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 from$n$to$\lceil\frac{n}{2}\rceil$, and from$n$to 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 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 fromnto⌈2n⌉, and fromnto 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.
Stabilizer states, along with Clifford manipulations (unitary transformations and measurements) thereof—despite being efficiently simulable on a classical computer—are an important tool in quantum information processing, with applications to quantum computing, error correction, and networking. Graph states, defined on a graph, are a special class of stabilizer states that are central to measurement-based quantum computing, all-photonic quantum repeaters, distributed quantum computing, and entanglement distribution in a network. All stabilizer states are local-Clifford equivalent to graph states. In this paper, we review the stabilizer framework and extend it by incorporating general stabilizer measurements such as multi-qubit joint projections. We provide an explicit procedure—using Karnaugh maps from Boolean algebra—for converting arbitrary stabilizer gates into tableau operations of the CHP formalism for efficient stabilizer manipulations. We derive graphical rules for arbitrary stabilizer manipulations of graph states, including multi-qubit stabilizer projections and unitaries. We implement the graphical rulebook resulting from above into a MATLAB simulator with a GUI. A user of this tool, e.g., for research in quantum networks, will not require any background in quantum information or the stabilizer framework.
Cluster states and graph states in general offer a useful model of the stabilizer formalism and a path toward the development of measurement-based quantum computation. Their defining structure – the stabilizer group – encodes all possible correlations that can be observed during measurement. The measurement outcomes which are consistent with the stabilizer structure make error correction possible. Here, we leverage both properties to design feasible families of states that can be used as robust building blocks of quantum computation. This procedure reduces the effect of experimentally relevant noise models on the extraction of smaller entangled states from the larger noisy graph state. In particular, we study the extraction of Bell pairs from linearly extended graph states – this has the immediate consequence for state teleportation across the graph. We show that robust entanglement can be extracted by proper design of the linear graph with only a minimal overhead of the physical qubits. This scenario is relevant to systems in which the entanglement can be created between neighboring sites. The results shown in this work provide a mathematical framework for noise reduction in measurement-based quantum computation. With proper connectivity structures, the effect of noise can be minimized for a large class of realistic noise processes.
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.
While stabilizer tableaus have proven exceptionally useful as a descriptive tool for additive quantum codes, they offer little guidance for concrete constructions or coding algorithm analysis. We introduce a representation of stabilizer codes as graphs with certain structures. Specifically, the graphs take a semi-bipartite form where input nodes map to output nodes, such that output nodes may connect to each other but input nodes may not. Intuitively, the graph's input-output edges represent information propagation of the encoder, while output-output edges represent the code's entanglement. We prove that this graph representation is in bijection with tableaus and give an efficient compilation algorithm that transforms tableaus into graphs. We show that this map is efficiently invertible, which gives a universal recipe for code construction by finding graphs with nice properties. The graph representation gives insight into both code construction and algorithms. To the former, we argue that graphs provide a flexible platform for building codes. We construct several constant-size codes and several infinite code families. We also use graphs to extend the quantum Gilbert-Varshamov bound to a three-way distance-rate-weight trade-off. To the latter, we show that key coding algorithms, distance approximation, weight reduction, and decoding, are unified as instances of a single optimization game on a graph. Moreover, key code properties such as distance, weight, and encoding circuit depth, are all controlled by the graph degree. We give efficient algorithms for producing encoding circuits whose depths scale with the degree and for implementing certain logical diagonal and Clifford gates with reduced depth. Finally, we find an efficient decoding algorithm for certain classes of graphs. These results give evidence that graphs are useful for the study of quantum computing and its implementations.
Graph states are present in quantum information and found applications ranging from quantum network protocols (like secret sharing) to measurement based quantum computing. In this paper, we extend the notion of graph states, which can be regarded as pure quantum graph states, or as homogeneous quadratic Boolean functions associated to simple undirected graphs, to quantum states based on mixed graphs (graphs which allow both directed and undirected edges), obtaining mixed quantum states, which are defined by matrices associated to the measurement of homogeneous quadratic Boolean functions in some (ancillary) variables. In our main result, we describe the extended graph state as the sum of terms of a commutative subgroup of the stabilizer group of the corresponding mixed graph with the edges' directions reversed.
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.