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arXiv:2610.10017

Moment Methods for Uniform Average Mixing on Strongly Regular Graphs

Musung Kang

quant-phmath.COmath.PRSubmitted Oct 7, 2026• Announced Oct 8, 2026• 0 citations
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Abstract

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.

Comments
26 pages, 1 table. Includes a correction of Theorem 5.1 and Lemma 5.2 in arXiv:1301.5889

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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 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.
Continuous-time quantum walks on graphs is a generalization of continuous-time Markov chains on discrete structures. Moore and Russell proved that the continuous-time quantum walk on the nnn-cube is instantaneous exactly uniform mixing but has no average mixing property. On complete (circulant) graphs KnK_{n}Kn​, the continuous-time quantum walk is neither instantaneous (except for n=2,3,4n=2,3,4n=2,3,4) nor average uniform mixing (except for n=2n=2n=2). We explore two natural {\em group-theoretic} generalizations of the nnn-cube as a GGG-circulant and as a bunkbed $G \rtimes \Int_{2}$, where GGG is a finite group. Analyses of these classes suggest that the nnn-cube might be special in having instantaneous uniform mixing and that non-uniform average mixing is pervasive, i.e., no memoryless property for the average limiting distribution; an implication of these graphs having zero spectral gap. But on the bunkbeds, we note a memoryless property with respect to the two partitions. We also analyze average mixing on complete paths, where the spectral gaps are nonzero.
We study the set of probability distributions visited by a continuous-time quantum walk on graphs. An edge-weighted graph GGG is {\em universal mixing} if the instantaneous or average probability distribution of the quantum walk on GGG ranges over all probability distributions on the vertices as the weights are varied over non-negative reals. The graph is {\em uniform} mixing if it visits the uniform distribution. Our results include the following: 1) All weighted complete multipartite graphs are instantaneous universal mixing. This is in contrast to the fact that no {\em unweighted} complete multipartite graphs are uniform mixing (except for the four-cycle K2,2K_{2,2}K2,2​). 2) For all n≥1n \ge 1n≥1, the weighted claw K1,nK_{1,n}K1,n​ is a minimally connected instantaneous universal mixing graph. In fact, as a corollary, the unweighted K1,nK_{1,n}K1,n​ is instantaneous uniform mixing. This adds a new family of uniform mixing graphs to a list that so far contains only the hypercubes. 3) Any weighted graph is average almost-uniform mixing unless its spectral type is sublinear in the size of the graph. This provides a nearly tight characterization for average uniform mixing on circulant graphs. 4) No weighted graphs are average universal mixing. This shows that weights do not help to achieve average universal mixing, unlike the instantaneous case. Our proofs exploit the spectra of the underlying weighted graphs and path collapsing arguments.
This paper studies uniform mixing in continuous-time quantum walks. We show that for some unitary signing σσσ, the complete graph KnσK^σ_nKnσ​ has probabilistic uniform mixing. In contrast, Ahmadi \etal (2003) proved that no complete graph has uniform mixing except for K2K_2K2​, K3K_3K3​, and K4K_4K4​. Our technique is based on a stopping rule for quantum walks which reduces global to local uniform mixing. As a corollary, we found an orientation of H(n,4)H(n,4)H(n,4) that mixes to uniform faster than any other Hamming graphs, which improves a result of Godsil and Zhan (2019). We also show that there are infinite families of oriented circulants with average uniform mixing. This is a chiral violation of a No-Go theorem due to Godsil (2013) which states that no graph has average uniform mixing except for K2K_2K2​.
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.
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.