Quantum variational solving of the Wheeler-DeWitt equation

TL;DR

This paper introduces a variational quantum computing method to solve the Wheeler-DeWitt equation for complex gravitational systems, demonstrated on a flat de Sitter universe using quantum simulators and real quantum hardware.

Contribution

It presents a novel quantum variational approach to solving the Wheeler-DeWitt equation applicable to complex gravitational configurations.

Findings

Method successfully applied to flat de Sitter universe

Implemented on both quantum emulator and IBM quantum computer

Discussed advantages and limitations of the quantum approach

Abstract

One of the central difficulties in the quantization of the gravitational interactions is that they are described by a set of constraints. The standard strategy for dealing with the problem is the Dirac quantization procedure, which leads to the Wheeler-DeWitt equation. However, solutions to the equation are known only for specific symmetry-reduced systems, including models of quantum cosmology. Novel methods, which enable solving the equation for complex gravitational configurations are, therefore, worth seeking. Here, we propose and investigate a new method of solving the Wheeler-DeWitt equation, which employs a variational quantum computing approach, and is possible to implement on quantum computers. For this purpose, the gravitational system is regularized, by performing spherical compactification of the phase space. This makes the system's Hilbert space finite-dimensional and allows to use $SU(2)$ variables, which are easy to handle in quantum computing. The validity of the method is examined in the case of the flat de Sitter universe. For the purpose of testing the method, both an emulator of a quantum computer and the IBM superconducting quantum computer have been used. The advantages and limitations of the approach are discussed.