Unitary Transformations

Julian Schwinger

The uncontrollable disturbance attendant upon a measurement implies that the act of measurement is indivisible. Any attempt to trace the history of a system during a measurement process usually changes the nature of the measurement that is being performed. Hence, to conceive of a given selective measurement M(a', b') as a compound measurement is without physical implication. All algebraic relations and adjoint connections among vectors and operators are preserved by unitary transformation. Two successive unitary transformations form a unitary transformation, and the inverse of a unitary transformation is unitary-unitary transformations form a group. In discussing successive unitary transformations, it must be recognized that a transformation which is specified by an array of numerical coefficients is symbolized by a unitary operator that depends upon the coordinate system to which it is applied. The continual repetition of an infinitesimal unitary transformation generates a finite unitary transformation.

David C. Collins, Alexander I. Lvovsky

Quantum algorithms are typically understood in terms of the evolution of a multi-qubit quantum system under a prescribed sequence of unitary transformations. The input to the algorithm prescribes some of the unitary transformations in the sequence with others remaining fixed. For oracle query algorithms, the input determines the oracle unitary transformation. Such algorithms can be regarded as devices for discriminating amongst a set of unitary transformations. The question arises: "Given a set of known oracle unitary transformations, to what extent is it possible to discriminate amongst them?" We investigate this for the Deutsch-Jozsa problem. The task of discriminating amongst the admissible oracle unitary transformations results in an exhaustive collection of algorithms which can solve the problem with certainty.

Zhu Ai-dong

The mesoscopic RLC circuit with a power source is quantized using the methods of canonical transformation and unitary transformation. By analyzing and comparing the calculation processes of several different unitary operators, we obtain the most appropriate scheme to build an unitary operator which enables the results of canonical transformation and unitary transformation to be same with each other.