Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1301.5889 · The Electronic Journal of Combinatorics

Uniform Mixing and Association Schemes

Chris D. Godsil, Natalie Mullin, Aidan Roy

Published Aug 11, 2017• 17 citations
arXivHTMLPDF

Abstract

We consider continuous-time quantum walks on distance-regular graphs. Using results about the existence of complex Hadamard matrices in association schemes, we determine which of these graphs have quantum walks that admit uniform mixing.First we apply a result due to Chan to show that the only strongly regular graphs that admit instantaneous uniform mixing are the Paley graph of order nine and certain graphs corresponding to regular symmetric Hadamard matrices with constant diagonal. Next we prove that if uniform mixing occurs on a bipartite graph $X$ with $n$ vertices, then $n$ is divisible by four. We also prove that if $X$ is bipartite and regular, then $n$ is the sum of two integer squares. Our work on bipartite graphs implies that uniform mixing does not occur on $C_{2m}$ for $m \geq 3$. Using a result of Haagerup, we show that uniform mixing does not occur on $C_p$ for any prime $p$ such that $p \geq 5$. In contrast to this result, we see that $\epsilon$-uniform mixing occurs on $C_p$ for all primes $p$.

DOI
10.37236/4745

Related papers

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.

•Sep 26, 2013•3 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.

arXiv:1305.5811•Algebraic Combinatorics•Jun 2, 2020•24 citations

Moment Methods for Uniform Average Mixing on Strongly Regular Graphs

Musung Kang

We study continuous-time quantum walks on connected strongly regular graphs that are not complete, observed at a random time drawn from a freely chosen probability law. Uniform average mixing (UAM) asks for a law under which every averaged transition probability equals $1/n$, where $n$ is the number of vertices. On a strongly regular graph this is equivalent to two affine constraints on three cosine moments. We construct a bounded, compactly supported time density for every strongly regular graph with nonintegral eigenvalues. For integral spectra we give an exact finite Toeplitz criterion and its Hankel form. Every averaged mixing matrix of such a graph is realized by at most two observation times. Three elementary inequalities on the moment line, which also give a short proof of Chan's classification of complex Hadamard matrices in the Bose-Mesner algebra, lead to a determination of all strongly regular graphs that admit UAM. Apart from the conference graphs of nonsquare order and the graphs with instantaneous uniform mixing, these are the members of two infinite families of parameter sets and their complements, and for them we give explicit laws with two observation times. The Petersen graph and its complement are the smallest members. A strongly regular graph with UAM admits a bounded time density exactly when it has no instantaneous uniform mixing. We also correct the classification of instantaneous uniform mixing on strongly regular graphs by Godsil, Mullin and Roy. Its sign condition excludes the halved $5$-cube, which mixes uniformly at time $π/4$. With the order $4θ^2$ read literally, its parity condition also excludes the Clebsch graph and includes the parameters $(36,14,4,6)$, for which no time law gives uniform average mixing.

quant-pharXiv:2610.10017•Oct 8, 2026•0 citations

Uniform mixing and εεε-uniform mixing on cycles

Xiwang Cao, Cuiwen Zhu

We study continuous-time quantum walks on cycles. We prove two complementary results. Firstly, the cycle $C_9$ does not admit uniform mixing at any time. Using the similar idea and Dickson polynomials, we prove that $C_{15}$ does not admit uniform mixing at any time neither. Secondly, for every prime $p$, we show that the cycle $C_{p^2}$ admits $ε$-uniform mixing.

math.COCross-listedarXiv:2607.04207•Jul 7, 2026•0 citations

On mixing in continuous-time quantum walks on some circulant graphs

A. Ahmadi, Ryan Belk, Christino Tamon +1 more

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous exactly uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs of the symmetric group lack this mixing property as well.

Quantum Information and Computation•Nov 1, 2003•54 citations

Mixing in Continuous Quantum Walks on Graphs

Ahmadi, Amir, Ryan Belk, Christino Tamon +1 more

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that a continuous quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs lack this mixing property as well.

arXiv:quant-ph/0209106•Sep 18, 2002•5 citations
We consider continuous-time quantum walks on distance-regular graphs. Using results about the existence of complex Hadamard matrices in association schemes, we determine which of these graphs have quantum walks that admit uniform mixing.First we apply a result due to Chan to show that the only strongly regular graphs that admit instantaneous uniform mixing are the Paley graph of order nine and certain graphs corresponding to regular symmetric Hadamard matrices with constant diagonal. Next we prove that if uniform mixing occurs on a bipartite graph XXX with nnn vertices, then nnn is divisible by four. We also prove that if XXX is bipartite and regular, then nnn is the sum of two integer squares. Our work on bipartite graphs implies that uniform mixing does not occur on C2mC_{2m}C2m​ for m≥3m \geq 3m≥3. Using a result of Haagerup, we show that uniform mixing does not occur on CpC_pCp​ for any prime ppp such that p≥5p \geq 5p≥5. In contrast to this result, we see that ϵ\epsilonϵ-uniform mixing occurs on CpC_pCp​ for all primes ppp.
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.
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 continuous-time quantum walks on connected strongly regular graphs that are not complete, observed at a random time drawn from a freely chosen probability law. Uniform average mixing (UAM) asks for a law under which every averaged transition probability equals 1/n1/n1/n, where nnn is the number of vertices. On a strongly regular graph this is equivalent to two affine constraints on three cosine moments. We construct a bounded, compactly supported time density for every strongly regular graph with nonintegral eigenvalues. For integral spectra we give an exact finite Toeplitz criterion and its Hankel form. Every averaged mixing matrix of such a graph is realized by at most two observation times. Three elementary inequalities on the moment line, which also give a short proof of Chan's classification of complex Hadamard matrices in the Bose-Mesner algebra, lead to a determination of all strongly regular graphs that admit UAM. Apart from the conference graphs of nonsquare order and the graphs with instantaneous uniform mixing, these are the members of two infinite families of parameter sets and their complements, and for them we give explicit laws with two observation times. The Petersen graph and its complement are the smallest members. A strongly regular graph with UAM admits a bounded time density exactly when it has no instantaneous uniform mixing. We also correct the classification of instantaneous uniform mixing on strongly regular graphs by Godsil, Mullin and Roy. Its sign condition excludes the halved 555-cube, which mixes uniformly at time π/4π/4π/4. With the order 4θ24θ^24θ2 read literally, its parity condition also excludes the Clebsch graph and includes the parameters (36,14,4,6)(36,14,4,6)(36,14,4,6), for which no time law gives uniform average mixing.
We study continuous-time quantum walks on cycles. We prove two complementary results. Firstly, the cycle C9C_9C9​ does not admit uniform mixing at any time. Using the similar idea and Dickson polynomials, we prove that C15C_{15}C15​ does not admit uniform mixing at any time neither. Secondly, for every prime ppp, we show that the cycle Cp2C_{p^2}Cp2​ admits εεε-uniform mixing.
Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous exactly uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs of the symmetric group lack this mixing property as well.
Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that a continuous quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs lack this mixing property as well.