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

TL;DR

This paper reviews the concept of quantum computational complexity, its geometric formulation, and its applications to understanding black hole interiors and holography, highlighting recent conjectures and their implications.

Contribution

It introduces a geometric approach to quantum complexity, applies it to quantum field theories, and explores its connection to gravitational physics within the holographic framework.

Findings

Complexity relates to black hole interior growth.

Conjectured links between complexity and gravitational quantities.

Complexity growth relates to chaos and scrambling in quantum systems.

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.