In this series of lectures, the notion of quantum chaos will be discussed. Specifically we will investigate how it may be defined and how it may be probed / diagnosed in a given physical system. Classical chaos is, by now, a well-understood concept and several features such as the exponential sensitivity to initial conditions and ergodic evolution in phase space can be seen as general features. Due to the Heisenberg uncertainty principle these notions are not well-defined upon quantizing the system, however. Nevertheless, they serve as inspiration for diagnostic tools such as the spectral form factor, out-of-time-ordered correlator and complexity, all of which will be discussed.
We will focus especially on quantum complexity due to a special role that it is conjectured to play in holographic theories. Specifically, it is conjectured that the growth of the black hole interior is dual to an appropriate notion of quantum complexity. Defining the complexity of quantum states is not unique and two different notions will be discussed, namely that of Nielsen complexity and Krylov complexity. Mathematical tools such as generalised coherent states (which have proven useful in the context of quantum complexity computations) will be unpacked.
A handful of select useful and accessible references dealing with these concepts:
A Bhattacharyya, W Chemissany, SS Haque, B Yan, ``Towards the Web of Quantum Chaos Diagnostics”, [arXiv:1909.01894]
R de Mello Koch, JH Huang, CT Ma, HJR van Zyl, ``Spectral Form Factor as an OTOC averaged over the Heisenberg Group”, [arXiv:1905.10981]
D Stanford, ``Many-body chaos at weak coupling”, [arXiv:1512.07687]
P Saad, SH Shenker, D Stanford, ``A semiclassical ramp in SYK and gravity”, [1806.06840]
N Chagnet, S Chapman, J de Boer, C Zukowski, ``Complexity for Conformal Field Theories in General Dimensions”, [arXiv:2103.06920]
S Chapman, G Policastro, ``Quantum Computational Complexity: From Quantum Information to Black Holes and Back”, [2110.14672]
P Caputa, JM Magan, D Patramanis, ``Geometry of Krylov Complexity”, [arXiv:2109.03824]
V Balasubramanian, P Caputa, JM Magan, Q Wu, ``Quantum chaos and the complexity of spread of states”, [arXiv:2202.06957]
AM Perelomov, ``Coherent states for arbitrary Lie group”, [arXiv:math-ph/0203002]