  Measurement-based quantum computing (MBQC) is a promising approach to
reducing circuit depth in noisy intermediate-scale quantum algorithms such as
the Variational Quantum Eigensolver (VQE). Unlike gate-based computing, MBQC
employs local measurements on a preprepared resource state, offering a
trade-off between circuit depth and qubit count. Ensuring determinism is
crucial to MBQC, particularly in the VQE context, as a lack of flow in
measurement patterns leads to evaluating the cost function at irrelevant
locations. This study introduces MBVQE-ans\"atze that respect determinism and
resemble the widely used problem-agnostic hardware-efficient VQE ansatz. We
evaluate our approach using ideal simulations on the Schwinger Hamiltonian and
$XY$-model and perform experiments on IBM hardware with an adaptive measurement
capability. In our use case, we find that ensuring determinism works better via
postselection than by adaptive measurements at the expense of increased
sampling cost. Additionally, we propose an efficient MBQC-inspired method to
prepare the resource state, specifically the cluster state, on hardware with
heavy-hex connectivity, requiring a single measurement round, and implement
this scheme on quantum computers with $27$ and $127$ qubits. We observe notable
improvements for larger cluster states, although direct gate-based
implementation achieves higher fidelity for smaller instances.
