TY - JOUR
T1 - Quantum Computational Complexity from Quantum Information to Black Holes and Back
AU - Chapman, Shira
AU - Policastro, Giuseppe
N1 - Funding Information:
We gratefully acknowledge discussions with Adam Chapman, Juan Hernandez and Shan-Ming Ruan. We are also grateful to our many collaborators in papers on the subject of complexity in the past few years. The work of SC is supported by the Israel Science Foundation (grant No. 1417/21) and by the German Research Foundation through a German-Israeli Project Cooperation (DIP) grant “Holography and the Swampland”. SC acknowledges the support of Carole and Marcus Weinstein through the BGU Presidential Faculty Recruitment Fund.
Publisher Copyright:
© 2022, The Author(s).
PY - 2022/2/10
Y1 - 2022/2/10
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85124835111&partnerID=8YFLogxK
U2 - 10.1140/epjc/s10052-022-10037-1
DO - 10.1140/epjc/s10052-022-10037-1
M3 - Article
AN - SCOPUS:85124835111
SN - 1434-6044
VL - 82
JO - European Physical Journal C
JF - European Physical Journal C
IS - 2
M1 - 128
ER -