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Perfect state transfer using Markovian quantum walk

Supriyo Dutta

TL;DR

A significantly powerful method is introduced based on the Markovian quantum walk to establish the quantum Perfect State Transfer between the extreme vertices of a path graph of arbitrary length.

Abstract

The quantum Perfect State Transfer (PST) is a fundamental tool of quantum communication in a network. It is not easy to achieve in practice. The original idea of PST depends on the fundamentals of the continuous-time quantum walk. A path graph with at most three vertices allows PST based on continuous-time quantum walk. Based on the Markovian quantum walk, we introduce a significantly powerful method for PST in this article. We establish PST between the extreme vertices of a path graph of arbitrary length. Moreover, any pair of symmetric vertices in a path graph allows PST under Markovian quantum walks. We extend our investigations for the cycle graphs. The cycle graphs with more than $4$ vertices do not allow the PST based on the continuous-time quantum walk. In contrast, a cycle graph with $2m$ vertices exhibits PST based on Markovian quantum walk between the vertices $j$ and $j + m$ for $j = 0, 1, \dots (m - 1)$, where $m > 0$ is an integer.

Perfect state transfer using Markovian quantum walk

TL;DR

A significantly powerful method is introduced based on the Markovian quantum walk to establish the quantum Perfect State Transfer between the extreme vertices of a path graph of arbitrary length.

Abstract

The quantum Perfect State Transfer (PST) is a fundamental tool of quantum communication in a network. It is not easy to achieve in practice. The original idea of PST depends on the fundamentals of the continuous-time quantum walk. A path graph with at most three vertices allows PST based on continuous-time quantum walk. Based on the Markovian quantum walk, we introduce a significantly powerful method for PST in this article. We establish PST between the extreme vertices of a path graph of arbitrary length. Moreover, any pair of symmetric vertices in a path graph allows PST under Markovian quantum walks. We extend our investigations for the cycle graphs. The cycle graphs with more than vertices do not allow the PST based on the continuous-time quantum walk. In contrast, a cycle graph with vertices exhibits PST based on Markovian quantum walk between the vertices and for , where is an integer.
Paper Structure (6 sections, 10 theorems, 40 equations, 7 figures)

This paper contains 6 sections, 10 theorems, 40 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Markovian quantum walk and perfect quantum state transfer
  3. State transfer on cycle graph
  4. State transfer on a path graph
  5. State transfer in glued path graph
  6. Conclusion

Key Result

Lemma 1

Let $G$ be a path or a cycle graph with vertices $0, 1, 2, \dots (n - 1)$. For $j \neq 0$ and $j \neq (n - 1)$ define $\ket{\psi_j} = \frac{1}{\sqrt{2}} \left( \ket{j(j + 1)} + \ket{j(j - 1)} \right)$ and $\Pi = \sum_{j \in V} \ket{\psi_j} \bra{\psi_j}$. Then

Figures (7)

  • Figure 1: Sub-figure (\ref{['undirected_6_cycle']}) presents a cycle graph $C_6$ with $6$ vertices. Note that, there is an edge between the vertices $j$ and $k$ if and only if $j - k \equiv 1(\mod 6)$. Figure (\ref{['cycle_6']}) is $C_6$ after orientation on the edges. Every edge in sub-figure (\ref{['undirected_6_cycle']}) generates two oppositely oriented edges in sub-figure (\ref{['cycle_6']}). Thus, every vertex is incident to two incoming edges and two outgoing edges, which means the out-degree of every vertex is $2$. Hence, $p_{j, k} = \frac{1}{2}$ for all possible edges $(j, k)$ in the cycle graph.
  • Figure 2: The red and green bars in the sub-figures indicate the probability of getting the walker at vertex $k, 0 \leq k \leq 5$ in $C_6$, at different time instances $t = 0, 1, 2$ and $3$ in continuous-time and Markovian quantum walks, respectively. Suppose the walker started at vertex $1$ at $t = 0$. At $t = 3$, it reaches vertex $4$ with full probability, under the Markovian quantum walks. Therefore, there is a PST between vertices $1$ and $4$ at time $t = 3$. In contrast, the continuous-quantum walks keep the walker at vertex $1$.
  • Figure 3: In this figure, we represent the probability of getting the walker at vertex $k, 0 \leq k \leq 4$ in a cycle graph $C_5$ with $5$ vertices at different time instances $t = 0, 1, \dots, 5$ with bar diagrams. Suppose the walker started at vertex $1$ at $t = 0$. At $t = 5$, it returns to the vertex $1$. Therefore, the probability of getting the walker at time $t = 0$ and $t = 5$ at vertex $1$ is $1$. Hence, the vertex $1$ is periodic with period $t = 5$.
  • Figure 4: We have a path graph with $6$ vertices in figure (a). For every edge, we generate two opposite orientations and draw the new graph in figure (b). Except for vertices $0$ and $5$, all other vertices have two outgoing edges. Hence, their out-degree is $2$. The out-degree of $1$ and $5$ is $1$. Therefore $p_{0, 1} = p_{5, 4} = 1$ and for all other edges $p_{j, k} = \frac{1}{2}$.
  • Figure 5: The red and green bars in the sub-figures indicate the probability of getting the walker at the vertex $k, 0 \leq k \leq 5$ in the path graph with $6$ vertices at different time instances $t = 0, 1, \dots, 5$ in continuous-time and Markovian quantum walks, respectively. Suppose the walker started at vertex $0$ at $t = 0$. At $t = 5$, the walker reaches the vertex $5$ with full probability under Markovian quantum walks. Therefore, there is a PST between vertices $1$ and $4$ at time $t = 5$. But the continuous-time quantum walk does not allow the walker to move to any vertex with full probability.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • ...and 10 more