While a number of researchers have previously investigated the relationship between graph theory and quantum probability, the current work explores a new perspective. The approach of this paper is to begin with an arbitrary graph having no previously established relationship to quantum probability and to use that graph to construct a quantum probability space in which moments of quantum random variables reveal information about the graph's structure. Given an arbitrary finite graph and arbitrary odd integer m ⩾ 3, fermion annihilation operators are used to construct a family of quantum random variables whose m th moments correspond to the graph's m -cycles. The approach is then generalized to recover a graph's m -cycles for any integer m ⩾ 3 by defining nilpotent adjacency operators in terms of null-square generators of an infinite-dimensional Abelian algebra. It is shown that ordering the vertices of a simple graph induces a canonical decomposition Ψ = Ψ + + Ψ − on any nilpotent adjacency operator Ψ. The work concludes with applications to Markov chains and random graphs.While a number of researchers have previously investigated the relationship between graph theory and quantum probability, the current work explores a new perspective. The approach of this paper is to begin with an arbitrary graph having no previously established relationship to quantum probability and to use that graph to construct a quantum probability space in which moments of quantum random variables reveal information about the graph's structure. Given an arbitrary finite graph and arbitrary odd integer m ⩾ 3, fermion annihilation operators are used to construct a family of quantum random variables whose m th moments correspond to the graph's m -cycles. The approach is then generalized to recover a graph's m -cycles for any integer m ⩾ 3 by defining nilpotent adjacency operators in terms of null-square generators of an infinite-dimensional Abelian algebra. It is shown that ordering the vertices of a simple graph induces a canonical decomposition Ψ = Ψ + + Ψ − on any nilpotent adjacency operator Ψ. The work concludes with applications to Markov chains and random graphs.

Journal of Physics A Mathematical and TheoreticalApr 2, 200817 citations

Weighted Graph Theory Representation of Quantum Information Inspired by Lie Algebras

Abdelilah Belhaj, A. Belhaj, Larbi Machkouri +2 more

Borrowing ideas from the relation between simply laced Lie algebras and Dynkin diagrams, a weighted graph theory representation of quantum information is addressed. In this way, the density matrix of a quantum state can be interpreted as a signless Laplacian matrix of an associated graph. Using similarities with root systems of simply laced Lie algebras, one-qubit theory is analyzed in some details and is found to be linked to a non-oriented weighted graph having two vertices. Moreover, this one-qubit theory is generalized to n-qubits. In this representation, quantum gates correspond to graph weight operations preserving the probability condition. A speculation from string theory, via D-brane quivers, is also given.Borrowing ideas from the relation between simply laced Lie algebras and Dynkin diagrams, a weighted graph theory representation of quantum information is addressed. In this way, the density matrix of a quantum state can be interpreted as a signless Laplacian matrix of an associated graph. Using similarities with root systems of simply laced Lie algebras, one-qubit theory is analyzed in some details and is found to be linked to a non-oriented weighted graph having two vertices. Moreover, this one-qubit theory is generalized to n-qubits. In this representation, quantum gates correspond to graph weight operations preserving the probability condition. A speculation from string theory, via D-brane quivers, is also given.

arXiv:1609.03534Sep 12, 20167 citations

Probability and Quantum Theory

Barry Loewer

Springer eBooksJan 1, 19943 citations

Arc search in graphs via Szegedy walks

Sho Kubota, Kiyoto Yoshino

This paper studies the search for a single arc in a graph using the Szegedy walk. Arc search can be interpreted as finding a quantum particle not only in its position but also with a specific internal state. The quantum walk employed in this study is essentially the model proposed by Segawa and Yoshie for the purpose of edge search. First, we investigate how the symmetry of a graph is reflected in its time evolution matrix, and provide a sufficient condition under which the success probability of the search is independent of the marked arc. In particular, we prove that if a graph is arc-transitive, the success probability is independent of the choice of the marked arc. Next, we analyze path and cycle graphs and show that the quantum search is ineffective for these graphs, whereas it performs well for complete bipartite graphs $K_{n,n}$ . These results provide a theoretical foundation for studying arc and edge searches on various graphs, while also suggesting new problems concerning the eigenvalue analysis of edge-signed graphs in spectral graph theory.This paper studies the search for a single arc in a graph using the Szegedy walk. Arc search can be interpreted as finding a quantum particle not only in its position but also with a specific internal state. The quantum walk employed in this study is essentially the model proposed by Segawa and Yoshie for the purpose of edge search. First, we investigate how the symmetry of a graph is reflected in its time evolution matrix, and provide a sufficient condition under which the success probability of the search is independent of the marked arc. In particular, we prove that if a graph is arc-transitive, the success probability is independent of the choice of the marked arc. Next, we analyze path and cycle graphs and show that the quantum search is ineffective for these graphs, whereas it performs well for complete bipartite graphs $K_{n,n}$. These results provide a theoretical foundation for studying arc and edge searches on various graphs, while also suggesting new problems concerning the eigenvalue analysis of edge-signed graphs in spectral graph theory.

quant-pharXiv:2604.19134Apr 22, 20260 citations

Probability and Quantum Theory

E. B. Davies

Oxford University Press eBooksJun 28, 20070 citations

K_{n,n}

. These results provide a theoretical foundation for studying arc and edge searches on various graphs, while also suggesting new problems concerning the eigenvalue analysis of edge-signed graphs in spectral graph theory.