[Paper Review] Perfect state transfer, integral circulants and join of graphs

Ricardo Javier Angeles-Canul, Rachael M. Norton+4 more|arXiv (Cornell University)|Jul 13, 2009

Quantum Computing Algorithms and Architecture6 references4 citations

TL;DR

This paper introduces new families of graphs exhibiting perfect state transfer in quantum networks using graph join operations and circulant generalizations. It constructs integral circulant graphs, such as ICGₙ({2, n/2ᵇ} ∪ Q) for n divisible by 16 and b ∈ {1,2}, and proves perfect state transfer in non-periodic double-cone graphs via the join of K̄₂ and regular graphs, answering an open question by Godsil.

ABSTRACT

We propose new families of graphs which exhibit quantum perfect state transfer. Our constructions are based on the join operator on graphs, its circulant generalizations, and the Cartesian product of graphs. We build upon the results of Bašić et al \cite{bps09,bp09} and construct new integral circulants and regular graphs with perfect state transfer. More specifically, we show that the integral circulant $ extsc{ICG}_{n}(\{2,n/2^{b}\} \cup Q)$ has perfect state transfer, where $b \in \{1,2\}$, $n$ is a multiple of 16 and $Q$ is a subset of the odd divisors of $n$. Using the standard join of graphs, we also show a family of double-cone graphs which are non-periodic but exhibit perfect state transfer. This class of graphs is constructed by simply taking the join of the empty two-vertex graph with a specific class of regular graphs. This answers a question posed by Godsil \cite{godsil08}.

Motivation & Objective

• To construct new families of graphs that exhibit perfect state transfer in unmodulated quantum spin networks.
• To generalize the graph join operation into a circulant join that preserves the circulant structure and enables perfect state transfer.
• To resolve an open question posed by Godsil regarding non-periodic graphs with perfect state transfer by constructing a double-cone family.
• To extend known results on perfect state transfer in integral circulants and Cartesian products to broader classes of graphs.
• To reduce the existence of perfect state transfer in composite graphs to conditions on their components, enabling systematic construction.

Proposed method

• Introduce the circulant join operation G+ₜC G, which connects two copies of a circulant graph G using a circulant matrix C, interpolating between standard join and Cartesian product.
• Prove that if cos(t*√(CᵀC)) = ±I and G has perfect state transfer at time t*, then G+ₜC G also has perfect state transfer at t*.
• Show that the circulant join produces a circulant graph when C is a palindrome circulant, enabling construction of new integral circulant graphs.
• Derive the family ICGₙ({2, n/2ᵇ} ∪ Q) for b ∈ {1,2}, n divisible by 16, and Q a subset of odd divisors of n, as a new class with perfect state transfer.
• Establish closure properties for the m-fold self-join of a graph and the Cartesian product of multiple graphs with the same perfect state transfer time.
• Reduce the perfect state transfer condition on G+H (join of two regular graphs) to spectral conditions on G and constraints on their sizes and degrees.

Experimental results

Research questions

• RQ1 Can the graph join operation be generalized to preserve the circulant structure and enable perfect state transfer in new graph families?
• RQ2 Do integral circulant graphs of the form ICGₙ({2, n/2ᵇ} ∪ Q) exhibit perfect state transfer for b ∈ {1,2} and n divisible by 16?
• RQ3 Can non-periodic graphs with perfect state transfer be constructed via the join of K̄₂ and regular graphs, answering Godsil’s open question?
• RQ4 Under what conditions does the m-fold self-join of a graph preserve perfect state transfer?
• RQ5 Can perfect state transfer in the Cartesian product of multiple graphs be guaranteed when all components share the same perfect state transfer time?

Key findings

• The integral circulant graph ICGₙ({2, n/2ᵇ} ∪ Q) exhibits perfect state transfer for b ∈ {1,2}, n divisible by 16, and Q a subset of the odd divisors of n.
• The circulant join construction G+ₜC G preserves perfect state transfer when cos(t*√(CᵀC)) = ±I and G has perfect state transfer at time t*.
• The double-cone graph K̄₂ + G, formed by joining the complement of K₂ with a regular graph G, exhibits perfect state transfer and is non-periodic.
• Perfect state transfer in the m-fold self-join ∑ₖ₌₁ᵐ G is reducible to conditions on G and additional spectral constraints, enabling systematic construction.
• The Cartesian product of multiple perfect state transfer graphs with identical transfer times also exhibits perfect state transfer.
• The paper resolves Godsil’s open question by constructing a new family of non-periodic graphs with perfect state transfer using the join operator.