Perfect quantum state transfer of hard-core bosons on weighted path graphs

The ability to accurately transfer quantum information through networks is an important primitive in distributed quantum systems. While perfect quantum state transfer (PST) can be effected by a single particle undergoing continuous-time quantum walks on a variety of graphs, it is not known if PST persists for many particles in the presence of interactions. We show that if single-particle PST occurs on one-dimensional weighted path graphs, then systems of hard-core bosons undergoing quantum walks on these paths also undergo PST. The analysis extends the Tonks-Girardeau ansatz to weighted graphs using techniques in algebraic graph theory. The results suggest that hard-core bosons do not generically undergo PST, even on graphs which exhibit single-particle PST.

Figure 1

Equitable mapping of the 3-cube ${\mathcal{Q}}_3$ to the weighted line ${\tilde{P}}_4$. The hypercube ${\mathcal{Q}}_n$ on $n$ vertices has the weighted path graph as a quotient graph, with weights $w(v,v+1)=\sqrt{v(n+1-v)}$ symmetric about the center. The blue shaded regions denote the cells in the equitable partitioning.

Figure 2

Two copies of the hypercubically weighted path graph on four vertices ${\tilde{P}}_4$ are shown in (a). Suppose that two distinguishable particles (yellow and red) are incident on the first and second sites; after undergoing unitary evolution for a time $\tau =\pi /2$, they are located at the fourth and third sites, respectively. The graph representation of their joint occupation space configuration is shown in (b), with indices labeling the possible positions of the two particles. The evolution described in (a) is represented by two orange-colored vertices of a single boson; the initial state is $(1,2)$, and the final state is $(4,3)$. The cell structure of the equitable partitioning from distinguishable to identical particles is depicted via the blue lines connecting vertices which are mirror symmetric with respect to the central diagonal (i.e., equivalent under a permutation of the indices). The resulting quotient graph representing two identical bosonic particles is shown in (c). The edge weights reflect the action of generalized equitable partitioning.

Figure 3

The effective graph representing two hard-core distinguishable bosons on the hypercubically weighted path graph ${\tilde{P}}_4$. Two copies of ${\tilde{P}}_4$ are shown in (a). The left shows two distinguishable hard-core particles (yellow and red) incident on the first and second sites; after undergoing unitary evolution for a time $\tau =\pi /2$, they are located at sites three and four, respectively (note that unlike noninteracting particles they cannot exchange their relative positions). The graph representation of their joint occupation space configuration is shown in (b). This is the Cartesian square ${\tilde{P}}_4^{\square 2}$ with vertices deleted whose indices have repeating labels, leading to two disconnected graphs (deleted vertices are indicated by the dashed circles on the diagonal). Input and output vertices under PST, corresponding to locations in (a), are shown in orange. The quotient disconnected graph components for identical hard-core bosons are shown in (c).

Figure 4

Occupation-space representation of the composite $\widehat{C}$ operator on a given configuration space site. The central arrow shows the composite operation directly as a whole while the side configurations exhibit the constituent operations, explicitly showing the commutativity of ${\widehat{\pi }}_{\perp }$ and $\widehat{P}$.

Figure 5

Configuration-space graph of two hard-core bosons on $\widetilde{P_4}$. (a) The mirror plane $M$ is shown as the dashed line on the antidiagonal. Vertices not in singleton cells of the partition are grouped in pairs and connected by the curved blue lines. The cells are symmetrically disposed about the antidiagonal line. (b) The resulting quotient graph.