The DOI matches its Crossref record; 68 of the 68 references it cites were found in a registry or an index.
Complexity, action, and black holes
Open accessAbstract
Our earlier paper ``Complexity Equals Action'' conjectured that the quantum computational complexity of a holographic state is given by the classical action of a region in the bulk (the ``Wheeler-DeWitt'' patch). We provide calculations for the results quoted in that paper, explain how it fits into a broader (tensor) network of ideas, and elaborate on the hypothesis that black holes are the fastest computers in nature.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
Susskind, L. (1994) · arXiv · doi:10.48550/arxiv.hep-th/9409089 · Cited by 4,085
Self-citationSusskind, L. (1995) · Journal of Mathematical Physics · doi:10.1063/1.531249 · Cited by 4,085
Self-citationMicroscopic origin of the Bekenstein-Hawking entropy
Strominger, A., & Vafa, C. (1996) · Physics Letters B · doi:10.1016/0370-2693(96)00345-0 · Cited by 3,072
Maldacena, J., Shenker, S. H., & Stanford, D. (2016) · Journal of High Energy Physics · doi:10.1007/jhep08(2016)106 · Cited by 2,313
Microscopic origin of the Bekenstein-Hawking entropy
Strominger, A., & Vafa, C. (1996) · CERN Document Server (European Organization for Nuclear Research) · Cited by 2,109
Average entropy of a subsystem
Page, D. N. (1993) · Physical Review Letters · doi:10.1103/physrevlett.71.1291 · Cited by 1,578
Eternal black holes in anti-de Sitter
Maldacena, J. (2003) · Journal of High Energy Physics · doi:10.1088/1126-6708/2003/04/021 · Cited by 1,555
Black holes and the butterfly effect
Shenker, S. H., & Stanford, D. (2014) · Journal of High Energy Physics · doi:10.1007/jhep03(2014)067 · Cited by 1,533
Black holes as mirrors: quantum information in random subsystems
Hayden, P., & Preskill, J. (2007) · Journal of High Energy Physics · doi:10.1088/1126-6708/2007/09/120 · Cited by 1,332
Cool horizons for entangled black holes
Maldacena, J., & Susskind, L. (2013) · Fortschritte der Physik · doi:10.1002/prop.201300020 · Cited by 1,304
Self-citationHartnoll, S. A., Herzog, C. P., & Horowitz, G. T. (2008) · Journal of High Energy Physics · doi:10.1088/1126-6708/2008/12/015 · Cited by 1,204
Universal upper bound on the entropy-to-energy ratio for bounded systems
Bekenstein, J. D. (1981) · Physical Review D · doi:10.1103/physrevd.23.287 · Cited by 1,182
Ultimate physical limits to computation
Lloyd, S. (2000) · Nature · doi:10.1038/35023282 · Cited by 1,120
Sekino, Y., & Susskind, L. (2008) · Journal of High Energy Physics · doi:10.1088/1126-6708/2008/10/065 · Cited by 1,112
Self-citationA Relativist's Toolkit: The Mathematics of Black-Hole Mechanics
Poisson, E. (2004) · doi:10.1017/cbo9780511606601 · Cited by 1,035
Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence
Pastawski, F., Yoshida, B., Harlow, D., & Preskill, J. (2015) · Journal of High Energy Physics · doi:10.1007/jhep06(2015)149 · Cited by 958
Entanglement renormalization and holography
Swingle, B. (2012) · Physical Review D · doi:10.1103/physrevd.86.065007 · Cited by 915
Self-citationThe maximum speed of dynamical evolution
Margolus, N., & Levitin, L. B. (1998) · Physica D: Nonlinear Phenomena · doi:10.1016/s0167-2789(98)00054-2 · Cited by 887
Holographic Complexity Equals Bulk Action?
