[Paper Review] Perfect state transfer of quantum walks on quotient graphs

Rachel Elizabeth Bachman, Eric Fredette+5 more|arXiv (Cornell University)|Aug 1, 2011

Quantum Computing Algorithms and Architecture1 citations

TL;DR

This paper establishes that perfect state transfer in quantum walks on quotient graphs preserves the property across equitable partitions, enabling construction of graphs with perfect state transfer between non-symmetric vertices. It proves that Cartesian products of quotient graphs are isomorphic to quotients of Cartesian products, providing an algebraic foundation for multi-boson quantum walk constructions.

ABSTRACT

We prove new results on perfect state transfer of quantum walks on quotient graphs. Since a graph $G$ has perfect state transfer if and only if its quotient $G/\pi$, under any equitable partition $\pi$, has perfect state transfer, we exhibit graphs with perfect state transfer between two vertices but which lack automorphism swapping them. This answers a question of Godsil (Discrete Mathematics 312(1):129-147, 2011). We also show that the Cartesian product of quotient graphs $\Box_{k} G_{k}/\pi_{k}$ is isomorphic to the quotient graph $\Box_{k} G_{k}/\pi$, for some equitable partition $\pi$. This provides an algebraic description of a construction due to Feder (Physical Review Letters 97, 180502, 2006) which is based on many-boson quantum walk.

Motivation & Objective

• To resolve a question posed by Godsil (2011) regarding the existence of graphs with perfect state transfer between vertices that are not swapped by any automorphism.
• To establish a correspondence between perfect state transfer in a graph and its quotient under any equitable partition.
• To provide an algebraic characterization of the Cartesian product of quotient graphs, linking it to the quotient of the Cartesian product.
• To offer a formal algebraic description of Feder's (2006) many-boson quantum walk construction using quotient graphs.

Proposed method

• Utilizes the theory of equitable partitions in graphs to define quotient graphs, preserving spectral properties relevant to quantum walks.
• Applies the equivalence between perfect state transfer in a graph and its quotient under any equitable partition, leveraging spectral graph theory.
• Employs the structure of Cartesian products of graphs to show that $\Box_k G_k / \pi$ is isomorphic to $\Box_k (G_k / \pi_k)$ for a suitable equitable partition $\pi$.
• Uses matrix algebra and eigenvalue analysis to demonstrate that the adjacency matrix of the quotient graph reflects the dynamics of the original graph.
• Relies on known results on perfect state transfer and spectral decomposition to prove the invariance of the property under quotienting.
• Connects the algebraic structure of quotient graphs to physical quantum walk models, particularly multi-boson systems as in Feder's work.

Experimental results

Research questions

• RQ1 Can perfect state transfer occur in a graph between two vertices that are not interchanged by any automorphism?
• RQ2 Does perfect state transfer in a graph imply perfect state transfer in its quotient under any equitable partition?
• RQ3 Is the Cartesian product of quotient graphs isomorphic to the quotient of the Cartesian product of the original graphs under a compatible equitable partition?
• RQ4 Can the construction of many-boson quantum walks via quotient graphs be given an algebraic characterization?
• RQ5 What is the relationship between the spectral properties of a graph and its quotient in the context of quantum walk dynamics?

Key findings

• The paper confirms that perfect state transfer can occur between vertices that are not automorphic images of each other, answering Godsil's question in the affirmative.
• It proves that a graph $G$ has perfect state transfer if and only if its quotient $G/\pi$ under any equitable partition $\pi$ also has perfect state transfer.
• The Cartesian product of quotient graphs $\Box_k G_k / \pi_k$ is isomorphic to the quotient graph $\Box_k G_k / \pi$ for a specific equitable partition $\pi$, establishing a structural equivalence.
• This isomorphism provides an algebraic framework for Feder's many-boson quantum walk construction, which was previously described via physical intuition.
• The results show that the dynamics of quantum walks on quotient graphs are fully determined by the spectral properties preserved under equitable partitioning.
• The framework enables the systematic construction of graphs with perfect state transfer even when no automorphism swaps the transfer vertices.