  We introduce an approximate description of an $N$-qubit state, which contains
sufficient information to estimate the expectation value of any observable to a
precision that is upper bounded by the ratio of a suitably-defined seminorm of
the observable to the square root of the number of the system's identical
preparations $M$, with no explicit dependence on $N$. We describe an
operational procedure for constructing the approximate description of the state
that requires, besides the quantum state preparation, only single-qubit
rotations followed by single-qubit measurements. We show that following this
procedure, the cardinality of the resulting description of the state grows as
$3MN$. We test the proposed method on Rigetti's quantum processor unit with 12,
16 and 25 qubits for random states and random observables, and find an
excellent agreement with the theory, despite experimental errors.