Brown, A. R., Roberts, D. A., Susskind, L., Swingle, B., & Zhao, Y. (2016) · Physical Review Letters · doi:10.1103/physrevlett.116.191301 · Cited by 852
Self-citationEntanglement as Topology: Hopf Linking as the Geometric Origin of Quantum Correlation
Novickis, A. (2026) · doi:10.5281/zenodo.20549411 · Cited by 821
Anandan, J., & Aharonov, Y. (1990) · Physical Review Letters · doi:10.1103/physrevlett.65.1697 · Cited by 794
Computational complexity and black hole horizons
Susskind, L. (2016) · Fortschritte der Physik · doi:10.1002/prop.201500092 · Cited by 767
Self-citationTime evolution of entanglement entropy from black hole interiors
Hartman, T., & Maldacena, J. (2013) · Journal of High Energy Physics · doi:10.1007/jhep05(2013)014 · Cited by 761
Complexity and shock wave geometries
Stanford, D., & Susskind, L. (2014) · Physical Review D · doi:10.1103/physrevd.90.126007 · Cited by 748
Self-citationD-brane approach to black hole quantum mechanics
Callan, C. G., & Maldacena, J. M. (1996) · Nuclear Physics B · doi:10.1016/0550-3213(96)00225-8 · Cited by 748
Hosur, P., Qi, X.-L., Roberts, D. A., & Yoshida, B. (2016) · Journal of High Energy Physics · doi:10.1007/jhep02(2016)004 · Cited by 717
Self-citationRoberts, D. A., Stanford, D., & Susskind, L. (2015) · Journal of High Energy Physics · doi:10.1007/jhep03(2015)051 · Cited by 652
Self-citationShenker, S. H., & Stanford, D. (2015) · Journal of High Energy Physics · doi:10.1007/jhep05(2015)132 · Cited by 604
Computational Capacity of the Universe
Lloyd, S. (2002) · Physical Review Letters · doi:10.1103/physrevlett.88.237901 · Cited by 571
Quantum Computation as Geometry
Nielsen, M. A., Dowling, M. R., Gu, M., & Doherty, A. C. (2006) · Science · doi:10.1126/science.1121541 · Cited by 539
Time in the Quantum Theory and the Uncertainty Relation for Time and Energy
Aharonov, Y., & Bohm, D. (1961) · Physical Review · doi:10.1103/physrev.122.1649 · Cited by 482
The maximum speed of dynamical evolution
Margolus, N., & Levitin, L. B. (1998) · Cited by 475
Shenker, S. H., & Stanford, D. (2014) · Journal of High Energy Physics · doi:10.1007/jhep12(2014)046 · Cited by 473
Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory
Roberts, D. A., & Stanford, D. (2015) · Physical Review Letters · doi:10.1103/physrevlett.115.131603 · Cited by 467
Self-citationThe gravitational shock wave of a massless particle
Dray, T., & 't Hooft, G. (1985) · Nuclear Physics B · doi:10.1016/0550-3213(85)90525-5 · Cited by 442
Towards the fast scrambling conjecture
Lashkari, N., Stanford, D., Hastings, M., Osborne, T., & Hayden, P. (2013) · Journal of High Energy Physics · doi:10.1007/jhep04(2013)022 · Cited by 428
Quasiclassical Method in the Theory of Superconductivity
Larkin, A. I., & Ovchinnikov, Y. N. (1969) · Journal of Experimental and Theoretical Physics · Cited by 426
Zero temperature limit of holographic superconductors
Horowitz, G. T., & Roberts, M. M. (2009) · Journal of High Energy Physics · doi:10.1088/1126-6708/2009/11/015 · Cited by 355
The Black Hole Singularity in AdS/CFT
Fidkowski, L., Hubeny, V., Kleban, M., & Shenker, S. (2004) · Journal of High Energy Physics · doi:10.1088/1126-6708/2004/02/014 · Cited by 330
Addendum to computational complexity and black hole horizons
Susskind, L. (2016) · Fortschritte der Physik · doi:10.1002/prop.201500093 · Cited by 324
Self-citationQuantum computation vs. firewalls
Harlow, D., & Hayden, P. (2013) · Journal of High Energy Physics · doi:10.1007/jhep06(2013)085 · Cited by 310
Charged black hole in a grand canonical ensemble
Braden, H. W., Brown, J. D., Whiting, B. F., & York, J. W. (1990) · Physical Review D · doi:10.1103/physrevd.42.3376 · Cited by 285
Relative entropy and the Bekenstein bound
Casini, H. (2008) · Classical and Quantum Gravity · doi:10.1088/0264-9381/25/20/205021 · Cited by 280
Duality between electric and magnetic black holes
Hawking, S. W., & Ross, S. F. (1995) · Physical Review D · doi:10.1103/physrevd.52.5865 · Cited by 268
Tensor Network States and Geometry
Evenbly, G., & Vidal, G. (2011) · Journal of Statistical Physics · doi:10.1007/s10955-011-0237-4 · Cited by 265
Alishahiha, M. (2015) · Physical Review D · doi:10.1103/physrevd.92.126009 · Cited by 263
The Black Hole Singularity in AdS/CFTPreprint
Fidkowski, L., Hubeny, V. E., Kleban, M., & Shenker, S. H. (2003) · Cited by 240
Switchbacks and the Bridge to NowherePreprint
Susskind, L., & Zhao, Y. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1408.2823 · Cited by 235
Self-citationIntegral geometry and holography
