I am proud to announce a new work in collaboration with Amos Chan and my PhD student Alexios Christopoulos. We consider the evolution of a wave function starting from a simple product state under a generic quantum circuit. We then analyse the coefficients in the expansion of the wavefunction on the computational basis of q-bits. It is known that for completely random states this distribution has a simple form named Porter-Thomas, which in the thermodynamic limit is simply obtained assuming that the coefficients are complex gaussian numbers. The question we ask is how this distribution is approached. We develop a generic formalism based on replicas and matching of Feynman histories, so that the problem can be rephrased in general terms as the partition function of a 1D gas of domain-walls in the permutation space. This approach highlights a novel kind of universality emerging in these quantities, alike what we had observed in the spectral form factors few years ago. So, we introduce a scaling limit where the shape of this distribution at large t (~ circuit depth) and L (~ system size) becomes completely universal. Only when t>>L, our family of distributions reduces to the known Porter-Thomas. I find this result particularly striking as it is

- largely universal
- ineherently quantum
- a manifestation of the many and extended degrees of freedom (so it would not be true for single-particle quantum chaos)
- experimentally verifiable, since modern platform allow simulating circuits and looking exactly at the distribution of overlaps