Engineering Perfect State Transfer Graphs via Givens Transformations

Abstract

Perfect quantum state transfer is achievable in different settings, including linear qubit chains, bi-dimensional arrays, ladders, etc. The most studied case contemplates transferring arbitrary one-qubit pure states in systems with homogeneous interactions. These restrictions allow finding numerous examples of systems that show perfect transfer but in geometries that are not implementable or are very difficult to implement in actual experimental settings. Relaxing the homogeneity of the interactions and inspired by the $XX$ qubit chains that show perfect transmission, we present a simple scheme based on the Givens Transformations to analyse and obtain a class of qubit graphs that possess perfect quantum state transmission. We present some simple examples and show how it is possible to generalize them for longer transmission lengths.

Figures (10)

• Figure 1: The cartoon depicts some properties of graphs. The circles and the straight lines correspond to the nodes and links of the graph, respectively. a) and b) show the same bipartite graph with two different labellings. The bipartite condition does not depend on the particular label employed on a single node. Labelling any single node as “A” or “B” and then labelling all nodes connected to it as “B” or “A,” respectively, makes it possible to assign labels to all nodes in the graph uniquely. c) shows that the addition of links to a bipartite graph leads to the loss of that condition if the link added is the red one. If the link added is the green one, the graph remains bipartite. Note that the nodes connected by the red link have three nodes connected to them before the introduction of the red bridge. d) shows a tree-like graph. The nodes on the graph belong to a layer, and the number of links that separate a given node from the original one labels the corresponding layer. The distance between two nodes is the smallest number of links needed to join them. Note that tree-like graphs are always bipartite.
• Figure 2: The cartoon depicts the graph proposed in Reference coutinho2016. The numbering of the nodes helps identify the relationship between the interaction coefficients and the entries of the Hamiltonian matrix. See Equation \ref{['eq-coutinho-6']}. The distance between nodes 1 and 5 is equal to 4, and the coefficients are $x,\,y,\,z$.
• Figure 3: The cartoon in the figure shows two graphs with more nodes and links than the one originally proposed by Coutinho coutinho2016. In panel (a), the largest distance between nodes is 4, while in panel (b) it is 5.
• Figure 4: ł(a) The cartoon shows how to generalize the graph originally proposed by Coutinho coutinho2016. The two tails, each formed by two nodes connected by a link (depicted in black), are connected to an internal set of nodes and links, such that at most two links reach at each internal node. The internal nodes and links correspond to the red rectangle, and all links connecting the internal nodes to the innermost tail nodes are represented by red wavy lines. (b) An example in which $n+3$ links connect to the innermost nodes of the tails.
• Figure 5: The cartoon in the figure shows a generalization of the graph originally proposed by Coutinho, following the prescription given in Fig. \ref{['fig-generalizing-coutinho']}(a). The internal set of nodes and links consists of two linear chains with $n$ nodes and $n-1$ links each. The interaction coefficients within this set are labeled from $w_1$ to $w_{n-1}$, while the interaction coefficients corresponding to the links connecting the tails to the chains are labeled $y$ and $z$. The interaction coefficient between the two nodes forming each tail is labeled $x$. See Lemma \ref{['lemma2']}..