Abstract
In this work, we propose a testing procedure to distinguish between the different approaches for computing complexity. Our test does not require a direct comparison between the approaches and thus avoids the issue of choice of gates, basis, etc. The proposed testing procedure employs the information-theoretic measures Loschmidt echo and Fidelity; the idea is to investigate the sensitivity of the complexity (derived from the different approaches) to the evolution of states. We discover that only circuit complexity obtained directly from the wave function is sensitive to time evolution, leaving us to claim that it surpasses the other approaches. We also demonstrate that circuit complexity displays a universal behaviour — the complexity is proportional to the number of distinct Hamiltonian evolutions that act on a reference state. Due to this fact, for a given number of Hamiltonians, we can always find the combination of states that provides the maximum complexity; consequently, other combinations involving a smaller number of evolutions will have less than maximum complexity and, hence, will have resources. Finally, we explore the evolution of complexity in non-local theories; we demonstrate the growth of complexity is sustained over a longer period of time as compared to a local theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [Int. J. Mod. Phys. D 19 (2010) 2429] [arXiv:1005.3035] [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].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [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].
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].
J.L.F. Barbon and E. Rabinovici, Holographic complexity and spacetime singularities, JHEP 01 (2016) 084 [arXiv:1509.09291] [INSPIRE].
M. Alishahiha, Holographic Complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].
W. Chemissany and T.J. Osborne, Holographic fluctuations and the principle of minimal complexity, JHEP 12 (2016) 055 [arXiv:1605.07768] [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].
A.R. Brown, L. Susskind and Y. Zhao, Quantum Complexity and Negative Curvature, Phys. Rev. D 95 (2017) 045010 [arXiv:1608.02612] [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].
R.-Q. Yang, Strong energy condition and complexity growth bound in holography, Phys. Rev. D 95 (2017) 086017 [arXiv:1610.05090] [INSPIRE].
S. Chapman, H. Marrochio and R.C. Myers, Complexity of Formation in Holography, JHEP 01 (2017) 062 [arXiv:1610.08063] [INSPIRE].
D. Carmi, R.C. Myers and P. Rath, Comments on Holographic Complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].
P. Rath, Holographic Complexity, Perimeter Scholars International essay (unpublished), (2016).
A. Reynolds and S.F. Ross, Divergences in Holographic Complexity, Class. Quant. Grav. 34 (2017) 105004 [arXiv:1612.05439] [INSPIRE].
A.R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE].
Y. Zhao, Complexity and Boost Symmetry, Phys. Rev. D 98 (2018) 086011 [arXiv:1702.03957] [INSPIRE].
M. Flory, A complexity/fidelity susceptibility g-theorem for AdS 3 /BCFT 2, JHEP 06 (2017) 131 [arXiv:1702.06386] [INSPIRE].
M. Alishahiha and A. Faraji Astaneh, Holographic Fidelity Susceptibility, Phys. Rev. D 96 (2017) 086004 [arXiv:1705.01834] [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].
J. Couch, S. Eccles, W. Fischler and M.-L. Xiao, Holographic complexity and noncommutative gauge theory, JHEP 03 (2018) 108 [arXiv:1710.07833] [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].
R. Abt et al., Topological Complexity in AdS 3 /CFT 2, Fortsch. Phys. 66 (2018) 1800034 [arXiv:1710.01327] [INSPIRE].
M. Moosa, Evolution of Complexity Following a Global Quench, JHEP 03 (2018) 031 [arXiv:1711.02668] [INSPIRE].
M. Moosa, Divergences in the rate of complexification, Phys. Rev. D 97 (2018) 106016 [arXiv:1712.07137] [INSPIRE].
B. Swingle and Y. Wang, Holographic Complexity of Einstein-Maxwell-Dilaton Gravity, JHEP 09 (2018) 106 [arXiv:1712.09826] [INSPIRE].
A.P. Reynolds and S.F. Ross, Complexity of the AdS Soliton, Class. Quant. Grav. 35 (2018) 095006 [arXiv:1712.03732] [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. D 97 (2018) 066022 [arXiv:1801.03638] [INSPIRE].
S. Bolognesi, E. Rabinovici and S.R. Roy, On Some Universal Features of the Holographic Quantum Complexity of Bulk Singularities, JHEP 06 (2018) 016 [arXiv:1802.02045] [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] [INSPIRE].
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, JHEP 02 (2019) 145 [arXiv:1804.01561] [INSPIRE].
R. Abt, J. Erdmenger, M. Gerbershagen, C.M. Melby-Thompson and C. Northe, Holographic Subregion Complexity from Kinematic Space, JHEP 01 (2019) 012 [arXiv:1805.10298] [INSPIRE].
