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.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.

arXiv:1902.10227Physical Review AJul 8, 20196 citations

Open quantum walks on graphs

Stéphane Attal, Francesco Petruccione, Ilya Sinayskiy

arXiv:1401.3305Physics Letters AMar 21, 201271 citations

Quantum walks with a one-dimensional coin

Alessandro Bisio, Giacomo Mauro D’Ariano, Marco Erba +2 more

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 $G$ . In the literature, investigations concerning infinite $G$ have been focused on graphs corresponding to $G={\mathbb{Z}}^{d}$ with a coin system of dimension 2, whereas for a one-dimensional coin (so-called scalar QWs) only the case of finite $G$ has been studied. Here we prove that the evolution of a scalar QW with $G$ 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 $G$ 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 $\mathbb{Z}$ 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.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 $G$. In the literature, investigations concerning infinite $G$ have been focused on graphs corresponding to $G={\mathbb{Z}}^{d}$ with a coin system of dimension 2, whereas for a one-dimensional coin (so-called scalar QWs) only the case of finite $G$ has been studied. Here we prove that the evolution of a scalar QW with $G$ 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 $G$ 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 $\mathbb{Z}$ 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.

arXiv:1603.07666Physical Review AJun 24, 201621 citations

Internal degrees of freedom from quantum walks without quantum coin

Giacomo Mauro D’Ariano, Marco Erba, Paolo Perinotti

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 to 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.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 to 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.

Feb 26, 20190 citations

G

. In the literature, investigations concerning infinite

G

have been focused on graphs corresponding to

G={\mathbb{Z}}^{d}

with a coin system of dimension 2, whereas for a one-dimensional coin (so-called scalar QWs) only the case of finite

G

has been studied. Here we prove that the evolution of a scalar QW with

G

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

G

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

\mathbb{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.