arXiv:0808.0510 · Physical Review A

Quantum networks on cubelike graphs

Anna Bernasconi, Chris D. Godsil, Simone Severini B

Published Nov 12, 2008• 87 citations

Abstract

Cubelike graphs are the Cayley graphs of the elementary Abelian group ${\mathbb{Z}}_{2}^{n}$ (e.g., the hypercube is a cubelike graph). We study perfect state transfer between two particles in quantum networks modeled by a large class of cubelike graphs. This generalizes the results of Christandl et al. [Phys. Rev. Lett. 92, 187902 (2004)] and Facer et al. [Phys. Rev. A 92, 187902 (2008)].

DOI: 10.1103/physreva.78.052320

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arXiv:0808.0510 · Physical Review A

Quantum networks on cubelike graphs

Anna Bernasconi, Chris D. Godsil, Simone Severini B

Published Nov 12, 2008• 87 citations

arXiv HTML PDF

Abstract

Cubelike graphs are the Cayley graphs of the elementary Abelian group

{\mathbb{Z}}_{2}^{n}

(e.g., the hypercube is a cubelike graph). We study perfect state transfer between two particles in quantum networks modeled by a large class of cubelike graphs. This generalizes the results of Christandl et al. [Phys. Rev. Lett. 92, 187902 (2004)] and Facer et al. [Phys. Rev. A 92, 187902 (2008)].

DOI: 10.1103/physreva.78.052320

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Perfect state transfer and Cayley presentations

Arnbjörg Soffía Árnadóttir, Krystal Guo

We study perfect state transfer on Cayley graphs from the point of view that state transfer is a property of a graph and not of a group. This paper is a bridge between the classical question about isomorphic Cayley graphs of non-isomorphic groups and quantum walks on graphs. We show that a Cayley graph of a group with an abelian subgroup of index two is a Cayley graph of an abelian group under any one of three hypotheses, two drawn from the theory of isomorphic Cayley graphs. A statement of the same kind holds for extraspecial groups: every Cayley graph of an extraspecial $p$ -group of order $p^{2n+1}$ with a conjugacy-closed connection set is a Cayley graph of $Z_p^{2n+1}$ . From these results we deduce that every explicit construction of perfect state transfer in the six papers we survey, on dihedral, dicyclic, generalized dihedral, $V_{8n}$ and extraspecial $2$ -groups, is a non-abelian presentation of an abelian Cayley graph. Moreover, we show that a non-abelian group with an abelian subgroup of index two admits a connected Cayley graph with perfect state transfer if and only if its order is divisible by four. Genuinely non-abelian examples do exist. We prove that, for every odd prime power $q\ge 5$ , the $SL(2,q)$ graph of Pantangi and Sin, which they showed to admit perfect state transfer, is a Cayley graph of no abelian group; to our knowledge, this is the first infinite family of Cayley graphs with perfect state transfer provably admitting no abelian Cayley presentation. We also construct an infinite family of Cayley graphs with peak state transfer and determine all regular subgroups of the automorphism group of every member. An appendix records a census of the connected vertex-transitive graphs with perfect state transfer on at most $30$ vertices.We study perfect state transfer on Cayley graphs from the point of view that state transfer is a property of a graph and not of a group. This paper is a bridge between the classical question about isomorphic Cayley graphs of non-isomorphic groups and quantum walks on graphs. We show that a Cayley graph of a group with an abelian subgroup of index two is a Cayley graph of an abelian group under any one of three hypotheses, two drawn from the theory of isomorphic Cayley graphs. A statement of the same kind holds for extraspecial groups: every Cayley graph of an extraspecial $p$-group of order $p^{2n+1}$ with a conjugacy-closed connection set is a Cayley graph of $Z_p^{2n+1}$. From these results we deduce that every explicit construction of perfect state transfer in the six papers we survey, on dihedral, dicyclic, generalized dihedral, $V_{8n}$ and extraspecial $2$-groups, is a non-abelian presentation of an abelian Cayley graph. Moreover, we show that a non-abelian group with an abelian subgroup of index two admits a connected Cayley graph with perfect state transfer if and only if its order is divisible by four. Genuinely non-abelian examples do exist. We prove that, for every odd prime power $q\ge 5$, the $SL(2,q)$ graph of Pantangi and Sin, which they showed to admit perfect state transfer, is a Cayley graph of no abelian group; to our knowledge, this is the first infinite family of Cayley graphs with perfect state transfer provably admitting no abelian Cayley presentation. We also construct an infinite family of Cayley graphs with peak state transfer and determine all regular subgroups of the automorphism group of every member. An appendix records a census of the connected vertex-transitive graphs with perfect state transfer on at most $30$ vertices.

