Fault Tolerant Quantum Computing Diagrammatic Approaches

Abstract

We carry out basic investigations into the study of fault-tolerant quantum computing in d-dimensions, for d an odd-prime. We focused particularly on magic state distillation and gate synthesis from a set of fault tolerant gates, and used ideas inspired by diagrammatic techniques. newlineWe identified qutrit states as some natural candidates for the endpoint of magic states distillation routines. We found the explicit form of non-stabilizer Clifford eigenstates of a single qutrit (d=3) and a single ququint (d=5). For a qutrit, there are four Non-degenerate eigenstates and two families of degenerate states. For a ququint, there are eight Non-degenerate eigenstates and three degenerate families. We derived these states through the use of conjugacy classes. We provided a diagrammatic representation of the symmetries of these states, that illustrates many of their otherwise difficult-to-visualize properties. We also observed some important simultaneous eigenvector called the qutrit strange state, which is distinguished into most magic qutrit state and most symmetric qutrit state. We proved that no similar state exists with properties of the qutrit strange state for prime dimension dgt3. newlineWe also obtained a normal form for single qutrit Clifford+T operators which generalizes the Matsumoto and Amano s normal form for qubits. We proved the optimality of this form with respect to T-count, and noted that any sequence of gates can be rewritten in this form in polynomial time. We described the ideas through provide a Unique expression for any random Clifford+T gate. We conjectured a generalization of this form to arbitrary prime dimensions. newline newline