  Reference arXiv:1904.03196 recently introduced an algorithm (QPS) for
simulating parton showers with intermediate flavor states using polynomial
resources on a digital quantum computer. We make use of a new quantum hardware
capability called dynamical quantum computing to improve the scaling of this
algorithm to significantly improve the method precision. In particular, we
modify the quantum parton shower circuit to incorporate mid-circuit qubit
measurements, resets, and quantum operations conditioned on classical
information. This reduces the computational depth from
$\mathcal{O}(N^5\log_2(N)^2)$ to $\mathcal{O}(N^3\log_2(N)^2)$ and the qubit
requirements are reduced from $\mathcal{O}(N\log_2(N))$ to $\mathcal{O}(N)$.
Using "matrix product state" statevector simulators, we demonstrate that the
improved algorithm yields expected results for 2, 3, 4, and 5-steps of the
algorithm. We compare absolute costs with the original QPS algorithm, and show
that dynamical quantum computing can significantly reduce costs in the class of
digital quantum algorithms representing quantum walks (which includes the QPS).
