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 fastest computers in nature.

We conjecture that the quantum complexity of a holographic state is dual to the action of a certain spacetime region that we call a Wheeler-DeWitt patch. We illustrate and test the conjecture in the context of neutral, charged, and rotating black holes in anti-de Sitter spacetime, as well as black holes perturbed with static shells and with shock waves. This conjecture evolved from a previous conjecture that complexity is dual to spatial volume, but appears to be a major improvement over the original. In light of our results, we discuss the hypothesis that black holes are the fastest computers in nature.

I show how recent progress in real space renormalization group methods can be used to define a generalized notion of holography inspired by holographic dualities in quantum gravity. The generalization is based upon organizing information in a quantum state in terms of scale and defining a higher dimensional geometry from this structure. While states with a finite correlation length typically give simple geometries, the state at a quantum critical point gives a discrete version of anti de Sitter space. Some finite temperature quantum states include black hole-like objects. The gross features of equal time correlation functions are also reproduced in this geometric framework. The relationship between this framework and better understood versions of holography is discussed. I. INTRODUCTIONHilbert space, the mathematical representation of possible states of a quantum system, is exponentially large when the system is a macroscopic piece of matter. The traditional theory of symmetry breaking reduces this overwhelming amount of information to three key quantities: the energy (or Hamiltonian), the symmetry of the Hamiltonian, and the pattern of symmetry breaking.However, the existence of exotic phases of matter not characterized by broken symmetry, as in the fractional quantum hall effect [1], demonstrates the need for a more general theory. Fractional quantum hall systems are distinguished by the presence of long range entanglement in the ground state, suggesting that important information is encoded in the spatial structure of entanglement.Here I show how such a "pattern" of entanglement can be defined and visualized using the geometry of an emergent holographic dimension. This picture connects two new tools in many body physics: entanglement renormalization and holographic gauge/gravity duality.Entanglement renormalization [2] is a combination real space renormalization group techniques and ideas from quantum information theory that grew out of attempts to describe quantum critical points. The key message of entanglement renormalization is that the removal of local entanglement is essential for defining a proper real space renormalization group transformation for quantum * Electronic address: bswingle@mit.edu states. This realization has permitted a compact description of some quantum critical points [3,4].Holographic gauge/gravity duality [5,6,7] is the proposal that certain quantum field theories without gravity are dual to theories of quantum gravity in a curved higher dimensional "bulk" geometry. Holography provides a way to compute field theory observables from a completely different point of view using a small amount of information encoded geometrically. Real space renormalization is also important in the holographic framework [8,9,10,11], thus hinting at a possible connection between holography and entanglement renormalization. We will begin with entanglement renormalization and build up to the full holographic picture. II. MANY BODY ENTANGLEMENTWe are interested in quantifying entanglement in many body systems...

General scaling arguments, and the behavior of the thermal entropy density, are shown to lead to an infrared metric holographically representing a compressible state with hidden Fermi surfaces. This metric is characterized by a general dynamic critical exponent, z, and a specific hyperscaling violation exponent, θ. The same metric exhibits a logarithmic violation of the area law of entanglement entropy, as shown recently by Ogawa et al. (arXiv:1111.1023). We study the dependence of the entanglement entropy on the shape of the entangling region(s), on the total charge density, on temperature, and on the presence of additional visible Fermi surfaces of gauge-neutral fermions;for the latter computations, we realize the needed metric in an Einstein-Maxwell-dilaton theory.All our results support the proposal that the holographic theory describes a metallic state with hidden Fermi surfaces of fermions carrying gauge charges of deconfined gauge fields.1 arXiv:1112.0573v3 [cond-mat.str-el]