Researches On Generalized Quantum Gates And Quantum Operations

In this paper,some questions about generalized quantum gates and quantum opera-tions are discussed by using the theory of operator and operator algebra. Some properties of restricted allowable generalized quantum gates are obtained. Some conclusions of com-plete positive maps and the matrix representations of the affine map are given.In addition, two decompositions of a Toffoli gate are given.At last,generalized controlled-NOT gates are introduced, a relationship between generalized controlled-NOT gates and generalized commutative gates is established. The main contents are as follows.In Chapter 1,some concepts of duality quantum computer are recalled, a relationship between divider operator and combiner operator in the duality mode are given.In Chapter 2, allowable generalized quantum gates and its realization condition are discussed, some properties of restricted allowable generalized quantum gates are obtained.In Chapter 3,at first,some related concepts of complete positive map are recalled,the necessary and sufficient condition of positive trace-preserving maps and complete positive trace-preserving maps are given. The linear combination of the quantum channels is also a quantum channel.At last,the matrix representations of the affine map are given.In Chapter 4, generalized controlled-NOT gates are introduced, then a relationship between generalized controlled-NOT gates and generalized commutative gates is estab-lished.The main results of this thesis are as follows.(1)Some properties of the divider operators and combiner operators are given, a relationship between divider operators and combiner operators in the duality mode are also given;(2)A necessary and sufficient condition for an allowable generalized quantum gate to be realizable is obtained;(3) Some properties of restricted allowable generalized quantum gates are given and it is proved that the set of its extreme points is just the set of unitary operators; (4) Necessary and sufficient conditions for a linear map on the matrix algebra to be positive trace-preserving and complete positive trace-preserving maps are given, respec-tively;(5) Two decompositions of a Toffoli gate are given;(6)Generalized controlled-NOT gates are introduced,a relationship between gener-alized controlled-NOT gates and generalized commutative gates is established.