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Constructive Quantum Shannon Decomposition from Cartan Involutions
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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.
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On the KAK Decomposition and Equivalence Classes
For SU(4), local equivalence classes under SU(2)⊗SU(2) multiplication are not geometrically represented by the Weyl chamber; that chamber appears only under projective-local equivalence that ignores global phases.
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