Abstract
We examine the circuit complexity of coherent states in a free scalar field theory, applying Nielsen’s geometric approach as in [1]. The complexity of the coherent states have the same UV divergences as the vacuum state complexity and so we consider the finite increase of the complexity of these states over the vacuum state. One observation is that generally, the optimal circuits introduce entanglement between the normal modes at intermediate stages even though our reference state and target states are not entangled in this basis. We also compare our results from Nielsen’s approach with those found using the Fubini-Study method of [2]. For general coherent states, we find that the complexities, as well as the optimal circuits, derived from these two approaches, are different.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].
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].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement ‘thermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].
T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from Entanglement in Holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
M. Ammon and J. Erdmenger, Gauge/gravity duality, Cambridge University Press, Cambridge (2015).
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, Lect. Notes Phys. 931 (2017) pp.1 [arXiv:1609.01287] [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].
L. Susskind and Y. Zhao, Switchbacks and the Bridge to Nowhere, arXiv:1408.2823 [INSPIRE].
L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [INSPIRE].
D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
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].
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].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
M. Alishahiha, Holographic Complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].
D. Carmi, R.C. Myers and P. Rath, Comments on Holographic Complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].
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].
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].
S. Chapman, H. Marrochio and R.C. Myers, Complexity of Formation in Holography, JHEP 01 (2017) 062 [arXiv:1610.08063] [INSPIRE].
A.R. Brown, L. Susskind and Y. Zhao, Quantum Complexity and Negative Curvature, Phys. Rev. D 95 (2017) 045010 [arXiv:1608.02612] [INSPIRE].
J. Couch, W. Fischler and P.H. Nguyen, Noether charge, black hole volume and complexity, JHEP 03 (2017) 119 [arXiv:1610.02038] [INSPIRE].
A.R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE].
A. Reynolds and S.F. Ross, Complexity in de Sitter Space, Class. Quant. Grav. 34 (2017) 175013 [arXiv:1706.03788] [INSPIRE].
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].
Y. Zhao, Uncomplexity and Black Hole Geometry, Phys. Rev. D 97 (2018) 126007 [arXiv:1711.03125] [INSPIRE].
M. Moosa, Evolution of Complexity Following a Global Quench, JHEP 03 (2018) 031 [arXiv:1711.02668] [INSPIRE].
B. Swingle and Y. Wang, Holographic Complexity of Einstein-Maxwell-Dilaton Gravity, arXiv:1712.09826 [INSPIRE].
Z. Fu, A. Maloney, D. Marolf, H. Maxfield and Z. Wang, Holographic complexity is nonlocal, JHEP 02 (2018) 072 [arXiv:1801.01137] [INSPIRE].
Y.-S. An and R.-H. Peng, Effect of the dilaton on holographic complexity growth, Phys. Rev. D97 (2018) 066022, [1801.03638].
M. Alishahiha, A. Faraji Astaneh, M.R. Mohammadi Mozaffar and A. Mollabashi, Complexity Growth with Lifshitz Scaling and Hyperscaling Violation, JHEP 07 (2018) 042 [arXiv:1802.06740] [INSPIRE].
B. Chen, W.-M. Li, R.-Q. Yang, C.-Y. Zhang and S.-J. Zhang, Holographic subregion complexity under a thermal quench, JHEP 07 (2018) 034 [arXiv:1803.06680].
S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part I, JHEP 06 (2018) 046 [arXiv:1804.07410] [INSPIRE].
C.A. Agón, M. Headrick and B. Swingle, Subsystem Complexity and Holography, arXiv:1804.01561 [INSPIRE].
S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part II, JHEP 06 (2018) 114 [arXiv:1805.07262] [INSPIRE].
J. Couch, S. Eccles, T. Jacobson and P. Nguyen, Holographic Complexity and Volume, arXiv:1807.02186 [INSPIRE].
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].
L. Hackl and R.C. Myers, Circuit complexity for free fermions, JHEP 07 (2018) 139 [arXiv:1803.10638] [INSPIRE].
R. Khan, C. Krishnan and S. Sharma, Circuit Complexity in Fermionic Field Theory, arXiv:1801.07620 [INSPIRE].
A.P. Reynolds and S.F. Ross, Complexity of the AdS Soliton, Class. Quant. Grav. 35 (2018) 095006 [arXiv:1712.03732] [INSPIRE].