K. Hashimoto, N. Iizuka and S. Sugishita, Thoughts on Holographic Complexity and its Basis-dependence, Phys. Rev. D 98 (2018) 046002 [arXiv:1805.04226] [INSPIRE].
S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part II, JHEP 06 (2018) 114 [arXiv:1805.07262] [INSPIRE].
M. Flory and N. Miekley, Complexity change under conformal transformations in AdS 3 /CFT 2, arXiv:1806.08376 [INSPIRE].
J. Couch, S. Eccles, T. Jacobson and P. Nguyen, Holographic Complexity and Volume, JHEP 11 (2018) 044 [arXiv:1807.02186] [INSPIRE].
S.A. Hosseini Mansoori, V. Jahnke, M.M. Qaemmaqami and Y.D. Olivas, Holographic complexity of anisotropic black branes, arXiv:1808.00067 [INSPIRE].
S. Mahapatra and P. Roy, On the time dependence of holographic complexity in a dynamical Einstein-dilaton model, JHEP 11 (2018) 138 [arXiv:1808.09917] [INSPIRE].
M. Ghodrati, Complexity growth rate during phase transitions, Phys. Rev. D 98 (2018) 106011 [arXiv:1808.08164] [INSPIRE].
Y. Ling, Y. Liu and C.-Y. Zhang, Holographic Subregion Complexity in Einstein-Born-Infeld theory, Eur. Phys. J. C 79 (2019) 194 [arXiv:1808.10169] [INSPIRE].
M. Alishahiha, K. Babaei Velni and M.R. Mohammadi Mozaffar, Subregion Action and Complexity, arXiv:1809.06031 [INSPIRE].
J. Jiang, Action growth rate for a higher curvature gravitational theory, Phys. Rev. D 98 (2018) 086018 [arXiv:1810.00758] [INSPIRE].
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].
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].
R. Khan, C. Krishnan and S. Sharma, Circuit Complexity in Fermionic Field Theory, Phys. Rev. D 98 (2018) 126001 [arXiv:1801.07620] [INSPIRE].
R.-Q. Yang, Y.-S. An, C. Niu, C.-Y. Zhang and K.-Y. Kim, Principles and symmetries of complexity in quantum field theory, Eur. Phys. J. C 79 (2019) 109 [arXiv:1803.01797] [INSPIRE].
L. Hackl and R.C. Myers, Circuit complexity for free fermions, JHEP 07 (2018) 139 [arXiv:1803.10638] [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].
J.M. Magán, Black holes, complexity and quantum chaos, JHEP 09 (2018) 043 [arXiv:1805.05839] [INSPIRE].
P. Caputa and J.M. Magán, Quantum Computation as Gravity, arXiv:1807.04422 [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, Phys. Rev. Lett. 122 (2019) 081601 [arXiv:1807.07075] [INSPIRE].
M. Guo, J. Hernandez, R.C. Myers and S.-M. Ruan, Circuit Complexity for Coherent States, JHEP 10 (2018) 011 [arXiv:1807.07677] [INSPIRE].
A. Bhattacharyya, A. Shekar and A. Sinha, Circuit complexity in interacting QFTs and RG flows, JHEP 10 (2018) 140 [arXiv:1808.03105] [INSPIRE].
R.-Q. Yang, Y.-S. An, C. Niu, C.-Y. Zhang and K.-Y. Kim, More on complexity of operators in quantum field theory, JHEP 03 (2019) 161 [arXiv:1809.06678] [INSPIRE].
J. Jiang, J. Shan and J. Yang, Circuit complexity for free Fermion with a mass quench, arXiv:1810.00537 [INSPIRE].
T. Ali, A. Bhattacharyya, S. Shajidul Haque, E.H. Kim and N. Moynihan, Post-Quench Evolution of Distance and Uncertainty in a Topological System: Complexity, Entanglement and Revivals, arXiv:1811.05985 [INSPIRE].
C. Bennett, Logical reversibility of computation, IBM J. Res. Dev. 17 (1973) 525.
A. Berthiaume and G. Brassard, The quantum challenge to structural complexity theory, proceedings of 7th IEEE Conference on Structure in Complexity Theory, (1992).
E. Bernstein and U. Vazirani, Quantum complexity theory, proceedings of ACM Symposium on Theory of Computing, (1993).
S. Lloyd, Ultimate physical Limits to computation, Nature 406 (2000) 1047 [quant-ph/9908043].
S. Arora and B. Barak, Computational Complexity: A Modern Approach, Cambridge University Press, (2009).
C. Moore and S. Mertens, The Nature of Computation, Oxford University Press, (2011).
S. Aaronson, The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes, arXiv:1607.05256 [INSPIRE].
J. Watrous, Quantum computational complexity, in Encyclopedia of complexity and systems science, Springer, (2009), pp. 7174–7201.
T.J. Osborne, Hamiltonian complexity, Rept. Prog. Phys. 75 (2012) 022001.
