Quantum computational path summation for relativistic quantum mechanics and a time dilation relation for a Dirac Hamiltonian generator on a qubit array

TL;DR

This paper presents a quantum algorithm encoding Dirac particle dynamics on a qubit array, maintaining Lorentz invariance at the Planck scale, and explores gravitational time dilation effects in this quantum simulation framework.

Contribution

It introduces a unitary path summation rule for simulating Dirac particles on a qubit array that preserves Lorentz invariance and models gravitational time dilation at the quantum level.

Findings

The model is Lorentz invariant down to the Planck scale.

It avoids the Fermi-sign problem in lattice simulations.

Time dilation effects analogous to black hole spacetime are demonstrated.

Abstract

Dirac particle dynamics is encoded as a unitary path summation rule and implemented on a qubit array, where the qubit array represents both spacetime and the fermions contained therein. The unitary path summation rule gives a quantum algorithm to model a many-body system of Dirac particles in a gauge field with Lorentz invariance down to the grid scale (Planck scale)--the lattice-based model neither suffers the Fermi-sign problem nor breaks Lorentz invariance. Yet, for the Dirac Hamiltonian to generate the unitary evolution of the 4-spinor field at the Planck scale, there is time dilation between the shortest observable time near a single space point and that time measured at long-wavelength scales. We find gravitational time dilation where the model space around each point (with an even number of qubits) is curved like the space around a Schwarzschild black hole.