  The paradigm of measurement-based quantum computing (MBQC) starts from a
highly entangled resource state on which unitary operations are executed
through adaptive measurements and corrections ensuring determinism. This is set
in contrast to the more common quantum circuit model, in which unitary
operations are directly implemented through quantum gates prior to final
measurements. In this work, we incorporate concepts from MBQC into the circuit
model to create a hybrid simulation technique, permitting us to split any
quantum circuit into a classically efficiently simulatable Clifford-part and a
second part consisting of a stabilizer state and local (adaptive) measurement
instructions, a so-called standard form, which is executed on a quantum
computer. We further process the stabilizer state with the graph state
formalism, thus enabling a significant decrease in circuit depth for certain
applications. We show that groups of fully commuting operators can be
implemented using fully-parallel, i.e., non-adaptive, measurements within our
protocol. In addition, we discuss how such circuits can be implemented in
constant quantum depths by employing quantum teleportation. Finally, we
demonstrate the utility of our technique on two examples of high practical
relevance: the Quantum Approximate Optimization Algorithm (QAOA) and the
Variational Quantum Eigensolver (VQE).
