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arXiv:1708.00826 · Physical Review A

Isotropic quantum walks on lattices and the Weyl equation

Giacomo Mauro D’Ariano, Marco Erba, Paolo Perinotti

Published Dec 1, 2017• 17 citations
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Abstract

We present a thorough classification of the isotropic quantum walks on lattices of dimension $d=1,2,3$ with a coin system of dimension $s=2$. For $d=3$ there exist two isotropic walks, namely, the Weyl quantum walks presented in the work of D'Ariano and Perinotti [G. M. D'Ariano and P. Perinotti, Phys. Rev. A 90, 062106 (2014)], resulting in the derivation of the Weyl equation from informational principles. The present analysis, via a crucial use of isotropy, is significantly shorter and avoids a superfluous technical assumption, making the result completely general.

DOI
10.1103/physreva.96.062101

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We present a thorough classification of the isotropic quantum walks on lattices of dimension d=1,2,3d=1,2,3d=1,2,3 with a coin system of dimension s=2s=2s=2. For d=3d=3d=3 there exist two isotropic walks, namely, the Weyl quantum walks presented in the work of D'Ariano and Perinotti [G. M. D'Ariano and P. Perinotti, Phys. Rev. A 90, 062106 (2014)], resulting in the derivation of the Weyl equation from informational principles. The present analysis, via a crucial use of isotropy, is significantly shorter and avoids a superfluous technical assumption, making the result completely general.
Quantum walks (QWs) describe particles evolving coherently on a graph. The internal degree of freedom corresponds to a Hilbert space, called a coin system. We consider QWs on Cayley graphs of some group GGG. In the literature, investigations concerning infinite GGG have been focused on graphs corresponding to G=ZdG={\mathbb{Z}}^{d}G=Zd with a coin system of dimension 2, whereas for a one-dimensional coin (so-called scalar QWs) only the case of finite GGG has been studied. Here we prove that the evolution of a scalar QW with GGG infinite Abelian is trivial, providing a thorough classification of this kind of walks. Then we consider the infinite dihedral group ${D}_{\ensuremath{\infty}}$, that is, the unique non-Abelian group GGG containing a subgroup $H\ensuremath{\cong}\mathbb{Z}$ with two cosets. We characterize the class of QWs on the Cayley graphs of ${D}_{\ensuremath{\infty}}$, and, via a coarse-graining technique, we show that it coincides with the class of spinorial walks on Z\mathbb{Z}Z which satisfies parity symmetry. This class of QWs includes the Weyl and the Dirac QWs. Remarkably, there exist also spinorial walks that are not coarse graining of a scalar QW, such as the Hadamard walk.
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Quantum walks (QWs) describe the evolution of quantum systems on graphs. An intrinsic degree of freedom---called the coin and represented by a finite-dimensional Hilbert space---is associated with each node. Scalar quantum walks are QWs with a one-dimensional coin. We propose a general strategy allowing one to construct scalar QWs on a broad variety of graphs, which admit embedding in Eulidean spaces, thus having a direct geometric interpretation. After reviewing the technique that allows one to regroup cells of nodes into new nodes, transforming finite spatial blocks into internal degrees of freedom, we prove that no QW with a two-dimensional coin can be derived from an isotropic scalar QW in this way. Finally, we show that the Weyl and Dirac QWs can be derived from scalar QWs in spaces of dimension up to three, via our construction.
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We consider quantum cellular automata on a body-centered cubic lattice and provide a simple derivation of the only two homogenous, local, isotropic, and unitary two-dimensional automata [G. M. D'Ariano and P. Perinotti, Phys. Rev. A 90, 062106 (2014)]. Our derivation relies on the notion of Gram matrix and emphasizes the link between the transition matrices that characterize the automata and the body-centered cubic lattice: The transition matrices essentially are the matrix representation of the vertices of the lattice's primitive cell. As expected, the dynamics of these two automata reduce to the Weyl equation in the limit of small wave vectors and continuous time. We also briefly examine the four-dimensional case, where we find two one-parameter families of automata that reduce to the Dirac equation in a suitable limit.
The analysis of a physical problem simplifies considerably when one uses a suitable coordinate system. We apply this approach to the discrete-time quantum walks with coins given by 2j+12j+12j+1-dimensional Wigner rotation matrices (Wigner walks), a model which was introduced in Miyazaki et al. [Phys. Rev. A 76, 012332 (2007)]. First, we show that from the three parameters of the coin operator only one is physically relevant for the limit density of the Wigner walk. Next, we construct a suitable basis of the coin space in which the limit density of the Wigner walk acquires a much simpler form. This allows us to identify various dynamical regimes which are otherwise hidden in the standard basis description. As an example, we show that it is possible to find an initial state which reduces the number of peaks in the probability distribution from generic 2j+12j+12j+1 to a single one. Moreover, the models with integer jjj lead to the trapping effect. The derived formula for the trapping probability reveals that it can be highly asymmetric and it deviates from purely exponential decay. Explicit results are given up to dimension 5.