Order from chaos in quantum walks on cyclic graphs

It has been shown classically that combining two chaotic random walks can yield an ordered (periodic) walk. Our aim in this paper is to find a quantum analog for this rather counterintuitive result. We study the chaotic and periodic nature of cyclic quantum walks and focus on a unique situation wherein a periodic quantum walk on 3-cycle graph is generated via a deterministic combination of two chaotic quantum walks on the same graph. We extend our results to even numbered cyclic graphs, specifically a 4-cycle graph too. Our results will be relevant in quantum cryptography and quantum chaos control.

Figure 1

3-cycle graph (a) and 4-cycle graph (b).

Figure 2

Probability for finding the walker at its initial site at $x=0$ for a 4-cycle graph [panel (a), ordered, i.e., periodic with period 8] and a 5-cycle graph [panel (b), chaotic]. Probability is plotted against time steps for the QW with a Hadamard coin $H=C_2(\frac{1}{2},0,0)$. Only even time steps are plotted.

Figure 3

The QW on a 3-cycle graph by repeatedly applying the unitary $A=U_3({\rho }_1,0,0)$, which results in a chaotic quantum walk.

Figure 4

A chaotic QW on a 3-cycle graph obtained by repeatedly applying the unitary $B=U_3({\rho }_2,0,0)$.

Figure 5

A chaotic QW on a 3-cycle graph by repeatedly applying the Parrondo sequence $ABAB...,$ plotting every second point.

Figure 6

An ordered QW on a 3-cycle graph by repeatedly applying the Parrondo sequence $AABB...$ Every fourth point is plotted. The quantum walk is periodic with a periodicity of 20.

Figure 7

An ordered QW on a 3-cycle graph by repeatedly applying the unitary $C$. Every fourth point is plotted. The quantum walk is periodic with a periodicity of 10.

Figure 8

A QW on a 4-cycle graph by repeatedly applying the unitary $A^{\prime }=U_4({\rho }_{41},0,0)$, which results in chaotic quantum walk.

Figure 9

A chaotic QW on a 4-cycle graph obtained by repeatedly applying the unitary $B^{\prime }=U_4({\rho }_{42},0,0)$.

Figure 10

An ordered QW on a 4-cycle graph by repeatedly applying the Parrondo sequence $A^{\prime }{A}^{\prime }{B}^{\prime }{B}^{\prime }...$ The quantum walk is periodic with a periodicity of 5.

Figure 11

An ordered QW on a 4-cycle graph by repeatedly applying the unitary $C^{\prime }$. The quantum walk is periodic with a periodicity of 10.