  In one-way quantum computation (1WQC) model, an initial highly entangled
state called a graph state is used to perform universal quantum computations by
a sequence of adaptive single-qubit measurements and post-measurement Pauli-X
and Pauli-Z corrections. The needed computations are organized as measurement
patterns, or simply patterns, in the 1WQC model. The entanglement operations in
a pattern can be shown by a graph which together with the set of its input and
output qubits is called the geometry of the pattern. Since a one-way quantum
computation pattern is based on quantum measurements, which are fundamentally
nondeterministic evolutions, there must be conditions over geometries to
guarantee determinism. Causal flow is a sufficient and generalized flow (gflow)
is a necessary and sufficient condition over geometries to identify a
dependency structure for the measurement sequences in order to achieve
determinism. Previously, three optimization methods have been proposed to
simplify 1WQC patterns which are called standardization, signal shifting and
Pauli simplification. These optimizations can be performed using measurement
calculus formalism by rewriting rules. However, maintaining and searching these
rules in the library can be complicated with respect to implementation.
Moreover, serial execution of these rules is time consuming due to executing
many ineffective commutation rules. To overcome this problem, in this paper, a
new scheme is proposed to perform optimization techniques on patterns with flow
or gflow only based on their geometries instead of using rewriting rules.
Furthermore, the proposed scheme obtains the maximally delayed gflow order for
geometries with flow. It is shown that the time complexity of the proposed
approach is improved over the previous ones.
