarities to bosonic circuit complexity, we develop a comprehensive mathematical framework to compute circuit complexity between arbitrary fermionic Gaussian states. We apply this framework to the free Dirac field in four dimensions where we compute the circuit complexity of the Dirac ground state with respect to several classes of spatially unentangled reference states. Moreover, we show that our methods can also be applied to compute the complexity of excited energy eigenstates of the free Dirac field. Finally, we discuss the relation of our results to alternative approaches based on the Fubini-Study metric, the relevance to holography and possible extensions.


AdS-CFT Correspondence Effective Field Theories 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    L. Susskind and Y. Zhao, Switchbacks and the Bridge to Nowhere, arXiv:1408.2823 [INSPIRE].
  3. [3]
    L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].ADSGoogle Scholar
  5. [5]
    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    M. Alishahiha, Holographic Complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].ADSMathSciNetGoogle Scholar
  8. [8]
    R.-G. Cai, S.-M. Ruan, S.-J. Wang, R.-Q. Yang and R.-H. Peng, Action growth for AdS black holes, JHEP 09 (2016) 161 [arXiv:1606.08307] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries, Phys. Rev. D 94 (2016) 084046 [arXiv:1609.00207] [INSPIRE].ADSMathSciNetGoogle Scholar
  10. [10]
    R.-Q. Yang, Strong energy condition and complexity growth bound in holography, Phys. Rev. D 95 (2017) 086017 [arXiv:1610.05090] [INSPIRE].ADSGoogle Scholar
  11. [11]
    S. Chapman, H. Marrochio and R.C. Myers, Complexity of Formation in Holography, JHEP 01 (2017) 062 [arXiv:1610.08063] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D. Carmi, R.C. Myers and P. Rath, Comments on Holographic Complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    D. Carmi, S. Chapman, H. Marrochio, R.C. Myers and S. Sugishita, On the Time Dependence of Holographic Complexity, JHEP 11 (2017) 188 [arXiv:1709.10184] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    A. Reynolds and S.F. Ross, Divergences in Holographic Complexity, Class. Quant. Grav. 34 (2017) 105004 [arXiv:1612.05439] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Y. Zhao, Complexity, boost symmetry and firewalls, arXiv:1702.03957 [INSPIRE].
  16. [16]
    A. Reynolds and S.F. Ross, Complexity in de Sitter Space, Class. Quant. Grav. 34 (2017) 175013 [arXiv:1706.03788] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    J. Couch, S. Eccles, W. Fischler and M.-L. Xiao, Holographic complexity and noncommutative gauge theory, JHEP 03 (2018) 108 [arXiv:1710.07833] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    B. Swingle and Y. Wang, Holographic Complexity of Einstein-Maxwell-Dilaton Gravity, arXiv:1712.09826 [INSPIRE].
  19. [19]
    Z. Fu, A. Maloney, D. Marolf, H. Maxfield and Z. Wang, Holographic complexity is nonlocal, JHEP 02 (2018) 072 [arXiv:1801.01137] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    X. Dong, A. Lewkowycz and M. Rangamani, Deriving covariant holographic entanglement, JHEP 11 (2016) 028 [arXiv:1607.07506] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    S.P. Jordan, K.S.M. Lee and J. Preskill, Quantum Algorithms for Quantum Field Theories, Science 336 (2012) 1130 [arXiv:1111.3633] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    S.P. Jordan, K.S.M. Lee and J. Preskill, Quantum Computation of Scattering in Scalar Quantum Field Theories, arXiv:1112.4833 [INSPIRE].
  25. [25]
    S.P. Jordan, K.S.M. Lee and J. Preskill, Quantum Algorithms for Fermionic Quantum Field Theories, arXiv:1404.7115 [INSPIRE].
  26. [26]
    S.P. Jordan, H. Krovi, K.S.M. Lee and J. Preskill, BQP-completeness of Scattering in Scalar Quantum Field Theory, arXiv:1703.00454 [INSPIRE].
  27. [27]
    T.J. Osborne, Hamiltonian complexity, Rept. Prog. Phys. 75 (2012) 022001.ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    S. Gharibian et al., Quantum hamiltonian complexity, Found. Trends Theor. Comput. Sci. 10 (2015) 159 [arXiv:1401.3916].MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    R. Orus, A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States, Annals Phys. 349 (2014) 117 [arXiv:1306.2164] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    G. Vidal, Entanglement Renormalization: an introduction, in Understanding Quantum Phase Transitions, L.D. Carr ed., CRC Press (2010) [arXiv:0912.1651].
  31. [31]
    K. Hashimoto, N. Iizuka and S. Sugishita, Time evolution of complexity in Abelian gauge theories, Phys. Rev. D 96 (2017) 126001 [arXiv:1707.03840] [INSPIRE].ADSGoogle Scholar
  32. [32]
    R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Toward a Definition of Complexity for Quantum Field Theory States, Phys. Rev. Lett. 120 (2018) 121602 [arXiv:1707.08582] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    R.-Q. Yang, Complexity for quantum field theory states and applications to thermofield double states, Phys. Rev. D 97 (2018) 066004 [arXiv:1709.00921] [INSPIRE].ADSGoogle Scholar
  35. [35]
    R.-Q. Yang, C. Niu, C.-Y. Zhang and K.-Y. Kim, Comparison of holographic and field theoretic complexities for time dependent thermofield double states, JHEP 02 (2018) 082 [arXiv:1710.00600] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    A.P. Reynolds and S.F. Ross, Complexity of the AdS Soliton, Class. Quant. Grav. 35 (2018) 095006 [arXiv:1712.03732] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    R. Khan, C. Krishnan and S. Sharma, Circuit Complexity in Fermionic Field Theory, arXiv:1801.07620 [INSPIRE].
