Implementation of a discrete-time quantum walk with a circulant matrix on a graph by optical polarizing elements

In this paper, we introduce a quantum walk whose local scattering at each vertex is denoted by a unitary circulant matrix, namely, the circulant quantum walk. We also introduce another quantum walk induced by the circulant quantum walk, namely, the optical quantum walk, whose underlying graph is a 2-regular directed graph and obtained by blowing up the original graph in some way. We propose a design of an optical circuit which implements the stationary state of the optical quantum walk. We show that if the induced optical quantum walk does not have $+1$ eigenvalue, then the stationary state of the optical quantum walk gives that of the original circulant quantum walk. From this result, we give a useful condition for the setting of the circulant quantum walks which can be implemented by this optical circuit.

Figure 1

The island of the blow-up graph (left side), the island mounted with optical elements (middle), and the simplified version of the island mounted with optical elements (right side). This figure corresponds to the islands of ${\tilde{G}}^{(\text{BU},\xi )}$ for $N=3$, where each island of ${\tilde{G}}^{(\text{BU},\xi )}$ is a directed graph with two inputs (i, s) and two outputs (o, s) (for example, on the left, we can see the vertices where i1 and s3 are inputs and o1 and s1 are outputs). The island on the right is a simplified equivalent of the middle optical island.

Figure 1

The island of the blow-up graph (left side), the island mounted with optical elements (middle), and the simplified version of the island mounted with optical elements (right side). This figure corresponds to the islands of ${\tilde{G}}^{(\text{BU},\xi )}$ for $N=3$, where each island of ${\tilde{G}}^{(\text{BU},\xi )}$ is a directed graph with two inputs (i, s) and two outputs (o, s) (for example, on the left, we can see the vertices where i1 and s3 are inputs and o1 and s1 are outputs). The island on the right is a simplified equivalent of the middle optical island.

Figure 2

The island implemented with the optical elements shown on the right of Fig. 1, with $N$ generalized(the figure shown here is for $N=8$, but it can be extended to general circumscribed polygons). Mirrors are eliminated and the angle of incidence and reflection at the PBS are not taken into account (generally, the angle of incidence and reflection would be ${90}^{\circ }$, but that is just a matter of changing the direction of the optical path). Each location of HWP corresponds to an individual vertex of the island. Each PBS plays two roles, both receiving the inflows with a superposition of H and V polarizations from the backward HWP and with H polarization from the outside, and sending the outflow with V polarization to the forward HWP and with H polarization to the outside.

Figure 2

The island implemented with the optical elements shown on the right of Fig. 1, with $N$ generalized(the figure shown here is for $N=8$, but it can be extended to general circumscribed polygons). Mirrors are eliminated and the angle of incidence and reflection at the PBS are not taken into account (generally, the angle of incidence and reflection would be ${90}^{\circ }$, but that is just a matter of changing the direction of the optical path). Each location of HWP corresponds to an individual vertex of the island. Each PBS plays two roles, both receiving the inflows with a superposition of H and V polarizations from the backward HWP and with H polarization from the outside, and sending the outflow with V polarization to the forward HWP and with H polarization to the outside.

Figure 3

The original graph $G_0$ (left) and its blow-up graph ${\tilde{G}}^{(\text{BU},\xi )}$ (right). The labeling $\xi$ is set as follows: ${\xi }_u\left(\text{tail}\right)=0, {\xi }_u\left((v,u)\right)=1, {\xi }_u\left((z,u)\right)=2$; ${\xi }_v\left(\text{tail}\right)=1, {\xi }_v\left((w,v)\right)=2, {\xi }_v\left((z,v)\right)=3, {\xi }_v\left((u,v)\right)=0$; ${\xi }_w\left(\text{tail}\right)=2, {\xi }_w\left((z,w)\right)=0, {\xi }_w\left((v,w)\right)=1$; ${\xi }_z\left(\text{tail}\right)=3, {\xi }_z\left((u,z)\right)=0, {\xi }_z\left((v,z)\right)=1, {\xi }_z\left((w.z)\right)=2$. For example; (i) there is an arc from $(u;2)$ to $(u;0)$ because the terminal vertices are commonly $u$ and $2+1=0$ in the modulus of $deg\left(u\right)=3$, which satisfies the connected condition (1) in Definition t4, and (ii) there are symmetric arcs between vertex $(v;2)$ and $(w;1)$ in ${\tilde{G}}_0^{(\text{BU};\xi )}$ because we can check that $o\left({\xi }_v^{-1}\left(2\right)\right)=o\left((w,v)\right)=w$ and $o\left({\xi }_w^{-1}\left(1\right)\right)=o\left((v,w)\right))=v$, which satisfies the connected condition (2) in Definition t4.

