Yang–Baxterizations, Universal Quantum Gates and Hamiltonians

Abstract

The unitary braiding operators describing topological entanglements can be viewed as universal quantum gates for quantum computation. With the help of the Brylinski's theorem, the unitary solutions of the quantum Yang-Baxter equation can be also related to universal quantum gates. This paper derives the unitary solutions of the quantum Yang-Baxter equation via Yang-Baxterization from the solutions of the braid relation. We study Yang-Baxterizations of the non-standard and standard representations of the six-vertex model and the complete solutions of the non-vanishing eight-vertex model. We construct Hamiltonians responsible for the time-evolution of the unitary braiding operators which lead to the Schrödinger equations.

Integrable quantum computation

“…In physics, hence, we define integrable quantum computation as quantum computing via the integrable condition so that we can obtain a unified description on both quantum computing via the Bethe ansatz [3] and quantum computing via the Yang-Baxter equation [7]. Note that we do not deal with quantum computing via the quantum Lax equation in this paper.…”

“…Section 4 is a brief sketch on quantum computing via the Yang-Baxter equation [7]. Given a nontrivial unitary solution of the Yang-Baxter equation, the Hamiltonian of an integrable system can be constructed in principle.…”

“…Given a nontrivial unitary solution of the Yang-Baxter equation, the Hamiltonian of an integrable system can be constructed in principle. Additionally, the paper [7] is written in the way to emphasize how to derive unitary solutions of the Yang-Baxter equation from unitary representations of the braid group. In view of the research [3], however, it has been distinct that integrable quantum computation is a research subject independent of quantum computing via unitary braid representations.…”

“…In view of the research [3], however, it has been distinct that integrable quantum computation is a research subject independent of quantum computing via unitary braid representations. Hence this section is a refinement of the research [7] from the viewpoint of integrable quantum computation.…”

“…On the other hand, it is easy to examine the delta-function interaction (1) as a suitable Hamiltonian underlying the two-qubit gate (14), but it is not so explicit to realize the delta-function interaction (1) from the formalism of the two-qubit gate (14). In [7], an approach for the construction of the Hamiltonian is presented via a two-qubit solutionŘ(x) of the Yang-Baxter equation (19). With an initial state ψ, the evolution state ψ(x) determined byŘ(x) is given by ψ(x) = ρ − 1 2Ř (x)ψ.…”