The Hessian in Spin Foam Models

We fill one of the remaining gaps in the asymptotic analysis of the vertex amplitudes of the Engle–Pereira–Rovelli–Livine (EPRL) spin foam models: We show that the Hessian is nondegenerate for the stationary points that corresponds to geometric nondegenerate 4 simplices. Our analysis covers the case when all faces are spacelike.

Spin Foam Vertex Amplitudes on Quantum Computer—Preliminary Results

Vertex amplitudes are elementary contributions to the transition amplitudes in the spin foam models of quantum gravity. The purpose of this article is to make the first step towards computing vertex amplitudes with the use of quantum algorithms. In our studies we are focused on a vertex amplitude of 3+1 D gravity, associated with a pentagram spin network. Furthermore, all spin labels of the spin network are assumed to be equal j = 1 / 2 , which is crucial for the introduction of the intertwiner qubits. A procedure of determining modulus squares of vertex amplitudes on universal quantum computers is proposed. Utility of the approach is tested with the use of: IBM’s ibmqx4 5-qubit quantum computer, simulator of quantum computer provided by the same company and QX quantum computer simulator. Finally, values of the vertex probability are determined employing both the QX and the IBM simulators with 20-qubit quantum register and compared with analytical predictions.