scholarly journals Pretty good state transfer on 1-sum of star graphs

2018 ◽  
Vol 16 (1) ◽  
pp. 1483-1489
Author(s):  
Hailong Hou ◽  
Rui Gu ◽  
Mengdi Tong
Keyword(s):  
Quantum State ◽  
Complex Number ◽  
Adjacency Matrix ◽  
Perfect State ◽  
Quantum State Transfer ◽  
Positive Real ◽  
Star Graphs ◽  
State Transfer ◽  
Good State ◽  
Perfect State Transfer

AbstractLetAbe the adjacency matrix of a graphGand supposeU(t) = exp(itA). We say that we have perfect state transfer inGfrom the vertexuto the vertexvat timetif there is a scalarγof unit modulus such thatU(t)eu=γ ev. It is known that perfect state transfer is rare. So C.Godsil gave a relaxation of this definition: we say that we have pretty good state transfer fromutovif there exists a complex numberγof unit modulus and, for each positive realϵthere is a timetsuch that ‖U(t)eu–γ ev‖ <ϵ. In this paper, the quantum state transfer on 1-sum of star graphsFk,lis explored. We show that there is no perfect state transfer onFk,l, but there is pretty good state transfer onFk,lif and only ifk=l.

scholarly journals Perfect state transfer on graphs with a potential

2017 ◽  
Vol 17 (3&4) ◽  
pp. 303-327
Author(s):  
Mark Kempton ◽  
Gabor Lippner ◽  
Shing-Tung Yau
Keyword(s):  
Potential Function ◽  
Quantum State ◽  
Quantum Tunneling ◽  
Perfect State ◽  
Graph Products ◽  
Quantum State Transfer ◽  
State Transfer ◽  
Vertex Set ◽  
Perfect State Transfer

In this paper we study quantum state transfer (also called quantum tunneling) on graphs when there is a potential function on the vertex set. We present two main results. First, we show that for paths of length greater than three, there is no potential on the vertices of the path for which perfect state transfer between the endpoints can occur. In particular, this answers a question raised by Godsil in Section 20 of [1]. Second, we show that if a graph has two vertices that share a common neighborhood, then there is a potential on the vertex set for which perfect state transfer will occur between those two vertices. This gives numerous examples where perfect state transfer does not occur without the potential, but adding a potential makes perfect state transfer possible. In addition, we investigate perfect state transfer on graph products, which gives further examples where perfect state transfer can occur.

scholarly journals PERFECT STATE TRANSFER IN XX CHAINS INDUCED BY BOUNDARY MAGNETIC FIELDS

2012 ◽  
Vol 10 (03) ◽  
pp. 1250029 ◽  
Author(s):  
THORBEN LINNEWEBER ◽  
JOACHIM STOLZE ◽  
GÖTZ S. UHRIG
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Strong Magnetic Field ◽  
Quantum State ◽  
Numerical Study ◽  
Perfect State ◽  
Quantum State Transfer ◽  
State Transfer ◽  
Perfect State Transfer

A recent numerical study of short chains found near-perfect quantum state transfer between the boundary sites of a spin-1/2 XX chain if a sufficiently strong magnetic field acts on these sites. We show that the phenomenon is based on a pair of states strongly localized at the boundaries of the system and provide a simple quantitative analytical explanation.

scholarly journals Pretty Good State Transfer on Circulant Graphs

10.37236/6388 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Hiranmoy Pal ◽  
Bikash Bhattacharjya
Keyword(s):  
Complex Number ◽  
Adjacency Matrix ◽  
Transition Matrix ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Real Numbers ◽  
Cartesian Products ◽  
Edge Disjoint ◽  
State Transfer ◽  
Good State

Let $G$ be a graph with adjacency matrix $A$. The transition matrix of $G$ relative to $A$ is defined by $H(t):=\exp{\left(-itA\right)}$, where $t\in {\mathbb R}$. The graph $G$ is said to admit pretty good state transfer between a pair of vertices $u$ and $v$ if there exists a sequence of real numbers $\{t_k\}$ and a complex number $\gamma$ of unit modulus such that $\lim\limits_{k\rightarrow\infty} H(t_k) e_u=\gamma e_v.$ We find that the cycle $C_n$ as well as its complement $\overline{C}_n$ admit pretty good state transfer if and only if $n$ is a power of two, and it occurs between every pair of antipodal vertices. In addition, we look for pretty good state transfer in more general circulant graphs. We prove that union (edge disjoint) of an integral circulant graph with a cycle, each on $2^k$ $(k\geq 3)$ vertices, admits pretty good state transfer. The complement of such union also admits pretty good state transfer. Using Cartesian products, we find some non-circulant graphs admitting pretty good state transfer.

scholarly journals EFFICIENT AND PERFECT STATE TRANSFER IN QUANTUM CHAINS

2006 ◽  
Vol 04 (03) ◽  
pp. 405-414 ◽  
Author(s):  
DANIEL BURGARTH ◽  
SOUGATO BOSE ◽  
VITTORIO GIOVANNETTI
Keyword(s):  
Quantum Wires ◽  
Perfect State ◽  
Heisenberg Chain ◽  
Quantum State Transfer ◽  
General Hypothesis ◽  
Heisenberg Spin ◽  
State Transfer ◽  
Perfect State Transfer ◽  
Heisenberg Spin Chains ◽  
Identical Quantum

