Localizing genuine multiparty entanglement in noisy stabilizer states

Characterizing large noisy multiparty quantum states using genuine multiparty entanglement is a challenging task. In this paper, we calculate lower bounds of genuine multiparty entanglement localized over a chosen multiparty subsystem of multi-qubit stabilizer states in the noiseless and noisy scenario. In the absence of noise, adopting a graph-based technique, we perform the calculation for arbitrary graph states as representatives of the stabilizer states, and show that the graph operations required for the calculation has a polynomial scaling with the system size. As demonstrations, we compute the localized genuine multiparty entanglement over subsystems of large graphs having linear, ladder, and square structures. We also extend the calculation for graph states subjected to single-qubit Markovian or non-Markovian Pauli noise on all qubits, and demonstrate, for a specific lower bound of the localizable genuine multiparty entanglement corresponding to a specific Pauli measurement setup, the existence of a critical noise strength beyond which all of the post measured states are biseparable. The calculation is also useful for arbitrary large stabilizer states under noise due to the local unitary connection between stabilizer states and graph states. We demonstrate this by considering a toric code defined on a square lattice, and computing a lower bound of localizable genuine multiparty entanglement over a non-trivial loop of the code. Similar to the graph states, we show the existence of the critical noise strength in this case also, and discuss its interesting features.Comment: 36 pages, 21 figures, 2 table

Hierarchical organization of functional connectivity in the mouse brain: a complex network approach

This paper represents a contribution to the study of the brain functional connectivity from the perspective of complex networks theory. More specifically, we apply graph theoretical analyses to provide evidence of the modular structure of the mouse brain and to shed light on its hierarchical organization. We propose a novel percolation analysis and we apply our approach to the analysis of a resting-state functional MRI data set from 41 mice. This approach reveals a robust hierarchical structure of modules persistent across different subjects. Importantly, we test this approach against a statistical benchmark (or null model) which constrains only the distributions of empirical correlations. Our results unambiguously show that the hierarchical character of the mouse brain modular structure is not trivially encoded into this lower-order constraint. Finally, we investigate the modular structure of the mouse brain by computing the Minimal Spanning Forest, a technique that identifies subnetworks characterized by the strongest internal correlations. This approach represents a faster alternative to other community detection methods and provides a means to rank modules on the basis of the strength of their internal edges.Comment: 11 pages, 9 figure

Sign and Basis Invariant Networks for Spectral Graph Representation Learning

Many machine learning tasks involve processing eigenvectors derived from data. Especially valuable are Laplacian eigenvectors, which capture useful structural information about graphs and other geometric objects. However, ambiguities arise when computing eigenvectors: for each eigenvector $v$, the sign flipped $-v$ is also an eigenvector. More generally, higher dimensional eigenspaces contain infinitely many choices of basis eigenvectors. These ambiguities make it a challenge to process eigenvectors and eigenspaces in a consistent way. In this work we introduce SignNet and BasisNet -- new neural architectures that are invariant to all requisite symmetries and hence process collections of eigenspaces in a principled manner. Our networks are universal, i.e., they can approximate any continuous function of eigenvectors with the proper invariances. They are also theoretically strong for graph representation learning -- they can approximate any spectral graph convolution, can compute spectral invariants that go beyond message passing neural networks, and can provably simulate previously proposed graph positional encodings. Experiments show the strength of our networks for molecular graph regression, learning expressive graph representations, and learning implicit neural representations on triangle meshes. Our code is available at https://github.com/cptq/SignNet-BasisNet .Comment: 35 page

Distributed Graph Neural Network Training with Periodic Stale Representation Synchronization

Despite the recent success of Graph Neural Networks, it remains challenging to train a GNN on large graphs with millions of nodes and billions of edges, which are prevalent in many graph-based applications. Traditional sampling-based methods accelerate GNN training by dropping edges and nodes, which impairs the graph integrity and model performance. Differently, distributed GNN algorithms accelerate GNN training by utilizing multiple computing devices and can be classified into two types: "partition-based" methods enjoy low communication costs but suffer from information loss due to dropped edges, while "propagation-based" methods avoid information loss but suffer from prohibitive communication overhead caused by the neighbor explosion. To jointly address these problems, this paper proposes DIGEST (DIstributed Graph reprEsentation SynchronizaTion), a novel distributed GNN training framework that synergizes the complementary strength of both categories of existing methods. We propose to allow each device to utilize the stale representations of its neighbors in other subgraphs during subgraph parallel training. This way, our method preserves global graph information from neighbors to avoid information loss and reduce communication costs. Our convergence analysis demonstrates that DIGEST enjoys a state-of-the-art convergence rate. Extensive experimental evaluation on large, real-world graph datasets shows that DIGEST achieves up to 21.82 speedups without compromising performance compared to state-of-the-art distributed GNN training frameworks.Comment: Preprint: 20 pages, 9 figure