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

Gareth Jay

Published Aug 23, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.26182/mrmr-0v98

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•Jul 31, 2015•0 citations
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.
Nowadays, quantum simulation schemes come in two flavours. Either they are continuous-time discrete-space models (a.k.a Hamiltonian-based), pertaining to non-relativistic quantum mechanics. Or they are discrete-spacetime models (a.k.a Quantum Walks or Quantum Cellular Automata-based) enjoying a relativistic continuous spacetime limit. We provide a first example of a quantum simulation scheme that unifies both approaches. The proposed scheme supports both a continuous-time discrete-space limit, leading to lattice fermions, and a continuous-spacetime limit, leading to the Dirac equation. The transition between the two can be thought of as a general relativistic change of coordinates, pushed to an extreme. As an emergent by-product of this procedure, we obtain a Hamiltonian for lattice-fermions in curved spacetime with synchronous coordinates.
A discrete-time Quantum Walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation), and thus provide us with discrete toy models of familiar particles (e.g. the electron). In this paper, we study the continuum limit of a wide class of QWs, and show that it leads to all those PDEs corresponding to the Hamiltonian form of the massive curved Dirac equation in (1 + 1) dimensions. Therefore a certain QW, which we make explicit, provides us with a unitary discrete toy model of the electron as a test particle in curved spacetime, in spite of the fixed background lattice. Mathematically we have introduced two novel ingredients for taking the continuum limit of a QW, but which apply to any quantum cellular automata: encoding and grouping.
A discrete-time Quantum Walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). In this paper, we study the continuum limit of a wide class of QWs, and show that it leads to an entire class of PDEs, encompassing the Hamiltonian form of the massive Dirac equation in (1+1)(1+1)(1+1) curved spacetime. Therefore a certain QW, which we make explicit, provides us with a unitary discrete toy model of a test particle in curved spacetime, in spite of the fixed background lattice. Mathematically we have introduced two novel ingredients for taking the continuum limit of a QW, but which apply to any quantum cellular automata: encoding and grouping.
Quantum simulation is an important way to study the Dirac particles in a general situation. Discrete quantum walk (DQW), is a powerful quantum simulation scheme, and implementable in well controllable table-top set-ups. We first identify that the conventional DQW can't exactly simulate Dirac Cellular Automaton (DCA), a discretized theory of free Dirac Hamiltonian (DH). We found some choice of coin parameters of the split-step (SS) DQW, a generalization of DQW can fully simulate single-particle DCA. Next we question whether the same SS-DQW can simulate dynamics of free Dirac particle with extra degrees of freedom like colors, flavors besides the spin or chirality. One such example is Neutrino oscillation. By moving from the U(2) coined SS-DQW to the U(6) coined SS-DQW we have simulated the exact probability profile of Neutrino flavor transitions. We further probe towards simulating single particle massive DH in presence of background potentials and space-time curvature. By using a SS-DQW with position-time dependent coin parameters, and we realize that it will give us an unbounded effective Hamiltonian, at the continuum limit of position-time. So we have introduced a modified version of SS-DQW which will produce a bounded effective Hamiltonian. This modified SS-DQW with U(2) coin operations produces single-particle massive DH in presence of abelian gauge potentials and space-time curvature. Introducing higher dimensional---U(N) coin operations in the modified SS-DQW we can include non-abelian potentials in the same DH. In order to simulate two-particle DH in presence of curved space-time and external potentials, we have used two particle modified SS-DQW, where the shift operations act separately on each particle, the coin operations which act simultaneously on both particles contain all kinds of interactions.
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 3D3D3D 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 3D3D3D 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 particular example is produced to prove that quantum walks can be used to simulate full-fledged discrete gauge theories. A new family of 2D2D2D walks is introduced and its continuous limit is shown to coincide with the dynamics of a Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic interpretation is extended beyond the continuous limit by proving that these DTQWs exhibit an exact discrete local U(1)U(1)U(1) gauge invariance and possess a discrete gauge-invariant conserved current. A discrete gauge-invariant electromagnetic field is also constructed and that field is coupled to the conserved current by a discrete generalization of Maxwell equations. The dynamics of the DTQWs under crossed electric and magnetic fields is finally explored outside the continuous limit by numerical simulations. Bloch oscillations and the so-called E×B{\bf E} \times {\bf B}E×B drift are recovered in the weak-field limit. Localization is observed for some values of the gauge fields.