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arXiv:0907.5359 · Nuclear Physics B

Direct computation of scattering matrices for general quantum graphs

Vincent Caudrelier, E. Ragoucy

Published Oct 14, 2009• 16 citations
arXivHTMLPDF
DOI
10.1016/j.nuclphysb.2009.10.012

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Representing graphs as quantum states is becoming an increasingly important approach to study entanglement of mixed states, alternate to the standard linear algebraic density matrix-based approach of study. In this paper, we propose a general weighted directed graph framework for investigating properties of a large class of quantum states which are defined by three types of Laplacian matrices associated with such graphs. We generalize the standard framework of defining density matrices from simple connected graphs to density matrices using both combinatorial and signless Laplacian matrices associated with weighted directed graphs with complex edge weights and with/without self-loops. We also introduce a new notion of Laplacian matrix, which we call signed Laplacian matrix associated with such graphs. We produce necessary and/or sufficient conditions for such graphs to correspond to pure and mixed quantum states. Using these criteria, we finally determine the graphs whose corresponding density matrices represent entangled pure states which are well known and important for quantum computation applications. We observe that all these entangled pure states share a common combinatorial structure.
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.
Spectral graph theory is a branch of mathematics that studies the relationships between the eigenvectors and eigenvalues of Laplacian and adjacency matrices and their associated graphs. The Variational Quantum Eigen- solver (VQE) algorithm was proposed as a hybrid quantum/classical algorithm that is used to quickly determine the ground state of a Hamiltonian, and more generally, the lowest eigenvalue of a matrix M ∈ Rnxn. There are many interesting problems associated with the spectral decompositions of associated matrices, such as partitioning, embedding, and the determination of other properties. In this paper, we will expand upon the VQE algorithm to analyze the spectra of directed and undirected graphs. We evaluate runtime and accuracy comparisons (empirically and theoretically) between different choices of ansatz parameters, graph sizes, graph densities, and matrix types, and demonstrate the effectiveness of our approach on Rigetti's QCS platform on graphs of up to 64 vertices, finding eigenvalues of adjacency and Laplacian matrices. We finally make direct comparisons to classical performance with the Quantum Virtual Machine (QVM) in the appendix, observing a superpolynomial runtime improvement of our algorithm when run using a quantum computer.*