9 research outputs found
Constructive Quantum Shannon Decomposition from Cartan Involutions
The work presented here extends upon the best known universal quantum
circuit, the Quantum Shannon Decomposition proposed in [Vivek V. Shende,
Stephen S. Bullock and Igor Markov, Synthesis of Quantum Logic Circuits, IEEE
Trans. Comput.-Aided Des. Integr. Circuits Syst. 25 (6): 1000-1010 (2006)]. We
obtain the basis of the circuit's design in a pair of Cartan decompositions.
This insight gives a simple constructive algorithm for obtaining the Quantum
Shannon Decomposition of a given unitary matrix in terms of the corresponding
Cartan involutions
Lie algebra decompositions with applications to quantum dynamics
Lie group decompositions are useful tools in the analysis and control of quantum systems. Several decompositions proposed in the literature are based on a recursive procedure that systematically uses the Cartan decomposition theorem. In this dissertation, we establish a link between Lie algebra gradings and recursive Lie algebra decompositions, and then we formulate a general scheme to generate Lie group decompositions. This scheme contains some procedures previously proposed as special cases and gives a virtually unbounded number of alternatives to factor elements of a Lie group