Quantum computing models for algebraic applications

Alexander Y. Vlasov

Here is discussed application of the Weyl pair to construction of universal set of quantum gates for high-dimensional quantum system. An application of Lie algebras (Hamiltonians) for construction of universal gates is revisited first. It is shown next, how for quantum computation with qubits can be used two-dimensional analog of this Cayley-Weyl matrix algebras, i.e. Clifford algebras, and discussed well known applications to product operator formalism in NMR, Jordan-Wigner construction in fermionic quantum computations. It is introduced universal set of quantum gates for higher dimensional system ("qudit"), as some generalization of these models. Finally it is briefly mentioned possible application of such algebraic methods to design of quantum processors (programmable gates arrays) and dsicussed generalization to quantum computation wiht continuous variables.

Peter P. Rohde

Jaroslav Hrdina, Dietmar Hildenbrand, Oliver Rettig

We present a real geometric algebra framework designed for the direct translation of the Dirac formalism into geometric algebra representations. Unlike previous approaches based on positive-definite signatures, QCA employs a split-signature construction that enables a natural realization of quantum states and operators while simplifying computational implementation. We further present an implementation of QCA using the \textit{GAALOP} software and show how quantum gates and multi-qubit systems can be efficiently represented and generated computationally. As an application, we demonstrate the use of QCA in quantum game theory, where the real-algebraic formulation provides computational advantages for modeling entangled strategies and quantum interactions. The proposed framework establishes a practical bridge between the abstract formalism of quantum computation and efficient geometric algebra implementations.

Mingsheng Ying

Camilo Jaramillo Correa

The idea of this work was to generalize the Donkor model (Quantum Information and Computation IX, edited by Eric Donkor, Andrew R. Pirich, Howard E. Brandt, Proc. of SPIE Vol. 8057) about the application of induced dipole-dipole forbidden transitions to quantum computation. Using computer algebra we were able to reproduce the original Donkor model. Then we applied some modifications to this model and obtained the respective solutions. It is expected that this model has applications for quantum computation.

Tetsuro Nishino