Probing scrambling and operator size distributions using random mixed states and local measurements

The dynamical spreading of quantum information through a many-body system, typically called scrambling, is a complex process that has proven to be essential to describe many properties of out-of-equilibrium quantum systems. Scrambling can, in principle, be fully characterized via the use of out-of-time-ordered correlation functions, which are notoriously hard to access experimentally. In this work, we put forward an alternative toolbox of measurement protocols to experimentally probe scrambling by accessing properties of the operator size probability distribution, which tracks the size of the support of observables in a many-body system over time. Our measurement protocols require the preparation of separable mixed states together with local operations and measurements, and combine the tools of randomized operations, a modern development of near-term quantum algorithms, with the use of mixed states, a standard tool in NMR experiments. We demonstrate how to efficiently probe the probability-generating function of the operator distribution and discuss the challenges associated with obtaining the moments of the operator distribution. We further show that manipulating the initial state of the protocol allows us to directly obtain the individual elements of the distribution for small system sizes.