Czech, B., Lamprou, L., McCandlish, S., & Sully, J. (2015) · Journal of High Energy Physics · doi:10.1007/jhep10(2015)175 · Cited by 219
Supersymmetry of the (2+1)-dimensional black holes
Coussaert, O., & Henneaux, M. (1994) · Physical Review Letters · doi:10.1103/physrevlett.72.183 · Cited by 205
On gravitational shock waves in curved spacetimes
Sfetsos, K. (1995) · Nuclear Physics B · doi:10.1016/0550-3213(94)00573-w · Cited by 174
Osborne, T. J. (2012) · Reports on Progress in Physics · doi:10.1088/0034-4885/75/2/022001 · Cited by 165
Bulk and transhorizon measurements in AdS/CFT
Heemskerk, I., Marolf, D., Polchinski, J., & Sully, J. (2012) · Journal of High Energy Physics · doi:10.1007/jhep10(2012)165 · Cited by 159
The effect of spherical shells of matter on the Schwarzschild black hole
Dray, T., & Hooft, G. ’. (1985) · Communications in Mathematical Physics · doi:10.1007/bf01215912 · Cited by 157
Bulk and transhorizon measurements in AdS/CFT
Heemskerk, I., Marolf, D., Polchinski, J., & Sully, J. (2012) · eScholarship (California Digital Library) · Cited by 126
A theory of quantum gravity based on quantum computationPreprint
Lloyd, S. (2005) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0501135 · Cited by 119
Spread of entanglement and causality
Casini, H., Liu, H., & Mezei, M. (2016) · Journal of High Energy Physics · doi:10.1007/jhep07(2016)077 · Cited by 118
The typical‐state paradox: diagnosing horizons with complexity
Susskind, L. (2016) · Fortschritte der Physik · doi:10.1002/prop.201500091 · Cited by 116
Self-citationBidirectional holographic codes and sub-AdS locality
Yang, Z., Hayden, P. M., & Qi, X.-L. (2016) · Journal of High Energy Physics · doi:10.1007/jhep01(2016)175 · Cited by 105
Notes on spacetime thermodynamics and the observer dependence of entropy
Marolf, D. M., Minić, D., & Ross, S. F. (2004) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.69.064006 · Cited by 102
Disrupting entanglement of black holes
Leichenauer, S. (2014) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.90.046009 · Cited by 99
Holographic complexity and spacetime singularities
Barbón, J. L. F., & Rabinovici, E. (2016) · Journal of High Energy Physics · doi:10.1007/jhep01(2016)084 · Cited by 86
The speed limit of quantum unitary evolution
Giovannetti, V., Lloyd, S., & Maccone, L. (2004) · Journal of Optics B: Quantum and Semiclassical Optics · doi:10.1088/1464-4266/6/8/028 · Cited by 62
Tensor network quotient takes the vacuum to the thermal state
Czech, B., Evenbly, G., Lamprou, L., McCandlish, S., Qi, X.-L., Sully, J., & Vidal, G. (2016) · Physical review. B./Physical review. B · doi:10.1103/physrevb.94.085101 · Cited by 54
The information theoretic interpretation of the length of a curve
Czech, B., Hayden, P., Lashkari, N., & Swingle, B. (2015) · Journal of High Energy Physics · doi:10.1007/jhep06(2015)157 · Cited by 45
Self-citationHolographic complexity of cold hyperbolic black holes
Barbón, J. L. F., & Martín-García, J. (2015) · Journal of High Energy Physics · doi:10.1007/jhep11(2015)181 · Cited by 32
Lachmann, M., Newman, M. E. J., & Moore, C. (2004) · American Journal of Physics · doi:10.1119/1.1773578 · Cited by 25
Physical Limits to Communication
Lloyd, S., Giovannetti, V., & Maccone, L. (2004) · Physical Review Letters · doi:10.1103/physrevlett.93.100501 · Cited by 23
Cited by
Holographic Complexity Equals Bulk Action?
Brown, A. R., Roberts, D. A., Susskind, L., Swingle, B., & Zhao, Y. (2016) · Physical Review Letters · doi:10.1103/physrevlett.116.191301 · Cited by 852
Self-citationEmergent Gravity and the Dark Universe
Verlinde, E. P. (2017) · SciPost Physics · doi:10.21468/scipostphys.2.3.016 · Cited by 554
Chaos and complexity by design
Roberts, D. A., & Yoshida, B. (2017) · Journal of High Energy Physics · doi:10.1007/jhep04(2017)121 · Cited by 455
Self-citationSecond law of quantum complexity
Brown, A. R., & Susskind, L. (2018) · Physical Review D · doi:10.1103/physrevd.97.086015 · Cited by 292
Self-citationComments on holographic complexity
Carmi, D., Myers, R. C., & Rath, P. (2017) · Journal of High Energy Physics · doi:10.1007/jhep03(2017)118 · Cited by 292
Metadata
- Authors
- Brown, A. R., Roberts, D. A., Susskind, L., Swingle, B., & Zhao, Y.
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2016-04-18
- Volume
- 93
- Issue
- 8
- ISSN
- 2470-0010 (ISSN-L)
- 2470-0029
- OpenAlex
- W2219721183
- Cited by
- 640
- References
- 68
- FWCI
- 39.60
- Type
- journal article
- Language
- en
- Open access
- green
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.