  Pauli-based computation (PBC) is driven by a sequence of adaptively chosen,
non-destructive measurements of Pauli observables. Any quantum circuit written
in terms of the Clifford+$T$ gate set and having $t$ $T$ gates can be compiled
into a PBC on $t$ qubits. Here we propose practical ways of implementing PBC as
adaptive quantum circuits and provide code to do the required classical
side-processing. Our schemes reduce the number of quantum gates to $O(t^2)$
(from a previous $O(t^3 / \log t)$ scaling) and space/time trade-offs are
discussed which lead to a reduction of the depth from $O(t \log t)$ to $O(t)$
within our schemes, at the cost of $t$ additional auxiliary qubits. We compile
examples of random and hidden-shift quantum circuits into adaptive PBC
circuits. We also simulate hybrid quantum computation, where a classical
computer effectively extends the working memory of a small quantum computer by
$k$ virtual qubits, at a cost exponential in $k$. Our results demonstrate the
practical advantage of PBC techniques for circuit compilation and hybrid
computation.