We present a communication protocol for chains of permanently coupled qubits which achieves perfect quantum state transfer and which is efficient with respect to the number of chains employed in the scheme. The system consists of M uncoupled identical quantum chains. Local control (gates, measurements) is only allowed at the sending/receiving end of the chains. Under a quite general hypothesis on the interaction Hamiltonian of the qubits, a theorem can be proved which shows that the receiver is able to asymptotically recover the messages by repetitive monitoring of his qubits. We show how two parallel Heisenberg spin chains can be used as quantum wires. Perfect state transfer with a probability of failure lower than P in a Heisenberg chain of N spin-1/2 particles can be achieved in a time scale of the order of [Formula: see text].

scholarly journals Periodic Graphs

10.37236/510 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Chris Godsil
Keyword(s):  
Adjacency Matrix ◽  
Infinite Family ◽  
Perfect State ◽  
State Transfer ◽  
Scalar Multiple ◽  
The Matrix ◽  
Vertex Transitive ◽  
Perfect State Transfer ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Let $X$ be a graph on $n$ vertices with adjacency matrix $A$ and let $H(t)$ denote the matrix-valued function $\exp(iAt)$. If $u$ and $v$ are distinct vertices in $X$, we say perfect state transfer from $u$ to $v$ occurs if there is a time $\tau$ such that $|H(\tau)_{u,v}|=1$. If $u\in V(X)$ and there is a time $\sigma$ such that $|H(\sigma)_{u,u}|=1$, we say $X$ is periodic at $u$ with period $\sigma$. It is not difficult to show that if the ratio of distinct non-zero eigenvalues of $X$ is always rational, then $X$ is periodic. We show that the converse holds, from which it follows that a regular graph is periodic if and only if its eigenvalues are distinct. For a class of graphs $X$ including all vertex-transitive graphs we prove that, if perfect state transfer occurs at time $\tau$, then $H(\tau)$ is a scalar multiple of a permutation matrix of order two with no fixed points. Using certain Hadamard matrices, we construct a new infinite family of graphs on which perfect state transfer occurs.

scholarly journals Some Results for the Periodicity and Perfect State Transfer

10.37236/671 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jiang Zhou ◽  
Changjiang Bu ◽  
Jihong Shen
Keyword(s):  
Adjacency Matrix ◽  
Sufficient Condition ◽  
Perfect State ◽  
Necessary And Sufficient Condition ◽  
State Transfer ◽  
Periodic Graph ◽  
Perfect State Transfer ◽  
Necessary And Sufficient

Let $G$ be a graph with adjacency matrix $A$, let $H(t)=\exp(itA)$. $G$ is called a periodic graph if there exists a time $\tau$ such that $H(\tau)$ is diagonal. If $u$ and $v$ are distinct vertices in $G$, we say that perfect state transfer occurs from $u$ to $v$ if there exists a time $\tau$ such that $|H(\tau)_{u,v}|=1$. A necessary and sufficient condition for $G$ is periodic is given. We give the existence for the perfect state transfer between antipodal vertices in graphs with extreme diameter.

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

2010 ◽  
Vol 08 (04) ◽  
pp. 641-676 ◽  
Author(s):  
ALASTAIR KAY
Keyword(s):  
Quantum State ◽  
Signal Amplification ◽  
Fixed Time ◽  
Perfect State ◽  
Entanglement Generation ◽  
State Transfer ◽  
Universal Quantum ◽  
The Subject ◽  
Perfect State Transfer ◽  
A Chain

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.

Quantum state transfer on unsymmetrical graphs via discrete-time quantum walk

2019 ◽  
Vol 34 (38) ◽  
pp. 1950317
Author(s):  
Wei-Feng Cao ◽  
Yu-Guang Yang ◽  
Dan Li ◽  
Jing-Ru Dong ◽  
Yi-Hua Zhou ◽  
...  
Keyword(s):  
Quantum State ◽  
Discrete Time ◽  
Complete Bipartite Graph ◽  
Quantum Walk ◽  
Star Graph ◽  
Perfect State ◽  
Quantum State Transfer ◽  
State Transfer ◽  
Butterfly Network ◽  
Time Quantum

Perfect state transfer can be achieved between two marked vertices of graphs like a star graph, a complete graph with self-loops and a complete bipartite graph, and two-dimensional Lattice by means of discrete-time quantum walk. In this paper, we investigate the quality of quantum state transfer between two marked vertices of an unsymmetrical graph like the butterfly network. Our numerical results support the conjecture that the fidelity of state transfer depends on the quantum state to be transferred dynamically. The butterfly network is a typical example studied in networking coding. Therefore, these results can provide a clue to the construction of quantum network coding schemes.

scholarly journals Quantum State Transfer in Coronas

10.37236/6145 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Ethan Ackelsberg ◽  
Zachary Brehm ◽  
Ada Chan ◽  
Joshua Mundinger ◽  
Christino Tamon
Keyword(s):  
Quantum State ◽  
Adjacency Matrix ◽  
Quantum Walks ◽  
Quantum State Transfer ◽  
State Transfer ◽  
Corona Product ◽  
Affect State

We study state transfer in quantum walks on graphs relative to the adjacency matrix. Our motivation is to understand how the addition of pendant subgraphs affect state transfer. For two graphs $G$ and $H$, the Frucht-Harary corona product $G \circ H$ is obtained by taking $|G|$ copies of the cone $K_{1} + H$ and by identifying the conical vertices according to $G$. Our work explores conditions under which the corona $G \circ H$ exhibits state transfer. We also describe new families of graphs with state transfer based on the corona product. Some of these constructions provide a generalization of related known results.

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