Complexity and Shock Wave Geometries

By D. Stanford, L. Susskind. 2014

763 citations recorded by the source corpus.

In this paper we refine a conjecture relating the time-dependent size of an Einstein-Rosen bridge (ERB) to the computational complexity of the dual quantum state. Our refinement states that the complexity is proportional to the spatial volume of the ERB. More precisely, up to an ambiguous numerical coefficient, we propose that the complexity is the regularized volume of the largest codimension one surface crossing the bridge, divided by G N l AdS . We test this conjecture against a wide variety of spherically symmetric shock wave geometries in different dimensions. We find detailed agreement.

Subfields: Astronomy and Astrophysics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/317fb39e091824f5879c6e3c96a63eebe09eaa81

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Citations in the corpus

On the time dependence of holographic complexity (2017, methodology): “Using the CV conjecture, the late time growth rate of the complexity also saturates, but this final rate is only proportional to the mass at high temperatures and with a coefficient that depends on the spacetime dimension [6, 26].”
Localized shocks (2014, methodology): “In [24], two of us checked a refinement of this relationship for a family of states, corresponding to the spherical shock wave geometries constructed in [25].”
Diving into traversable wormholes (2017, background): “As a side remark, note that the complexity = volume conjecture [32, 33] here would say that complexity = (distance between the two boundaries).”
The quantum gravity dynamics of near extremal black holes (2018, background): “5 Einstein-Rosen Bridge The Einstein-Rosen Bridge in a classical wormhole keeps growing linearly with time and this behavior was conjectured to related with the growth of computational complexity of the dual quantum state [28].”
Holographic Complexity Equals Bulk Action? (2015, background): “[3], the coefficient in the rate of growth of complexity depended both on the size of the black hole and on the number of dimensions, which made it impossible to saturate a universal bound of the form of Eq.”
Switchbacks and the Bridge to Nowhere (2014, methodology): “Within a numerical factor discussed in [1] A is the area of the horizon.”
Comments on holographic complexity (2016, methodology): “These coefficients also contain dimensionless parameters characterizing the underlying CFT, e.g., CT ∼ Ld−1/GN in the present case where the bulk is described by Einstein gravity.11 The leading divergence appearing with the CV duality is proportional to the volume of the boundary time slice, i.e., C”
On volumes of subregions in holography and complexity (2016, background): “In the following we briefly review the concept of quantum complexity, and its proposed holographic conjectures as given in [24–31].”
Divergences in holographic complexity (2016, background): “This proposal was refined in [3] into the conjecture that the computational complexity of the boundary state at a given time (on some spacelike slice of the boundary) could be identified with the volume of a maximal volume spacelike slice in the bulk, ending on the given boundary slice.”
Entanglement is not enough (2014, background): “Among these slices there is one of maximum spatial volume [7, 8].”
Linear growth of quantum circuit complexity (2021, methodology): “We design the new block from Clifford gates such that two operations are equivalent: (1) applying P to the input qubits before the extended blocks and (2) applying the extended block, then a single-site Z.”
Circuit complexity for coherent states (2018, background): “Of course, the latter is also the conjectured dual of complexity according to the CV proposal [15, 16].”
Second law of quantum complexity (2017, background): “(ii) Hayden-Preskill circuits exhibit an effect called the switchback effect [2,3].”
Quantum chaos and the complexity of spread of states (2022, positive): “…to use these results, in particular the time evolved TFD wavefunction, to better understand the physics of the black hole interior, expanding on [95], and perhaps the conjectures relating complexity in quantum ﬁeld theory to spatial volumes and action in dual theory of gravity [44–46 , 96].”
Quantum epidemiology: operator growth, thermal effects, and SYK (2018, neutral): “Pn [ ψ (t) ρ ] ≡ ∣∣ 〈n|ψ 1 (t)|TFD〉∣∣2) to the size wave-function 3If we replace t→ −t, then this is the precursor state ψ 1 (−t) |TFD〉, where the “boundary” operator ψ 1 acted upon the thermofield double state at time −t [3, 12,39,40].”
Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT (2017, background): “on some spacelike slice, say Σ, of the boundary, was first proposed in [21, 22] to be identified with the volume of a maximal volume space-like slice, say MΣ, in the bulk where the”
Holographic complexity in Vaidya spacetimes. Part I (2018, methodology): “Further, this constant rate matches the late time rate of growth found for the eternal black hole in [28, 46].”
Circuit complexity for free fermions (2018, background): “The complexity=volume (CV) conjecture suggests that complexity is dual to the volume of an extremal (codimensionone) bulk surface anchored to a certain time slice in the boundary [1, 4].”
Complexity, action, and black holes (2015, methodology): “From the boundary/complexity perspective, this has been called the “switchback” effect [6].”