S. Gharibian et al., Quantum hamiltonian complexity, Foundations and Trends in Theoretical Computer Science 10 (2015) 159.
R. Raz and A. Tal, Oracle Separation of BQP and PH, Electronic Colloquium on Computational Complexity, Report No. 107 (2018).
S. Aaronson, BQP and the polynomial hierarchy, in Proceedings of the 42nd ACM symposium on Theory of computing — STOC’10 STOC 2010: 141–150.
S. Aaronson and A. Ambainis, Forrelation: A Problem that Optimally Separates Quantum from Classical Computing, in Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing — STOC’15 STOC 2015: 307–316.
C.H. Bennett and J. Gill, Relative to a Random Oracle A, P A ≠ NP A ≠ co-NP A with Probability 1, SIAM J. Comput. 10 (1981) 96.
S.P. Jordan, K.S.M. Lee and J. Preskill, Quantum Algorithms for Quantum Field Theories, Science 336 (2012) 1130 [arXiv:1111.3633] [INSPIRE].
S.P. Jordan, K.S.M. Lee and J. Preskill, Quantum Computation of Scattering in Scalar Quantum Field Theories, arXiv:1112.4833 [INSPIRE].
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].
M.A. Nielsen, A geometric approach to quantum circuit lower bounds, quant-ph/0502070.
M.A. Nielsen, M.R. Dowling, M. Gu and A.M. Doherty, Quantum Computation as Geometry, Science 311 (2006) 1133 [quant-ph/0603161].
M.A. Nielsen and M.R. Dowling, The geometry of quantum computation, quant-ph/0701004.
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].
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].
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].
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].
T. Takayanagi, Holographic Spacetimes as Quantum Circuits of Path-Integrations, JHEP 12 (2018) 048 [arXiv:1808.09072] [INSPIRE].
T. Gorin, T. Prosen, T.H. Seligman and M. Znidaric, Dynamics of Loschmidt echoes and fidelity decay, Phys. Rept. 435 (2006) 33.
A. Goussev, R.A. Jalabert, H.M. Pastawski and D. Wisniacki, Loschmidt Echo, Scholarpedia 7 (2012) 11687 [arXiv:1206.6348].
W.P. Su, J.R. Schrieffer and A.J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42 (1979) 1698 [INSPIRE].
C.L. Kane and T.C. Lubensky, Topological boundary modes in isostatic lattices, Nature Phys. 10 (2014) 39.
A.M. Perelomov, Generalized Coherent States and Their Applications, Springer, Berlin, Heidelberg, Germany, (1986).
I. Bengtsson and K. Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, Cambridge University Press, (2006).
F.M. Cucchietti, The Loschmidt echo in classically chaotic systems: Quantum chaos, irreversibility and decoherence, Ph.D. Thesis, quant-ph/0410121.
D. Petz, An Invitation to the Algebra of Canonical Commutation Relations, Leuven University Press, Leuven, Belgium, (1990).
B. Swingle and N. Yunger Halpern, Resilience of scrambling measurements, Phys. Rev. A 97 (2018) 062113 [arXiv:1802.01587] [INSPIRE].
B. Swingle, G. Bentsen, M. Schleier-Smith and P. Hayden, Measuring the scrambling of quantum information, Phys. Rev. A 94 (2016) 040302 [arXiv:1602.06271] [INSPIRE].
N. Shiba and T. Takayanagi, Volume Law for the Entanglement Entropy in Non-local QFTs, JHEP 02 (2014) 033 [arXiv:1311.1643] [INSPIRE].
T. Ali, A. Bhattacharyya, S.S. Haque, E. Kim and N. Moynihan, Complexity vs entanglement growth: local vs non-local theory, in progress.
A. Milsted, J. Haegeman and T.J. Osborne, Matrix product states and variational methods applied to critical quantum field theory, Phys. Rev. D 88 (2013) 085030 [arXiv:1302.5582] [INSPIRE].
A. Bhattacharyya, L. Cheng, L.-Y. Hung, S. Ning and Z. Yang, Notes on the Causal Structure in a Tensor Network, arXiv:1805.03071 [INSPIRE].
A. Milsted and G. Vidal, Tensor networks as conformal transformations, arXiv:1805.12524 [INSPIRE].
A. Milsted and G. Vidal, Tensor networks as path integral geometry, arXiv:1807.02501 [INSPIRE].
Q. Hu, A. Franco-Rubio and G. Vidal, Continuous tensor network renormalization for quantum fields, arXiv:1809.05176 [INSPIRE].
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: 1810.02734
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
Ali, T., Bhattacharyya, A., Haque, S.S. et al. Time evolution of complexity: a critique of three methods. J. High Energ. Phys. 2019, 87 (2019). https://doi.org/10.1007/JHEP04(2019)087
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2019)087