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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).