# Pretty good state transfer on double stars

@article{Fan2012PrettyGS, title={Pretty good state transfer on double stars}, author={Xiaoxia Fan and Chris D. Godsil}, journal={arXiv: Combinatorics}, year={2012} }

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 $\tau$ if there is a scalar $\gamma$ such that $U(\tau)e_u = \gamma e_v$. (Since $U(t)$ is unitary, $\norm\gamma=1$.) For example, if $X$ is the $d$-cube and $u$ and $v$ are at distance $d$ then we have perfect state…

## 25 Citations

Pretty Good State Transfer on Circulant Graphs

- Mathematics, Computer ScienceElectron. J. Comb.
- 2017

It is proved 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 and the complement of such union also admits prettyGood state transfer.

Pretty good state transfer on some NEPS

- Mathematics, Computer ScienceDiscret. Math.
- 2017

The find that NEPS (Non-complete Extended P-Sum) of the path on three vertices with basis containing tuples with hamming weights of both parities do not exhibit perfect state transfer, but these NEPS admit pretty good state transfer with an additional condition.

Universal State Transfer on Graphs

- Mathematics, Physics
- 2013

A continuous-time quantum walk on a graph $G$ is given by the unitary matrix $U(t) = \exp(-itA)$, where $A$ is the Hermitian adjacency matrix of $G$. We say $G$ has pretty good state transfer between…

Strongly Cospectral Vertices

- Mathematics
- 2017

Two vertices $a$ and $b$ in a graph $X$ are cospectral if the vertex-deleted subgraphs $X\setminus a$ and $X\setminus b$ have the same characteristic polynomial. In this paper we investigate a…

More circulant graphs exhibiting pretty good state transfer

- Mathematics, Computer ScienceDiscret. Math.
- 2018

Some circulant graphs exhibiting or not exhibiting pretty good state transfer is classified, to generalize several pre-existing results on circULant graphs admitting pretty goodState transfer.

Quantum State Transfer in Coronas

- Physics, Computer ScienceElectron. J. Comb.
- 2017

This work explores conditions under which the corona of graphs G \circ H exhibits state transfer and describes new families of graphs with state transfer based on the Frucht-Harary corona product.

Pretty good state transfer on 1-sum of star graphs

- MathematicsOpen Mathematics
- 2018

Abstract Let A be the adjacency matrix of a graph G and suppose U(t) = exp(itA). We say that we have perfect state transfer in G from the vertex u to the vertex v at time t if there is a scalar γ of…

Pretty good state transfer on Cayley graphs over dihedral groups

- Computer Science, MathematicsDiscret. Math.
- 2020

If n is a power of 2, then Cay ( D n, S ) exhibits pretty good state transfer for some subset S in D n , some concrete constructions are provided and it is shown that this is basically the only case for a non-integral Cayley graph Cay (D n , S ) to have PGST.

Continuous Time Quantum Walks on Graphs: Group State Transfer

- Physics, Mathematics
- 2021

We introduce the concept of group state transfer on graphs, summarize its relationship to other concepts in the theory of quantum walks, set up a basic theory, and discuss examples. Let X be a graph…

Designing pretty good state transfer via isospectral reductions

- Physics, Mathematics
- 2020

We present an algorithm to design networks that feature pretty good state transfer (PGST), which is of interest for high-fidelity transfer of information in quantum computing. Realizations of PGST…

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A survey of some of the work on perfect state transfer and related questions is offered, almost entirely on the mathematics, and the interactions between graph theory andperfect state transfer are surveyed.

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It is shown that, for a class of graphs X including all vertex-transitive graphs, if perfect state transfer occurs at time τ , then H(τ) is a scalar multiple of a permutation matrix of order two with no fixed points.

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This work determines the exact number of qubits in unmodulated chains (with an XY Hamiltonian) that permit transfer with a fidelity arbitrarily close to 1, a phenomenon called pretty good state transfer and highlights the potential of quantum spin system dynamics for reinterpreting questions about the arithmetic structure of integers and primality.

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It is shown that 2log3N is the maximal perfect communication distance for hypercube geometries if one allows fixed but different couplings between the qubits, then perfect state transfer can be achieved over arbitrarily long distances in a linear chain.

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The natural notion of almost perfect state transfer (APST) is examined. It is applied to the modelling of efficient quantum wires with the help of $XX$ spin chains. It is shown that APST occurs in…

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The means for simultaneous scouring of metal surfaces contains a waste product in manufacture of fodder yeast, citric acid, ammonium citrate, aqueous solution of sodium gluconate, sulphonated ricinic…