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Due to the rapid development of quantum computing, the compact representationof quantum operations based on decision diagrams has been received more andmore attraction. Since variable orders have a significant impact on the size ofthe decision diagram, identifying a good variable order is of paramountimportance. In this paper, we integrate linear transformations into anefficient and canonical form of quantum computing: Quantum Multiple-ValuedDecision Diagrams (QMDDs) and develop a novel canonical representation, namelylinearly transformed QMDDs (LTQMDDs). We design a linear sifting algorithm forLTQMDDs that search a good linear transformation to obtain a more compact formof quantum function. Experimental results show that the linear siftingalgorithm is able to generate decision diagrams that are significantly improvedcompared with the original sifting algorithm. Moreover, for certain types ofcircuits, linear sifting algorithm have good performance whereas siftingalgorithm does not decrease the size of QMDDs.

Quantum Multiple-Valued Decision Diagrams with Linear Transformations