  The use of quantum processing units (QPUs) promises speed-ups for solving
computational problems, but the quantum devices currently available possess
only a very limited number of qubits and suffer from considerable
imperfections. One possibility to progress towards practical utility is to use
a co-design approach: Problem formulation and algorithm, but also the physical
QPU properties are tailored to the specific application. Since QPUs will likely
be used as accelerators for classical computers, details of systemic
integration into existing architectures are another lever to influence and
improve the practical utility of QPUs.
  In this work, we investigate the influence of different parameters on the
runtime of quantum programs on tailored hybrid CPU-QPU-systems. We study the
influence of communication times between CPU and QPU, how adapting QPU designs
influences quantum and overall execution performance, and how these factors
interact. Using a simple model that allows for estimating which design choices
should be subjected to optimisation for a given task, we provide an intuition
to the HPC community on potentials and limitations of co-design approaches. We
also discuss physical limitations for implementing the proposed changes on real
quantum hardware devices.
