Abstract

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. These results lead to a conjecture: quantum chaotic systems have Lanczos coefficients whose local means and covariances are described by RMTs. I will apply this formalism to the Double Scaled SYK (DSSYK) model with is dual to JT gravity and explain that: (a) wormhole length in JT gravity is dual to spread complexity in the DSSYK theory, (b) the spread complexity increases linearly at early times matching the classical JT geometry, (c) the spread complexity saturates at late time implying that the classical description of the bulk theory fails because the wavefunction has become delocalized in the configuration space of gravity.