D.W.F. Alves and G. Camilo, Evolution of complexity following a quantum quench in free field theory, JHEP 06 (2018) 029 [arXiv:1804.00107] [INSPIRE].
H.A. Camargo, P. Caputa, D. Das, M.P. Heller and R. Jefferson, Complexity as a novel probe of quantum quenches: universal scalings and purifications, arXiv:1807.07075 [INSPIRE].
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].
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].
S. Chapman et al., Circuit Complexity for Thermofield Double States, in preparation (2018).
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].
R. Abt et al., Topological Complexity in AdS 3 /CFT 2, Fortsch. Phys. 66 (2018) 1800034 [arXiv:1710.01327] [INSPIRE].
R. Abt, J. Erdmenger, M. Gerbershagen, C.M. Melby-Thompson and C. Northe, Holographic Subregion Complexity from Kinematic Space, arXiv:1805.10298 [INSPIRE].
J. Molina-Vilaplana and A. Del Campo, Complexity Functionals and Complexity Growth Limits in Continuous MERA Circuits, JHEP 08 (2018) 012 [arXiv:1803.02356] [INSPIRE].
M.A. Nielsen, M.R. Dowling, M. Gu and A.C. Doherty, Quantum computation as geometry, Science 311 (2006) 1133 [quant-ph/0603161].
M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, Quant. Inf. Comput. 8 (2008) 861.
M.A. Nielsen, A geometric approach to quantum circuit lower bounds, Quant. Inf. Comput. 6 (2006) 213.
P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT, JHEP 11 (2017) 097 [arXiv:1706.07056] [INSPIRE].
B. Czech, Einstein Equations from Varying Complexity, Phys. Rev. Lett. 120 (2018) 031601 [arXiv:1706.00965] [INSPIRE].
P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 071602 [arXiv:1703.00456] [INSPIRE].
A. Bhattacharyya, P. Caputa, S.R. Das, N. Kundu, M. Miyaji and T. Takayanagi, Path-Integral Complexity for Perturbed CFTs, JHEP 07 (2018) 086 [arXiv:1804.01999] [INSPIRE].
P. Caputa and J.M. Magan, Quantum Computation as Gravity, arXiv:1807.04422 [INSPIRE].
S. Chapman, L. Hackl, M.P. Heller, H. Marrochio and R.C. Myers, Geometry of circuit complexity: Nielsen versus fubini-study, in preparation (2018).
R. Bhatia, Matrix analysis, vol. 169, Springer Science & Business Media (2013).
J. Watrous, Theory of Quantum Information, Cambridge University Press (2018).
M.I. Gil, Operator functions and localization of spectra, Springer (2003).
L. Hackl, Notes on circuit complexity of bosonic and fermionic gaussian states, unpublished.
M.A. Nielsen and I. Chuang, Quantum computation and quantum information, Cambridge University Press (2002).
I. Bengtsson and K. Życzkowski, Geometry of quantum states: an introduction to quantum entanglement, Cambridge University Press (2017).
M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Distance between Quantum States and Gauge-Gravity Duality, Phys. Rev. Lett. 115 (2015) 261602 [arXiv:1507.07555] [INSPIRE].
N. Lashkari and M. Van Raamsdonk, Canonical Energy is Quantum Fisher Information, JHEP 04 (2016) 153 [arXiv:1508.00897] [INSPIRE].
S. Banerjee, J. Erdmenger and D. Sarkar, Connecting Fisher information to bulk entanglement in holography, JHEP 08 (2018) 001 [arXiv:1701.02319] [INSPIRE].
M. Alishahiha and A. Faraji Astaneh, Holographic Fidelity Susceptibility, Phys. Rev. D 96 (2017) 086004 [arXiv:1705.01834] [INSPIRE].
A. Belin, A. Lewkowycz and G. Sárosi, The boundary dual of the bulk symplectic form, arXiv:1806.10144 [INSPIRE].
A. Bernamonti, F. Galli, J. Hernandez, R.C. Myers, J. Simon and S.-M. Ruan, The First Law of Complexity, in preparation (2018).
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1807.07677
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Guo, M., Hernandez, J., Myers, R.C. et al. Circuit complexity for coherent states. J. High Energ. Phys. 2018, 11 (2018). https://doi.org/10.1007/JHEP10(2018)011
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2018)011