Optimization of One-Way Quantum Computation Measurement Patterns

In one-way quantum computation (1WQC), 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. 1WQC computation can be represented by a measurement pattern (or simply a pattern). The entanglement operations in a pattern can be shown by a graph which together with the identified set of its input and output qubits is called the geometry of the pattern. Since a pattern is based on quantum measurements, which are fundamentally nondeterministic evolutions, there must be conditions over geometries to guarantee determinism. These conditions are formalized by the notions of flow and generalized flow (gflow). Previously, three optimization methods have been proposed to optimize 1WQC patterns which can be performed using the measurement calculus formalism by rewriting rules. However, the serial implementation 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 the optimization techniques simultaneously on patterns with flow and only gflow based on their geometries. 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.