Fast scrambling on sparse graphs
The fast scrambling conjecture is that quantum many-body systems with few-body interactions can scramble information among N degrees of freedom in a time that grows no slower than log(N). I will review this conjecture and some simple cartoons which justify it. By studying 2-local Hamiltonians with general connectivity graphs, I will then clarify why arbitrarily fast information scrambling is generally impossible. Using a mix of formal perspectives and simple models of chaotic quantum dynamics including random unitary circuits and generalized Sachdev-Ye-Kitaev models, I will describe both the spread of entanglement between initially unentangled subsystems and the growth of (out-of-time-ordered) correlation functions. On many graphs, I will prove the fast scrambling conjecture for 2-local Hamiltonians and give a simple graph-theoretic diagnostic for determining whether a given graph can host a fast scrambler. I will propose new ways to look for fast scramblers in cold atom experiments.