We consider the continuous-time quantum walk defined on the adjacency matrix of a graph. At each instant, the walk defines a mixing matrix which is doubly-stochastic. The average of the mixing matrices contains relevant information about the quantum walk and about the graph. We show that it is the matrix of transformation of the orthogonal projection onto the commutant algebra of the adjacency matrix, restricted to diagonal matrices. Using this formulation of the average mixing matrix, we find connections between its rank and automorphisms of the graph.We consider the continuous-time quantum walk defined on the adjacency matrix of a graph. At each instant, the walk defines a mixing matrix which is doubly-stochastic. The average of the mixing matrices contains relevant information about the quantum walk and about the graph. We show that it is the matrix of transformation of the orthogonal projection onto the commutant algebra of the adjacency matrix, restricted to diagonal matrices. Using this formulation of the average mixing matrix, we find connections between its rank and automorphisms of the graph.

The Electronic Journal of CombinatoricsOct 19, 20185 citations

Diagonal entries of the average mixing matrix

Chris D. Godsil, Krystal Guo, Mariia Sobchuk

We study the diagonal entries of the average mixing matrix of continuous quantum walks. The average mixing matrix is a graph invariant; it is the sum of the Schur squares of spectral idempotents of the Hamiltonian. It is non-negative, doubly stochastic and positive semi-definite. We investigate the diagonal entries of this matrix. We study the graphs for which the trace of the average mixing matrix is maximum or minimum and we classify those which are maximum. We give two constructions of graphs whose average mixing matrices have constant diagonal.We study the diagonal entries of the average mixing matrix of continuous quantum walks. The average mixing matrix is a graph invariant; it is the sum of the Schur squares of spectral idempotents of the Hamiltonian. It is non-negative, doubly stochastic and positive semi-definite. We investigate the diagonal entries of this matrix. We study the graphs for which the trace of the average mixing matrix is maximum or minimum and we classify those which are maximum. We give two constructions of graphs whose average mixing matrices have constant diagonal.

arXiv:1910.02039Oct 4, 20191 citation

Average mixing of continuous quantum walks

Chris D. Godsil

If $X$ is a graph with adjacency matrix $A$ , then we define $H(t)$ to be the operator $\exp(itA)$ . The Schur (or entrywise) product $H(t)\circ H(-t)$ is a doubly stochastic matrix and, because of work related to quantum computing, we are concerned the \textsl{average mixing matrix}. This can be defined as the limit of $C^{-1} \int_0^C H(t)\circ H(-t)\dt$ as $C\to\infty$ . We establish some of the basic properties of this matrix, showing that it is positive semidefinite and that its entries are always rational. We find that for paths and cycles this matrix takes on a surprisingly simple form, thus for the path it is a linear combination of $I$ , $J$ (the all-ones matrix), and a permutation matrix.If $X$ is a graph with adjacency matrix $A$, then we define $H(t)$ to be the operator $\exp(itA)$. The Schur (or entrywise) product $H(t)\circ H(-t)$ is a doubly stochastic matrix and, because of work related to quantum computing, we are concerned the \textsl{average mixing matrix}. This can be defined as the limit of $C^{-1} \int_0^C H(t)\circ H(-t)\dt$ as $C\to\infty$. We establish some of the basic properties of this matrix, showing that it is positive semidefinite and that its entries are always rational. We find that for paths and cycles this matrix takes on a surprisingly simple form, thus for the path it is a linear combination of $I$, $J$ (the all-ones matrix), and a permutation matrix.

arXiv:1103.2578Mar 14, 20115 citations

Irrational Quantum Walks

Gabriel Coutinho, Pedro Ferreira Baptista, Chris D. Godsil +2 more

Abstract. The adjacency matrix of a graph [Formula: see text] is the Hamiltonian for a continuous-time quantum walk on the vertices of [Formula: see text]. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behavior of the walk is typically not periodic. In this paper, we develop a theory to exactly study any quantum walk generated by an integral Hamiltonian, and we put emphasis on those with irrational eigenvalues—what we call irrational quantum walks. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost perfect) state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix and discuss possible applications of these results. Throughout the paper, we emphasize the interplay between different fields of mathematics applied to the study of quantum walks.Abstract. The adjacency matrix of a graph [Formula: see text] is the Hamiltonian for a continuous-time quantum walk on the vertices of [Formula: see text]. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behavior of the walk is typically not periodic. In this paper, we develop a theory to exactly study any quantum walk generated by an integral Hamiltonian, and we put emphasis on those with irrational eigenvalues—what we call irrational quantum walks. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost perfect) state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix and discuss possible applications of these results. Throughout the paper, we emphasize the interplay between different fields of mathematics applied to the study of quantum walks.

arXiv:2208.08971SIAM Journal on Applied Algebra and GeometryJul 25, 20235 citations

