  In dynamic quantum circuits, classical information from mid-circuit
measurements is fed forward during circuit execution. This emerging capability
of quantum computers confers numerous advantages that can enable more efficient
and powerful protocols by drastically reducing the resource requirements for
certain core algorithmic primitives. In particular, in the case of the
$n$-qubit quantum Fourier transform followed immediately by measurement, the
scaling of resource requirements is reduced from $O(n^2)$ two-qubit gates in an
all-to-all connectivity in the standard unitary formulation to $O(n)$
mid-circuit measurements in its dynamic counterpart without any connectivity
constraints. Here, we demonstrate the advantage of dynamic quantum circuits for
the quantum Fourier transform on IBM's superconducting quantum hardware with
certified process fidelities of $>50\%$ on up to $16$ qubits and $>1\%$ on up
to $37$ qubits, exceeding previous reports across all quantum computing
platforms. These results are enabled by our contribution of an efficient method
for certifying the process fidelity, as well as of a dynamical decoupling
protocol for error suppression during mid-circuit measurements and feed-forward
within a dynamic quantum circuit that we call ``feed-forward-compensated
dynamical decoupling" (FC-DD). Our results demonstrate the advantages of
leveraging dynamic circuits in optimizing the compilation of quantum
algorithms.
