Abstract: The aim of this talk is to outline the results obtained by me (during my PhD studies) and my collaborators.

In quantum information theory, there is a rich collection of analytic tools to study tensor product of Hilbert spaces. The principal objective of the first set of study, supervised by T Barron, is to investigate how the geometry of the space influences the quantum information theoretic aspects of the Hilbert space and vice versa. For instance, in one of the works, we presented an asymptotic result for the average entropy over all the pure states on the Hilbert space H^0(M_1, L_1^⊗N) ⊗ H^0(M_2,L_2^⊗N), where L_j is a Hermitian ample line bundle over a compact complex manifold M_j. In another work, we associated states with subsets of a product of two integral Kahler manifolds using a quantum line bundle in a particular manner. We proved that the state associated this way is separable when the subset is a finite union of products.

In the second part of the talk, we present a work on quantum circuit synthesis, joint with A.R. Kalra, D. Valluri, S. Winnick and J. Yard. In classical computing, we choose a small set of special gates (known as a universal gate set) and make circuits using these gates to generate any classical gates (Boolean functions). However, the quantum version of circuit synthesis is a bit more complicated. In this talk, we will introduce what it means to be a universal gate set in quantum computing and see some examples. Finally, we shall present our exact synthesis algorithm of unitaries in the groups U_{3^n} (Z [1/3, e^2πi/3]) and U_{3^n}(Z[1/3, e^2π𝑖/9]) over the multi-qutrit Clifford+T universal gate set with the of help of ancilla.