  A central challenge in the verification of quantum computers is benchmarking
their performance as a whole and demonstrating their computational
capabilities. In this work, we find a universal model of quantum computation,
Bell sampling, that can be used for both of those tasks and thus provides an
ideal stepping stone towards fault-tolerance. In Bell sampling, we measure two
copies of a state prepared by a quantum circuit in the transversal Bell basis.
We show that the Bell samples are classically intractable to produce and at the
same time constitute what we call a circuit shadow: from the Bell samples we
can efficiently extract information about the quantum circuit preparing the
state, as well as diagnose circuit errors. In addition to known properties that
can be efficiently extracted from Bell samples, we give several new and
efficient protocols: an estimator of state fidelity, a test for the depth of
the circuit and an algorithm to estimate a lower bound to the number of T gates
in the circuit. With some additional measurements, our algorithm learns a full
description of states prepared by circuits with low T-count.