  38. [38]
    R.-Q. Yang, Y.-S. An, C. Niu, C.-Y. Zhang and K.-Y. Kim, Axiomatic complexity in quantum field theory and its applications, arXiv:1803.01797 [INSPIRE].
  39. [39]
    S. Chapman et al., Circuit Complexity for Thermofield Double States, to appear (2018).Google Scholar
  40. [40]
    M.A. Nielsen, A geometric approach to quantum circuit lower bounds, quant-ph/0502070.
  41. [41]
    M.A. Nielsen, M.R. Dowling, M. Gu and A.M. Doherty, Quantum Computation as Geometry, Science 311 (2006) 1133 [quant-ph/0603161].
  42. [42]
    M.A. Nielsen and M.R. Dowling, The geometry of quantum computation, quant-ph/0701004.
  43. [43]
    A.R. Brown, L. Susskind and Y. Zhao, Quantum Complexity and Negative Curvature, Phys. Rev. D 95 (2017) 045010 [arXiv:1608.02612] [INSPIRE].ADSMathSciNetGoogle Scholar
  44. [44]
    A.R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE].ADSGoogle Scholar
  45. [45]
    J. Alvarez and C. Gómez, A comment on fisher information and quantum algorithms, quant-ph/9910115.
  46. [46]
    S. Arora and B. Barak, Computational Complexity: A Modern Approach, Cambridge University Press (2009).Google Scholar
  47. [47]
    C. Moore and S. Mertens, The Nature of Computation, Oxford University Press (2011).Google Scholar
  48. [48]
    S. Aaronson, The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes, arXiv:1607.05256 [INSPIRE].
  49. [49]
    J. Watrous, Quantum computational complexity, in Encyclopedia of complexity and systems science, Springer (2009), pp. 7174–7201.Google Scholar
  50. [50]
    M. Mimura and H. Toda, Topology of Lie groups, I and II, vol. 91, American Mathematical Society (1991).Google Scholar
  51. [51]
    C. Weedbrook et al., Gaussian quantum information, Rev. Mod. Phys. 84 (2012) 621 [arXiv:1110.3234].ADSCrossRefGoogle Scholar
  52. [52]
    E. Bianchi and L. Hackl, Bosonic and fermionic Gaussian states from Kähler structures, to appear (2018).Google Scholar
  53. [53]
    E. Bianchi, L. Hackl and N. Yokomizo, Linear growth of the entanglement entropy and the Kolmogorov-Sinai rate, JHEP 03 (2018) 025 [arXiv:1709.00427] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  54. [54]
    L. Vidmar, L. Hackl, E. Bianchi and M. Rigol, Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians, Phys. Rev. Lett. 119 (2017) 020601 [arXiv:1703.02979] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    E. Bianchi, L. Hackl and N. Yokomizo, Entanglement entropy of squeezed vacua on a lattice, Phys. Rev. D 92 (2015) 085045 [arXiv:1507.01567] [INSPIRE].ADSMathSciNetGoogle Scholar
  56. [56]
    D. Bump, Lie groups, Springer (2004).Google Scholar
  57. [57]
    B. Dutta et al., The real symplectic groups in quantum mechanics and optics, Pramana 45 (1995) 471.ADSCrossRefGoogle Scholar
  58. [58]
    A. Kirillov, An introduction to Lie groups and Lie algebras, vol. 113, Cambridge University Press (2008).Google Scholar
  59. [59]
    M.E. Peskin and D.V. Schroeder, An introduction to quantum field theory, Addison-Wesley Publishing Company (1995).Google Scholar
  60. [60]
    R. Bhatia, Matrix analysis, vol. 169, Springer Science & Business Media (2013).Google Scholar
  61. [61]
    J. Watrous, Theory of Quantum Information, Cambridge University Press (2018).Google Scholar
  62. [62]
    L. Hackl, Notes on circuit complexity of bosonic and fermionic gaussian states, unpublished.Google Scholar
  63. [63]
    S. Chapman, L. Hackl, M.P. Heller, H. Marrochio and R.C. Myers, Geometry of circuit complexity: Nielsen versus fubini-study, in preparation.Google Scholar
  64. [64]
    Z. Fu, A. Maloney, D. Marolf, H. Maxfield and Z. Wang, Holographic complexity is nonlocal, JHEP 02 (2018) 072 [arXiv:1801.01137] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    G. Fubini, Sulle metriche definite da una forma hermitiana: nota, Office graf. C. Ferrari (1904).Google Scholar
  66. [66]
    E. Study, Shortest paths in the complex domain (in German), Math. Ann. 60 (1905) 321 [INSPIRE].
  67. [67]
    D.S. Abrams and S. Lloyd, Simulation of many body Fermi systems on a universal quantum computer, Phys. Rev. Lett. 79 (1997) 2586 [quant-ph/9703054] [INSPIRE].
  68. [68]
    S.B. Bravyi and A.Y. Kitaev, Fermionic quantum computation, Annals Phys. 298 (2002) 210.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    S.P. Jordan, K.S.M. Lee and J. Preskill, Quantum Algorithms for Quantum Field Theories, Science 336 (2012) 1130 [arXiv:1111.3633] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    D.J. Gross and A. Neveu, Dynamical Symmetry Breaking in Asymptotically Free Field Theories, Phys. Rev. D 10 (1974) 3235 [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Institute for Gravitation and the Cosmos & Physics DepartmentPenn StateUniversity ParkU.S.A.
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada

Personalised recommendations