Abstract
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.
Highlights
The basic objective of theories of quantum gravity is to calculate transition amplitudes between configurations of the gravitational field
The purpose of this article was to explore the possibility of computing vertex amplitudes in the spin foam models of quantum gravity with the use of quantum algorithms
The notion of intertwiner qubit being crucial to implement the vertex amplitudes on quantum computers has been pedagogically introduced. It has been shown how one of the two basis states of the intertwiner qubit can be implemented with the use available IBM 5-qubit quantum computer
Summary
The basic objective of theories of quantum gravity is to calculate transition amplitudes between configurations of the gravitational field. While the formula (1) is easy to write, it is not very practical for the case of continuous gravitational field, characterized by infinite number of degrees of freedom. One of the approaches to determine (1) utilizes discretization of the gravitational field associated with some cut-off scale. The expectation is that continuous limit of such discretized theory can be recovered at the second order phase transition [1,2]. The essential step in this challenge is to generate different discrete space-time configurations (triangulations) contributing to the path integral (1). An advantage of the method is that in the large N (color) limit of such theories, symmetry factors associated with given triangulations can be recovered [5]
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