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We completely classify connected Cayley graphs over finite abelian groups that exhibit perfect state transfer in Grover walks. First, we show that a Cayley graph over any finite group exhibiting perfect state transfer has valency $1$ , $2$ , or $4$ . For valency $4$ , we prove that an abelian Cayley graph $\operatorname{Cay}(Γ,S)$ exhibits perfect state transfer if and only if $S=\{a,-a,a+h,-a+h\}$ , where $h$ has order $2$ , and $4\nmid |Γ|/2$ . Combining this characterization with the valency-one and valency-two cases, we explicitly determine all pairs $(Γ,S)$ for which $\operatorname{Cay}(Γ,S)$ exhibits perfect state transfer. Our results extend the classification of circulant graphs of valency at most four obtained by Kubota and Yoshino [Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks, \emph{J. Comb. Theory Ser. A.}, 216 (2025) 106064] to Cayley graphs over arbitrary finite abelian groups and to all valencies.We completely classify connected Cayley graphs over finite abelian groups that exhibit perfect state transfer in Grover walks. First, we show that a Cayley graph over any finite group exhibiting perfect state transfer has valency $1$, $2$, or $4$. For valency $4$, we prove that an abelian Cayley graph $\operatorname{Cay}(Γ,S)$ exhibits perfect state transfer if and only if $S=\{a,-a,a+h,-a+h\}$, where $h$ has order $2$, and $4\nmid |Γ|/2$. Combining this characterization with the valency-one and valency-two cases, we explicitly determine all pairs $(Γ,S)$ for which $\operatorname{Cay}(Γ,S)$ exhibits perfect state transfer. Our results extend the classification of circulant graphs of valency at most four obtained by Kubota and Yoshino [Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks, \emph{J. Comb. Theory Ser. A.}, 216 (2025) 106064] to Cayley graphs over arbitrary finite abelian groups and to all valencies.

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We propose a class of qubit networks that admit perfect state transfer of any two-dimensional quantum state in a fixed period of time. We further show that such networks can distribute arbitrary entangled states between two distant parties, and can, by using such systems in parallel, transmit the higher-dimensional systems states across the network. Unlike many other schemes for quantum computation and communication, these networks do not require qubit couplings to be switched on and off. When restricted to $N$ -qubit spin networks of identical qubit couplings, we show that $2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{3}N$ is the maximal perfect communication distance for hypercube geometries. Moreover, 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. This paper expands and extends the work done by Christandl et al., Phys. Rev. Lett. 92, 187902 (2004).We propose a class of qubit networks that admit perfect state transfer of any two-dimensional quantum state in a fixed period of time. We further show that such networks can distribute arbitrary entangled states between two distant parties, and can, by using such systems in parallel, transmit the higher-dimensional systems states across the network. Unlike many other schemes for quantum computation and communication, these networks do not require qubit couplings to be switched on and off. When restricted to $N$-qubit spin networks of identical qubit couplings, we show that $2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{3}N$ is the maximal perfect communication distance for hypercube geometries. Moreover, 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. This paper expands and extends the work done by Christandl et al., Phys. Rev. Lett. 92, 187902 (2004).

arXiv:quant-ph/0411020Physical Review AMar 14, 2005519 citations

Cubelike graphs are the Cayley graphs of the elementary Abelian group

{\mathbb{Z}}_{2}^{n}

p

-group of order

p^{2n+1}

with a conjugacy-closed connection set is a Cayley graph of

Z_p^{2n+1}

. From these results we deduce that every explicit construction of perfect state transfer in the six papers we survey, on dihedral, dicyclic, generalized dihedral,

V_{8n}

and extraspecial

2

-groups, is a non-abelian presentation of an abelian Cayley graph. Moreover, we show that a non-abelian group with an abelian subgroup of index two admits a connected Cayley graph with perfect state transfer if and only if its order is divisible by four. Genuinely non-abelian examples do exist. We prove that, for every odd prime power

q\ge 5

, the

SL(2,q)

graph of Pantangi and Sin, which they showed to admit perfect state transfer, is a Cayley graph of no abelian group; to our knowledge, this is the first infinite family of Cayley graphs with perfect state transfer provably admitting no abelian Cayley presentation. We also construct an infinite family of Cayley graphs with peak state transfer and determine all regular subgroups of the automorphism group of every member. An appendix records a census of the connected vertex-transitive graphs with perfect state transfer on at most

30

vertices.

\mathcal{U}(t) = e^{iAt}

, where

A

is the adjacency matrix of the graph. An important notion in the quantum random walk is the transfer of a quantum state from one vertex to another. If the fidelity of the transfer is unity, we call it a perfect state transfer. Many graph families have been shown to admit PST or periodicity, including cubelike graphs. These graphs are unweighted. In this paper, we generalize the PST or periodicity of cubelike graphs to that of weighted cubelike graphs. We characterize the weights for which they admit PST or show periodicity, both at time

t=\fracπ{2}

We develop the work of Christandl et al. [Phys. Rev. A 71, 032312 (2005)], to show how a

d

-hypercube homogenous network can be dressed by additional links to perfectly route quantum information between any given input and output nodes in a duration that is independent of the routing chosen and, surprisingly, the size of the network.

1

2

, or

4

. For valency

4

, we prove that an abelian Cayley graph

\operatorname{Cay}(Γ,S)

exhibits perfect state transfer if and only if

S=\{a,-a,a+h,-a+h\}

, where

h

has order

2

, and

4\nmid |Γ|/2

. Combining this characterization with the valency-one and valency-two cases, we explicitly determine all pairs

(Γ,S)

for which

\operatorname{Cay}(Γ,S)

exhibits perfect state transfer. Our results extend the classification of circulant graphs of valency at most four obtained by Kubota and Yoshino [Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks, \emph{J. Comb. Theory Ser. A.}, 216 (2025) 106064] to Cayley graphs over arbitrary finite abelian groups and to all valencies.

N

-qubit spin networks of identical qubit couplings, we show that

2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{3}N

is the maximal perfect communication distance for hypercube geometries. Moreover, 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. This paper expands and extends the work done by Christandl et al., Phys. Rev. Lett. 92, 187902 (2004).