A quantum walk whose continuous limit coincides with Dirac equation is usually called a Dirac Quantum Walk (DQW). A new systematic method to build DQWs coupled to electromagnetic (EM) fields is introduced and put to test on several examples of increasing difficulty. It is first used to derive the EM coupling of a well-known $3D$ walk on the cubic lattice. Recently introduced DQWs on the triangular and honeycomb lattice are then re-derived, showing for the first time that these are the only DQWs that can be defined with spinors living on the vertices of these lattices. As a third example of the method's effectiveness, a new $3D$ walk on a parallelepiped lattice is derived. As a fourth, negative example, it is shown that certain lattices like the rhombohedral lattice cannot be used to build DQWs. The effect of changing representation in the Dirac equation is also discussed.A quantum walk whose continuous limit coincides with Dirac equation is usually called a Dirac Quantum Walk (DQW). A new systematic method to build DQWs coupled to electromagnetic (EM) fields is introduced and put to test on several examples of increasing difficulty. It is first used to derive the EM coupling of a well-known $3D$ walk on the cubic lattice. Recently introduced DQWs on the triangular and honeycomb lattice are then re-derived, showing for the first time that these are the only DQWs that can be defined with spinors living on the vertices of these lattices. As a third example of the method's effectiveness, a new $3D$ walk on a parallelepiped lattice is derived. As a fourth, negative example, it is shown that certain lattices like the rhombohedral lattice cannot be used to build DQWs. The effect of changing representation in the Dirac equation is also discussed.

arXiv:1812.06729Dec 17, 20181 citation

APPROACHES TO QUANTUM THEORIES FOLLOWING DIRAC

WORLD SCIENTIFIC eBooksJun 1, 19950 citations

Quantum simulation of the Dirac equation

R. Gerritsma, Gerhard Kirchmair, F. Zähringer +3 more

arXiv:0909.0674NatureJan 1, 2010725 citations

Dirac quantum walks with conserved angular momentum

Gareth Jay, Pablo Arnault, Fabrice Debbasch

A Quantum Walk (QW) simulating the flat $(1 + 2)$ D Dirac Eq.\ on a spatial polar grid is constructed. Because fermions are represented by spinors, which do not constitute a representation of the rotation group, but rather of its double cover, the QW can only be defined globally on an extended spacetime where the polar angle extends from $0$ to $4 π$ . The coupling of the QW with arbitrary electromagnetic fields is also presented. Finally, the cylindrical relativistic Landau levels of the Dirac Eq.\ are computed explicitly and simulated by the QW.A Quantum Walk (QW) simulating the flat $(1 + 2)$D Dirac Eq.\ on a spatial polar grid is constructed. Because fermions are represented by spinors, which do not constitute a representation of the rotation group, but rather of its double cover, the QW can only be defined globally on an extended spacetime where the polar angle extends from $0$ to $4 π$. The coupling of the QW with arbitrary electromagnetic fields is also presented. Finally, the cylindrical relativistic Landau levels of the Dirac Eq.\ are computed explicitly and simulated by the QW.

arXiv:2007.06211Jul 13, 20200 citations

Discrete Dirac Dynamics: a quantum walk-based framework in discrete spacetime

Gareth Jay

In this thesis a study of a special type of quantum cellular automata called Dirac quantum walks (DQWs) is presented. At the continuous limit, these DQWs coincide with the Dirac equation. A new systematic method to build DQWs, with particular focus on non-square or cube shaped grids, is constructed and is put to test on several examples of increasing difficulty. Ultimately, a quantum walk simulating the Dirac equation on a spatial polar grid is constructed with a discrete analogue to the conservation of angular momentum. Cylindrical relativistic Landau levels of the Dirac equation are computed explicitly and simulated.In this thesis a study of a special type of quantum cellular automata called Dirac quantum walks (DQWs) is presented. At the continuous limit, these DQWs coincide with the Dirac equation. A new systematic method to build DQWs, with particular focus on non-square or cube shaped grids, is constructed and is put to test on several examples of increasing difficulty. Ultimately, a quantum walk simulating the Dirac equation on a spatial polar grid is constructed with a discrete analogue to the conservation of angular momentum. Cylindrical relativistic Landau levels of the Dirac equation are computed explicitly and simulated.

Aug 23, 20210 citations

Fundamentals of quantum walks

Giuseppe Di Molfetta

Institute of Physics eBooksSep 1, 20240 citations

3D

walk on the cubic lattice. Recently introduced DQWs on the triangular and honeycomb lattice are then re-derived, showing for the first time that these are the only DQWs that can be defined with spinors living on the vertices of these lattices. As a third example of the method's effectiveness, a new

3D

walk on a parallelepiped lattice is derived. As a fourth, negative example, it is shown that certain lattices like the rhombohedral lattice cannot be used to build DQWs. The effect of changing representation in the Dirac equation is also discussed.

A Quantum Walk (QW) simulating the flat

(1 + 2)

D Dirac Eq.\ on a spatial polar grid is constructed. Because fermions are represented by spinors, which do not constitute a representation of the rotation group, but rather of its double cover, the QW can only be defined globally on an extended spacetime where the polar angle extends from

0

4 π

. The coupling of the QW with arbitrary electromagnetic fields is also presented. Finally, the cylindrical relativistic Landau levels of the Dirac Eq.\ are computed explicitly and simulated by the QW.