Quantum Chaos, Random Matrices, and Spread Complexity of Time Evolution

I will describe a measure of quantum state complexity defined by minimizing the spread of the wavefunction over all choices of basis. We can efficiently compute this measure, which displays universal behavior for diverse chaotic systems including spin chains, the SYK model, and quantum billiards. In the minimizing basis, the Hamiltonian is tridiagonal, thus representing the dynamics as if they unfold on a one-dimensional chain. The recurrent and hopping matrix elements of this chain comprise the Lanczos coefficients, which I will relate through an integral formula to the density of states. For Random Matrix Theories (RMTs), which are believed to describe the energy level statistics of chaotic systems, I will also derive an integral formula for the covariances of the Lanczos coefficients. I will apply this formalism to the Double Scaled SYK (DSSYK) model, where recent work has proposed a duality at early times between spread complexity and wormhole length in JT gravity. I will argue that in this context the saturation of spread complexity at late times, which we can compute explicitly, implies that the classical description of the bulk theory fails because the wavefunction has become delocalized in the configuration space of gravity.