Figure 3

The original graph $G_0$ (left) and its blow-up graph ${\tilde{G}}^{(\text{BU},\xi )}$ (right). The labeling $\xi$ is set as follows: ${\xi }_u\left(\text{tail}\right)=0, {\xi }_u\left((v,u)\right)=1, {\xi }_u\left((z,u)\right)=2$; ${\xi }_v\left(\text{tail}\right)=1, {\xi }_v\left((w,v)\right)=2, {\xi }_v\left((z,v)\right)=3, {\xi }_v\left((u,v)\right)=0$; ${\xi }_w\left(\text{tail}\right)=2, {\xi }_w\left((z,w)\right)=0, {\xi }_w\left((v,w)\right)=1$; ${\xi }_z\left(\text{tail}\right)=3, {\xi }_z\left((u,z)\right)=0, {\xi }_z\left((v,z)\right)=1, {\xi }_z\left((w.z)\right)=2$. For example; (i) there is an arc from $(u;2)$ to $(u;0)$ because the terminal vertices are commonly $u$ and $2+1=0$ in the modulus of $deg\left(u\right)=3$, which satisfies the connected condition (1) in Definition t4, and (ii) there are symmetric arcs between vertex $(v;2)$ and $(w;1)$ in ${\tilde{G}}_0^{(\text{BU};\xi )}$ because we can check that $o\left({\xi }_v^{-1}\left(2\right)\right)=o\left((w,v)\right)=w$ and $o\left({\xi }_w^{-1}\left(1\right)\right)=o\left((v,w)\right))=v$, which satisfies the connected condition (2) in Definition t4.

Figure 4

$G^{\prime }$ (left) and the design of the optical circuit (right). To highlight the act of drawing the circuit, we include ${\tilde{G}}^{(\text{BU},\xi )}$ in gray on the left. We draw the circumscribed polygons around each island (blue arcs) and place the vertices on the corners and in the midpoint of each side. Then, the sides of the circumscribed polygon correspond to $A_0^{\text{cycle}}$. On the other hand, all the arcs in ${\tilde{A}}_0$ are reconnected as depicted on the left. For example, the symmetric arcs between $(u;1)$ and $(v;0)$ are reconnected by changing the origin and terminal vertices to the corresponding corners of the circumscribed polygon. We set the half-wave plate on each black vertex and the polarizing beam splitter on each white vertex as depicted on the right.

Figure 4

$G^{\prime }$ (left) and the design of the optical circuit (right). To highlight the act of drawing the circuit, we include ${\tilde{G}}^{(\text{BU},\xi )}$ in gray on the left. We draw the circumscribed polygons around each island (blue arcs) and place the vertices on the corners and in the midpoint of each side. Then, the sides of the circumscribed polygon correspond to $A_0^{\text{cycle}}$. On the other hand, all the arcs in ${\tilde{A}}_0$ are reconnected as depicted on the left. For example, the symmetric arcs between $(u;1)$ and $(v;0)$ are reconnected by changing the origin and terminal vertices to the corresponding corners of the circumscribed polygon. We set the half-wave plate on each black vertex and the polarizing beam splitter on each white vertex as depicted on the right.

Figure 5

Time course of ${\mu }_n$ for the circulant QW with a marked vertex in the setting of Example 1: The horizontal and vertical lines are the time step and the relative probability of each vertex. The blue (upper) line is the time course of the relative probability of the marked vertex, the other lines are the time courses of the other vertices.

Figure 5

Time course of ${\mu }_n$ for the circulant QW with a marked vertex in the setting of Example 1: The horizontal and vertical lines are the time step and the relative probability of each vertex. The blue (upper) line is the time course of the relative probability of the marked vertex, the other lines are the time courses of the other vertices.

Figure 6

Time course of ${\mu }_n^{\text{BU}}$ for the optical QW with a marked vertex induced by the left circulant QW: The blue (upper) line is the time course of the relative probability of the marked vertex, the other lines are the time courses of the other vertices.

Figure 6

Time course of ${\mu }_n^{\text{BU}}$ for the optical QW with a marked vertex induced by the left circulant QW: The blue (upper) line is the time course of the relative probability of the marked vertex, the other lines are the time courses of the other vertices.

Figure 7

Time course of the relative probabilities of the uniform circulant QW in the setting of Example 2.

Figure 7

Time course of the relative probabilities of the uniform circulant QW in the setting of Example 2.

Figure 8

Time course of the relative probabilities of the uniform optical QW induced by the left circulant QW.

Figure 8

Time course of the relative probabilities of the uniform optical QW induced by the left circulant QW.

Figure 9

The symmetry breaking design in Theorem t11 (i) for the implementation.

Figure 9

The symmetry breaking design in Theorem t11 (i) for the implementation.

Figure 10

The symmetry-breaking design in Theorem t11 (ii) for the implementation.

Figure 10

The symmetry-breaking design in Theorem t11 (ii) for the implementation.

Figure 11

The vertex colored white is the vertex connecting to a tail. The eigenvector of eigenvalue 1 does not overlap to $B_u$, for any island $u$.

Figure 12

Labeling of the arcs of island $u$ in the proof of Lemma t15.