A continuous-time quantum walk on a graph is a matrix-valued function $\exp(-\mathtt{i} At)$ over the reals, where $A$ is the adjacency matrix of the graph. Such a quantum walk has universal perfect state transfer if for all vertices $u,v$ , there is a time where the $(v,u)$ entry of the matrix exponential has unit magnitude. We prove new characterizations of graphs with universal perfect state transfer. This extends results of Cameron et al. (Linear Algebra and Its Applications, 455:115-142, 2014). Also, we construct non-circulant families of graphs with universal perfect state transfer. All prior known constructions were circulants. Moreover, we prove that if a circulant, whose order is prime, prime squared, or a power of two, has universal perfect state transfer then its underlying graph must be complete. This is nearly tight since there are universal perfect state transfer circulants with non-prime-power order where some edges are missing.A continuous-time quantum walk on a graph is a matrix-valued function $\exp(-\mathtt{i} At)$ over the reals, where $A$ is the adjacency matrix of the graph. Such a quantum walk has universal perfect state transfer if for all vertices $u,v$, there is a time where the $(v,u)$ entry of the matrix exponential has unit magnitude. We prove new characterizations of graphs with universal perfect state transfer. This extends results of Cameron et al. (Linear Algebra and Its Applications, 455:115-142, 2014). Also, we construct non-circulant families of graphs with universal perfect state transfer. All prior known constructions were circulants. Moreover, we prove that if a circulant, whose order is prime, prime squared, or a power of two, has universal perfect state transfer then its underlying graph must be complete. This is nearly tight since there are universal perfect state transfer circulants with non-prime-power order where some edges are missing.

arXiv:1701.04145Jan 16, 20170 citations

Perfect state transfer on Cayley graphs over a non-abelian group of order $8n$ Perfect state transfer on Cayley graphs over a non-abelian group of order $8n$

Akash Kalita, Bikash Bhattacharjya

The \textit{transition matrix} of a graph $Γ$ with adjacency matrix $A$ is defined by $H(τ) := \exp(-\mathbf{i}τA)$ , where $τ\in \mathbb{R}$ and $\mathbf{i} = \sqrt{-1}$ . The graph $Γ$ exhibits \textit{perfect state transfer} (PST) between the vertices $u$ and $v$ if there exists $τ_0(>0)\in \mathbb{R}$ such that $\lvert H(τ_0)_{uv} \rvert = 1$ . For a positive integer $n$ , the group $V_{8n}$ is defined as $V_{8n} := \langle a,b \colon a^{2n} = b^{4} = 1, ba = a^{-1}b^{-1}, b^{-1}a = a^{-1}b \rangle$ . In this paper, we study the existence of perfect state transfer on Cayley graphs $\text{Cay}(V_{8n}, S)$ . We present some necessary and sufficient conditions for the existence of perfect state transfer on $\text{Cay}(V_{8n}, S)$ .The \textit{transition matrix} of a graph $Γ$ with adjacency matrix $A$ is defined by $H(τ) := \exp(-\mathbf{i}τA)$, where $τ\in \mathbb{R}$ and $\mathbf{i} = \sqrt{-1}$. The graph $Γ$ exhibits \textit{perfect state transfer} (PST) between the vertices $u$ and $v$ if there exists $τ_0(>0)\in \mathbb{R}$ such that $\lvert H(τ_0)_{uv} \rvert = 1$. For a positive integer $n$, the group $V_{8n}$ is defined as $V_{8n} := \langle a,b \colon a^{2n} = b^{4} = 1, ba = a^{-1}b^{-1}, b^{-1}a = a^{-1}b \rangle$. In this paper, we study the existence of perfect state transfer on Cayley graphs $\text{Cay}(V_{8n}, S)$. We present some necessary and sufficient conditions for the existence of perfect state transfer on $\text{Cay}(V_{8n}, S)$.

arXiv:2405.02122May 3, 20240 citations

State-Transfer Collisions

Alan C. Gallagher

Jan 1, 19701 citation

PERFECT, EFFICIENT, STATE TRANSFER AND ITS APPLICATION AS A CONSTRUCTIVE TOOL

Alastair Kay

We review the subject of perfect state transfer — how one designs the (fixed) interactions of a chain of spins so that a quantum state, initially inserted on one end of the chain, is perfectly transferred to the opposite end in a fixed time. The perfect state transfer systems are then used as a constructive tool to design Hamiltonian implementations of other primitive protocols such as entanglement generation and signal amplification in measurements, before showing that, in fact, universal quantum computation can be implemented in this way.We review the subject of perfect state transfer — how one designs the (fixed) interactions of a chain of spins so that a quantum state, initially inserted on one end of the chain, is perfectly transferred to the opposite end in a fixed time. The perfect state transfer systems are then used as a constructive tool to design Hamiltonian implementations of other primitive protocols such as entanglement generation and signal amplification in measurements, before showing that, in fact, universal quantum computation can be implemented in this way.

arXiv:0903.4274International Journal of Quantum InformationJun 1, 2010320 citations

Universality Under Conditions of Self-tuning

Ole Peters, Michelle Girvan

arXiv:0902.1956Journal of Statistical PhysicsAug 19, 20100 citations

A continuous-time quantum walk on a graph is a matrix-valued function

\exp(-\mathtt{i} At)

over the reals, where

A

is the adjacency matrix of the graph. Such a quantum walk has universal perfect state transfer if for all vertices

u,v

, there is a time where the

(v,u)

entry of the matrix exponential has unit magnitude. We prove new characterizations of graphs with universal perfect state transfer. This extends results of Cameron et al. (Linear Algebra and Its Applications, 455:115-142, 2014). Also, we construct non-circulant families of graphs with universal perfect state transfer. All prior known constructions were circulants. Moreover, we prove that if a circulant, whose order is prime, prime squared, or a power of two, has universal perfect state transfer then its underlying graph must be complete. This is nearly tight since there are universal perfect state transfer circulants with non-prime-power order where some edges are missing.

Perfect state transfer on Cayley graphs over a non-abelian group of order

8n

The \textit{transition matrix} of a graph

Γ

with adjacency matrix

A

is defined by

H(τ) := \exp(-\mathbf{i}τA)

, where

τ\in \mathbb{R}

and

\mathbf{i} = \sqrt{-1}

. The graph

Γ

exhibits \textit{perfect state transfer} (PST) between the vertices

u

and

v

if there exists

τ_0(>0)\in \mathbb{R}

such that

\lvert H(τ_0)_{uv} \rvert = 1

. For a positive integer

n

, the group

V_{8n}

is defined as

V_{8n} := \langle a,b \colon a^{2n} = b^{4} = 1, ba = a^{-1}b^{-1}, b^{-1}a = a^{-1}b \rangle

. In this paper, we study the existence of perfect state transfer on Cayley graphs

\text{Cay}(V_{8n}, S)

. We present some necessary and sufficient conditions for the existence of perfect state transfer on

\text{Cay}(V_{8n}, S)