Algorithm for the solution of the Dirac equation on digital quantum computers

TL;DR

This paper presents a quantum algorithm for solving the Dirac equation, leveraging operator splitting and quantum walks, achieving exponential speedup over classical methods under certain conditions.

Contribution

It introduces a novel quantum algorithm for the Dirac equation with explicit gate decomposition and analyzes its potential for implementation on current quantum hardware.

Findings

Exponential speedup over classical schemes under specific conditions

Explicit gate decomposition for the quantum algorithm

Feasibility of proof-of-principle calculations with current quantum tech

Abstract

A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator splitting decomposition technique that allows for a mapping of the Dirac operator to a quantum walk supplemented by unitary rotation steps in spinor space. Every step of the splitting method is decomposed into sets of quantum gates. It is demonstrated that the algorithm has an exponential speedup over the implementation of the same numerical scheme on a classical computer, as long as certain conditions are satisfied. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out to determine the resource requirements. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.