  Increasingly sophisticated programmable quantum simulators and quantum
computers are opening unprecedented opportunities for exploring and exploiting
the properties of highly entangled complex quantum systems. The complexity of
large quantum systems is the source of their power, but also makes them
difficult to control precisely or characterize accurately using measured
classical data. We review recently developed protocols for probing the
properties of complex many-qubit systems using measurement schemes that are
practical using today's quantum platforms. In all these protocols, a quantum
state is repeatedly prepared and measured in a randomly chosen basis; then a
classical computer processes the measurement outcomes to estimate the desired
property. The randomization of the measurement procedure has distinct
advantages; for example, a single data set can be employed multiple times to
pursue a variety of applications, and imperfections in the measurements are
mapped to a simplified noise model that can more easily be mitigated. We
discuss a range of use cases that have already been realized in quantum
devices, including Hamiltonian simulation tasks, probes of quantum chaos,
measurements of nonlocal order parameters, and comparison of quantum states
produced in distantly separated laboratories. By providing a workable method
for translating a complex quantum state into a succinct classical
representation that preserves a rich variety of relevant physical properties,
the randomized measurement toolbox strengthens our ability to grasp and control
the quantum world.
