[Paper Review] Continuous Time Quantum Walks on Graphs: Group State Transfer

Luke C. Brown, William J. Martin+1 more|arXiv (Cornell University)|Mar 16, 2021

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces group state transfer (GST) in continuous-time quantum walks on graphs, generalizing perfect state transfer and fractional revival by allowing transfer of quantum states from one set of vertices to another. The key contribution is a structure theorem showing bijective GST is monogamous and rare, with non-trivial examples found in symmetric double stars, bipartite graphs with integer eigenvalues, and graph joins via Cartesian products.

ABSTRACT

We introduce the concept of group state transfer on graphs, summarize its relationship to other concepts in the theory of quantum walks, set up a basic theory, and discuss examples. Let $X$ be a graph with adjacency matrix $A$ and consider quantum walks on the vertex set $V(X)$ governed by the continuous time-dependent unitary transition operator $U(t)= \exp(itA)$. For $S,T\subseteq V(X)$, we says $X$ admits "group state transfer" from $S$ to $T$ at time $τ$ if the submatrix of $U(τ)$ obtained by restricting to columns in $S$ and rows not in $T$ is the all-zero matrix. As a generalization of perfect state transfer, fractional revival and periodicity, group state transfer satisfies natural monotonicity and transitivity properties. Yet non-trivial group state transfer is still rare; using a compactness argument, we prove that bijective group state transfer (the optimal case where $|S|=|T|$) is absent for almost all $t$. Focusing on this bijective case, we obtain a structure theorem, prove that bijective group state transfer is "monogamous", and study the relationship between the projections of $S$ and $T$ into each eigenspace of the graph. Group state transfer is obviously preserved by graph automorphisms and this gives us information about the relationship between the setwise stabilizer of $S\subseteq V(X)$ and the stabilizers of naturally defined subsets obtained by spreading $S$ out over time and crudely reversing this process. These operations are sufficiently well-behaved to give us a topology on $V(X)$ which is likely to be simply the topology of subsets for which bijective group state transfer occurs at that time. We illustrate non-trivial group state transfer in bipartite graphs with integer eigenvalues, in joins of graphs, and in symmetric double stars. The Cartesian product allows us to build new examples from old ones.

Motivation & Objective

• To formalize and generalize the concept of state transfer in quantum walks beyond single-vertex transfer to sets of vertices.
• To investigate the conditions under which quantum state transfer occurs between subsets of vertices, termed group state transfer (GST).
• To establish structural and topological properties of GST, including monotonicity, transitivity, and the rarity of non-trivial bijective GST.
• To explore the relationship between eigenstructure of the graph's adjacency matrix and the occurrence of GST, particularly in symmetric and bipartite graphs.
• To construct new examples of GST using graph operations such as joins and Cartesian products, and to analyze their spectral properties.

Proposed method

• Define GST from set $S$ to set $T$ at time $ au$ as the condition that the submatrix of $U( au) = /exp(itA)$ with rows not in $T$ and columns in $S$ is zero.
• Use spectral decomposition of the adjacency matrix $A$ to analyze $U( au)$, expressing it in terms of projections onto eigenspaces.
• Apply a compactness argument to show that bijective GST (where $|S| = |T|$) occurs for only finitely many times $t$ in any interval.
• Prove that bijective GST is monogamous: a set $S$ can be transferred to at most one other set $T$ of equal size.
• Leverage graph automorphisms and eigenspace projections to derive symmetries and invariants of GST configurations.
• Construct explicit examples using symmetric double stars, joins of graphs, and Cartesian products, verifying GST via eigenvalue and projection analysis.

Experimental results

Research questions

• RQ1 Which graph products preserve group state transfer, and can new GST examples be systematically constructed from known ones?
• RQ2 For a path graph, can all pairs of vertex subsets $S, T$ admitting bijective GST be completely classified?
• RQ3 Do all $t$-closed vertex subsets arise from bijective GST, and what is the topological structure of such sets?
• RQ4 Under what conditions does a weighted quotient graph inherit perfect state transfer from a GST configuration?
• RQ5 Does the number of vertices in a graph with GST between sets $S$ and $T$ of minimum distance $\delta$ grow exponentially with $\delta$?

Key findings

• Bijective group state transfer is monogamous: a set $S$ can be transferred to at most one other set $T$ of equal size, excluding trivial cases.
• A compactness argument proves that for all but finitely many $t$ in any finite interval, only trivial GST pairs $(\emptyset,\emptyset)$ and $(V(X),V(X))$ are maximal.
• In symmetric double stars with $m$ vertices of degree one, $(S,S)$-GST occurs at $\tau = 2\pi / \sqrt{4k+1}$ for $S = \{1,2\}$, verified via eigenvalue and projection analysis.
• For the join of two complete graphs $K_m$ with $m \geq 3$, non-trivial GST occurs at time $\tau = 2\pi / \sqrt{4k+1}$, with $S$ and $T$ being the two vertex sets.
• The Cartesian product of graphs preserves GST, enabling construction of new examples from existing ones.
• In bipartite graphs with integer eigenvalues, non-trivial GST is possible, and the eigenvalue structure ensures that certain projection terms vanish at specific times, enabling GST.