Uniform Mixing of Quantum Walks and Association Schemes

Natalie Mullin

In recent years quantum algorithms have become a popular area of mathematical research. Farhi and Gutmann introduced the concept of a quantum walk in 1998. In this thesis we investigate mixing properties of continuous-time quantum walks from a mathematical perspective. We focus on the connections between mixing properties and association schemes. \n \nThere are three main goals of this thesis. Our primary goal is to develop the algebraic groundwork necessary to systematically study mixing properties of continuous-time quantum walks on regular graphs. Using these tools we achieve two additional goals: we construct new families of graphs that admit uniform mixing, and we prove that other families of graphs never admit uniform mixing. \n \nWe begin by introducing association schemes and continuous-time quantum walks. Within this framework we develop specific algebraic machinery to tackle the uniform mixing problem. Our main algebraic result shows that if a graph has an irrational eigenvalue, then its transition matrix has at least one transcendental coordinate at all nonzero times. \n \nNext we study algebraic varieties related to uniform mixing to determine information about the coordinates of the corresponding transition matrices. Combining this with our main algebraic result we prove that uniform mixing does not occur on even cycles or prime cycles. However, we show that the probability distribution of a quantum walk on a prime cycle gets arbitrarily close to uniform. \n \nFinally we consider uniform mixing on Cayley graphs of elementary abelian groups. We utilize graph quotients to connect the mixing properties of these graphs to Hamming graphs. This enables us to find new results about uniform mixing on Cayley graphs of certain elementary abelian groups.In recent years quantum algorithms have become a popular area of mathematical research. Farhi and Gutmann introduced the concept of a quantum walk in 1998. In this thesis we investigate mixing properties of continuous-time quantum walks from a mathematical perspective. We focus on the connections between mixing properties and association schemes. \n \nThere are three main goals of this thesis. Our primary goal is to develop the algebraic groundwork necessary to systematically study mixing properties of continuous-time quantum walks on regular graphs. Using these tools we achieve two additional goals: we construct new families of graphs that admit uniform mixing, and we prove that other families of graphs never admit uniform mixing. \n \nWe begin by introducing association schemes and continuous-time quantum walks. Within this framework we develop specific algebraic machinery to tackle the uniform mixing problem. Our main algebraic result shows that if a graph has an irrational eigenvalue, then its transition matrix has at least one transcendental coordinate at all nonzero times. \n \nNext we study algebraic varieties related to uniform mixing to determine information about the coordinates of the corresponding transition matrices. Combining this with our main algebraic result we prove that uniform mixing does not occur on even cycles or prime cycles. However, we show that the probability distribution of a quantum walk on a prime cycle gets arbitrarily close to uniform. \n \nFinally we consider uniform mixing on Cayley graphs of elementary abelian groups. We utilize graph quotients to connect the mixing properties of these graphs to Hamming graphs. This enables us to find new results about uniform mixing on Cayley graphs of certain elementary abelian groups.

Sep 26, 20133 citations

Complex Hadamard matrices, instantaneous uniform mixing and cubes

Ada Chan

We study the continuous-time quantum walks on graphs in the adjacency algebra of the n -cube and its related distance regular graphs. For k ≥ 2 , we find graphs in the adjacency algebra of ( 2 k + 2 - 8 ) -cube that admit instantaneous uniform mixing at time π / 2 k and graphs that have perfect state transfer at time π / 2 k . We characterize the folded n -cubes, the halved n -cubes and the folded halved n -cubes whose adjacency algebra contains a complex Hadamard matrix. We obtain the same conditions for the characterization of these graphs admitting instantaneous uniform mixing.We study the continuous-time quantum walks on graphs in the adjacency algebra of the n -cube and its related distance regular graphs. For k ≥ 2 , we find graphs in the adjacency algebra of ( 2 k + 2 - 8 ) -cube that admit instantaneous uniform mixing at time π / 2 k and graphs that have perfect state transfer at time π / 2 k . We characterize the folded n -cubes, the halved n -cubes and the folded halved n -cubes whose adjacency algebra contains a complex Hadamard matrix. We obtain the same conditions for the characterization of these graphs admitting instantaneous uniform mixing.

arXiv:1305.5811Algebraic CombinatoricsJun 2, 202024 citations

X

is a graph with adjacency matrix

A

, then we define

H(t)

to be the operator

\exp(itA)

. The Schur (or entrywise) product

H(t)\circ H(-t)

is a doubly stochastic matrix and, because of work related to quantum computing, we are concerned the \textsl{average mixing matrix}. This can be defined as the limit of $C^{-1} \int_0^C H(t)\circ H(-t)\dt$ as

C\to\infty

. We establish some of the basic properties of this matrix, showing that it is positive semidefinite and that its entries are always rational. We find that for paths and cycles this matrix takes on a surprisingly simple form, thus for the path it is a linear combination of

I

J

(the all-ones matrix), and a permutation matrix.