Quantum Computational Complexity -- From Quantum Information to Black Holes and Back

Abstract

Quantum computational complexity estimates the difficulty of constructing quantum states from elementary operations, a problem of prime importance for quantum computation. Surprisingly, this quantity can also serve to study a completely different physical problem -that of information processing inside black holes. Quantum computational complexity was suggested as a new entry in the holographic dictionary, which extends the connection between geometry and information and resolves the puzzle of why black hole interiors keep growing for a very long time. In this pedagogical review, we present the geometric approach to complexity advocated by Nielsen and show how it can be used to define complexity for generic quantum systems; in particular, we focus on Gaussian states in QFT, both pure and mixed, and on certain classes of CFT states. We then present the conjectured relation to gravitational quantities within the holographic correspondence and discuss several examples in which different versions of the conjectures have been tested. We highlight the relation between complexity, chaos and scrambling in chaotic systems. We conclude with a discussion of open problems and future directions. This article was written for the special issue of EPJ-C Frontiers in Holographic Duality.

Quantum Information Scrambling: From Holography to Quantum Simulators

“…This complexity basically measures the growth of the size of the operator under time evolution. Some recent progress has been made towards understanding complexity for the dual field theory [229][230][231][232][233][234][235][236][237][238][239][240][241][242], see [243] for a recent review. However, it is in its early stage of development.…”

Path Integral Complexity and Kasner singularities

“…and we consider time-like and space-like Q's separately. 4 In the notation used in [18] for spacelike Q the boost parameter was written as η0. See Fig.…”

“…These conjectures have by now been studied intensely in various static and dynamic geometries (see e.g. review [4]). On the field theory side, various notions of complexity have been explored and their relations to each other and to holographic notions have been discussed (see e.g.…”

Path integral complexity and Kasner singularities

“…These conjectures have by now been studied intensely in various static and dynamic geometries (see e.g. review [5]). On the field theory side, various notions of complexity have been explored and their relations to each other and to holographic notions have been discussed (see e.g.…”