[Paper Review] Pretty good state transfer on double stars

Xiaoxia Fan, Chris Godsil|arXiv (Cornell University)|Jun 1, 2012

Quantum Computing Algorithms and Architecture9 references4 citations

TL;DR

This paper establishes conditions for pretty good state transfer (PGST) in double-star graphs $S_{k,\ell}$, proving PGST occurs between end vertices in $S_{2,\ell}$ for $\ell > 2$, and between the central vertices in $S_{k,k}$ if and only if $4k+1$ is not a perfect square. The results reveal a deep connection between quantum state transfer and number-theoretic conditions, extending the known families of graphs with PGST beyond paths.

ABSTRACT

Let A be the adjacency matrix of a graph $X$ and suppose U(t)=exp(itA). We view A as acting on $\cx^{V(X)}$ and take the standard basis of this space to be the vectors $e_u$ for $u$ in $V(X)$. Physicists say that we have perfect state transfer from vertex $u$ to $v$ at time $τ$ if there is a scalar $γ$ such that $U(τ)e_u = γe_v$. (Since $U(t)$ is unitary, $ ormγ=1$.) For example, if $X$ is the $d$-cube and $u$ and $v$ are at distance $d$ then we have perfect state transfer from $u$ to $v$ at time $π/2$. Despite the existence of this nice family, it has become clear that perfect state transfer is rare. Hence we consider a relaxation: we say that we have pretty good state transfer from $u$ to $v$ if there is a complex number $γ$ and, for each positive real $ε$ there is a time $t$ such that $ orm{U(t)e_u - γe_v} < ε$. Again we necessarily have $|γ|=1$. Godsil, Kirkland, Severini and Smith showed that we have have pretty good state transfer between the end vertices of the path $P_n$ if and only $n+1$ is a power of two, a prime, or twice a prime. (There is perfect state transfer between the end vertices only for $P_2$ and $P_3$.) It is something of a surprise that the occurrence of pretty good state transfer is characterized by a number-theoretic condition. In this paper we study double-star graphs, which are trees with two vertices of degree $k+1$ and all other vertices with degree one. We prove that there is never perfect state transfer between the two vertices of degree $k+1$, and that there is pretty good state transfer between them if and only if $4k+1$ is a perfect square.

Motivation & Objective

• To investigate the existence of pretty good state transfer (PGST) in double-star graphs $S_{k,\ell}$, a class of trees with two central vertices of degrees $k+1$ and $\ell+1$.
• To determine whether PGST occurs between end vertices or central vertices in these graphs.
• To establish a number-theoretic condition for PGST, analogous to the known condition for paths $P_n$.
• To show that perfect state transfer does not occur in any double-star graph, but PGST can still occur.

Proposed method

• Use the continuous-time quantum walk defined by $U(t) = \exp(itA)$, where $A$ is the adjacency matrix of the graph.
• Analyze the dynamics of $U(t)$ on the standard basis vectors $e_u$ and $e_v$ to assess state transfer from vertex $u$ to $v$.
• Apply spectral decomposition and eigenvalue analysis to determine conditions under which $\|U(t)e_u - \gamma e_v\| < \epsilon$ for arbitrarily small $\epsilon > 0$.
• Use the symmetrized quotient graph technique to reduce the problem to analyzing a smaller system with fewer eigenvalues.
• Leverage Kronecker’s approximation theorem to show density of orbits when $\alpha$ is irrational, enabling approximation of desired state transfer.
• Prove that the closure of the unitary group $\overline{G}$ is compact and that PGST implies recurrence in time intervals, ensuring regularity of approximate state transfer.

Experimental results

Research questions

• RQ1 Does pretty good state transfer occur between the end vertices of $S_{2,\ell}$ for $\ell > 2$?
• RQ2 Under what conditions does pretty good state transfer occur between the two central vertices in $S_{k,k}$?
• RQ3 Why does perfect state transfer not occur in any double-star graph?
• RQ4 Is there a number-theoretic characterization for PGST in double-stars, similar to the $n+1$ being a power of two or prime for paths?
• RQ5 Does PGST in double-stars imply a regular recurrence pattern in time?

Key findings

• Perfect state transfer does not occur in any double-star graph $S_{k,\ell}$, regardless of $k$ and $\ell$.
• Pretty good state transfer occurs between the end vertices of $S_{2,\ell}$ for all $\ell > 2$, regardless of the value of $\ell$.
• For $S_{k,k}$ with $k > 2$, pretty good state transfer occurs between the two central vertices if and only if $4k+1$ is not a perfect square.
• When $4k+1$ is a perfect square, the system is periodic and PGST does not occur, even though the eigenvalues are rational multiples of $\pi$.
• The existence of PGST in $S_{k,k}$ is determined solely by the number-theoretic condition that $4k+1$ is not a perfect square.
• PGST implies that the time evolution is approximately periodic, and for any $\epsilon > 0$, there exists a time $T$ such that every interval of length $T$ contains a time where $U(t)$ is within $\epsilon$ of a state achieving PGST.