  Predicting properties of large-scale quantum systems is crucial for the
development of quantum science and technology. Shadow estimation is an
efficient method for this task based on randomized measurements, where
many-qubit random Clifford circuits are used for estimating global properties
like quantum fidelity. Here we introduce the minimal Clifford measurement (MCM)
to reduce the number of possible random circuits to the minimum, while keeping
the effective post-processing channel in shadow estimation. In particular, we
show that MCM requires $2^n+1$ distinct Clifford circuits, and it can be
realized by Mutually Unbiased Bases (MUB), with $n$ as the total qubit number.
By applying the Z-Tableau formalism, this ensemble of circuits can be
synthesized to the $\mathrm{-S-CZ-H-}$ structure, which can be composed by
$2n-1$ fixed circuit modules, and the total circuit depth is at most $n+1$.
Compared to the original Clifford measurements, our MCM significantly reduces
the circuit complexity and the compilation costs. In addition, we find the
sampling advantage of MCM on estimating off-diagonal operators, and extend this
observation to the biased-MCM scheme to enhance the sampling improvement
further.
