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arXiv:1504.03302 · Physical Review A

Group structures and representations of graph states

Jun-Yi Wu, Hermann Kampermann, Dagmar Bruß

Published Jul 20, 2015• 5 citations
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Abstract

A special configuration of graph state stabilizers, which contains only Pauli ${\ensuremath{\sigma}}_{X}$ operators, is studied. The vertex sets $\ensuremath{\xi}$ associated with such configurations are defined as what we call $X$ chains of graph states. The $X$ chains of a general graph state can be determined efficiently. They form a group structure such that one can obtain the explicit representation of graph states in the $X$ basis via the so-called $X$-chain factorization diagram. We show that graph states with different $X$-chain groups can have different probability distributions of $X$-measurement outcomes, which allows one to distinguish certain graph states with $X$ measurements. We provide an approach to find the Schmidt decomposition of graph states in the $X$ basis. The existence of $X$ chains in a subsystem facilitates error correction in the entanglement localization of graph states. In all of these applications, the difficulty of the task decreases with increasing number of $X$ chains. Furthermore, we show that the overlap of two graph states can be efficiently determined via $X$ chains, while its computational complexity with other known methods increases exponentially.

DOI
10.1103/physreva.92.012322

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A special configuration of graph state stabilizers, which contains only Pauli \ensuremathσX{\ensuremath{\sigma}}_{X}\ensuremathσX​ operators, is studied. The vertex sets \ensuremathξ\ensuremath{\xi}\ensuremathξ associated with such configurations are defined as what we call XXX chains of graph states. The XXX chains of a general graph state can be determined efficiently. They form a group structure such that one can obtain the explicit representation of graph states in the XXX basis via the so-called XXX-chain factorization diagram. We show that graph states with different XXX-chain groups can have different probability distributions of XXX-measurement outcomes, which allows one to distinguish certain graph states with XXX measurements. We provide an approach to find the Schmidt decomposition of graph states in the XXX basis. The existence of XXX chains in a subsystem facilitates error correction in the entanglement localization of graph states. In all of these applications, the difficulty of the task decreases with increasing number of XXX chains. Furthermore, we show that the overlap of two graph states can be efficiently determined via XXX chains, while its computational complexity with other known methods increases exponentially.
A special configuration of graph state stabilizers, which contains only Pauli σX\sigma_XσX​ operators, is studied. The vertex sets ξ\xiξ associated with such configurations are defined as the X-chains of graph states. The X-chains of a general graph state can be determined efficiently. With the help of X-chains, one obtains the explicit representation of graph states in the X-basis via the so-called X-chain factorization diagram. We show that graph states with different X-chains can have different probability distributions of X-measurement outcomes, which allows to distinguish certain graph states with X-measurements. We provide an approach to find the Schmidt decomposition of graph states in the X-basis. The existence of X-chains in a subsystem facilitates error correction in the entanglement localization of graph states. In all these applications, the difficulty of the task decreases with increasing number of X-chains. Furthermore, we show that the overlap of two graph states can be efficiently determined via X-chains, while its computational complexity with other known methods increases exponentially.
Abstract The representation of graph states in the X -basis as well as the calculation of graph state overlaps can efficiently be performed by using the concept of X -chains (Wu et al 2015 Phys. Rev. A 92 012322 ). We present a necessary and sufficient criterion for X -chains and show that they can efficiently be determined by the Bareiss algorithm. An analytical approach for searching X -chain groups of a graph state is proposed. Furthermore we generalize the concept of X -chains to so-called Euler chains, whose induced subgraphs are Eulerian. This approach helps to determine if a given vertex set is an X -chain and we show how Euler chains can be used in the construction of multipartite Bell inequalities for graph states.
Graph states are the main computational building blocks of measurement-based computation and a useful tool for error correction in the gate model architecture. The graph states form a class of quantum states which are eigenvectors for the abelian group of stabilizer operators. They own topological properties, arising from their graph structure, including the presence of highly connected nodes, called hubs. Starting from hub nodes, we demonstrate how to efficiently express a graph state through the generators of the stabilizer group. We provide examples by expressing the ring and the star topology, for which the number of stabilizers reduces fromnnnto⌈n2⌉\lceil\frac{n}{2}\rceil⌈2n​⌉, and fromnnnto 1, respectively. We demonstrate that the graph states can be generated by a subgroup of the stabilizer group. Therefore, we provide an algebraic framework to manipulate the graph states with a reduced number of stabilizers.
Graph states are widely used in quantum information theory, including entanglement theory, quantum error correction, and one-way quantum computing. Graph states have a nice structure related to a certain graph, which is given by either a stabilizer group or an encoding circuit, both can be directly given by the graph. To generalize graph states, whose stabilizer groups are abelian subgroups of the Pauli group, one approach taken is to study non-abelian stabilizers. In this work, we propose to generalize graph states based on the encoding circuit, which is completely determined by the graph and a Hadamard matrix. We study the entanglement structures of these generalized graph states and show that they are all maximally mixed locally. We also explore the relationship between the equivalence of Hadamard matrices and local equivalence of the corresponding generalized graph states. This leads to a natural generalization of the Pauli (X, Z) pairs, which characterizes the local symmetries of these generalized graph states. Our approach is also naturally generalized to construct graph quantum codes which are beyond stabilizer codes.
Graph states are multiparticle entangled states that correspond to mathematical graphs, where the vertices of the graph take the role of quantum spin systems and edges represent Ising interactions. They are many-body spin states of distributed quantum systems that play a significant role in quantum error correction, multiparty quantum communication, and quantum computation within the framework of the one-way quantum computer. We characterize and quantify the genuine multiparticle entanglement of such graph states in terms of the Schmidt measure, to which we provide upper and lower bounds in graph theoretical terms. Several examples and classes of graphs will be discussed, where these bounds coincide. These examples include trees, cluster states of different dimensions, graphs that occur in quantum error correction, such as the concatenated [7,1,3]-CSS code, and a graph associated with the quantum Fourier transform in the one-way computer. We also present general transformation rules for graphs when local Pauli measurements are applied, and give criteria for the equivalence of two graphs up to local unitary transformations, employing the stabilizer formalism. For graphs of up to seven vertices we provide complete characterization modulo local unitary transformations and graph isomorphisms.
Recently, there are tremendous developments on the number of controllable qubits in several quantum computing systems. For these implementations, it is crucial to determine the entanglement structure of the prepared multipartite quantum state as a basis for further information processing tasks. In reality, evaluation of a multipartite state is in general a very challenging task owing to the exponential increase of the Hilbert space with respect to the number of system components. In this work, we propose a systematic method using very few local measurements to detect multipartite entanglement structures based on the graph state --- one of the most important classes of quantum states for quantum information processing. Thanks to the close connection between the Schmidt coefficient and quantum entropy in graph states, we develop a family of efficient witness operators to detect the entanglement between subsystems under any partitions and hence the entanglement intactness. We show that the number of local measurements equals to the chromatic number of the underlying graph, which is a constant number, independent of the number of qubits. In reality, the optimization problem involved in the witnesses can be challenging with large system size. For several widely-used graph states, such as 1-D and 2-D cluster states and the Greenberger-Horne-Zeilinger state, by taking advantage of the area law of entanglement entropy, we derive analytical solutions for the witnesses, which only employ two local measurements. Our method offers a standard tool for entanglement structure detection to benchmark multipartite quantum systems.