  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.
