Complex Hadamard Matrices, Instantaneous Uniform Mixing and Cubes

Chan, Ada

doi:10.48550/arxiv.1305.5811

Cited by 11 publications

(16 citation statements)

References 13 publications

(22 reference statements)

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“…Theorem 8.5, Theorem 8.6 and Theorem 9.1 show that, for an arbitrarily large integer k, we can construct families of Cayley graphs over Z d q that admit uniform mixing at time 2π/q k , where q = 3, 4. These examples extend the results of Mullin [9] and Chan [3].…”

Section: Introductionsupporting

confidence: 88%

“…When d is clear from the context, we drop the superscript and write p r (s). The following theorem due to Chan [3] shows that whether a graph admits uniform mixing depends only on its spectrum and the eigenvalues of the Bose-Mesner algebra containing its adjacency matrix.…”

Section: Hamming Schemesmentioning

confidence: 99%

“…It turns out that the conditions on the eigenvalues can be further simplified using Lemma 6.2. From now on, we focus on the Hamming schemes H(d, 3) and H(d, 4), as the examples in H(d, 2) are already given in [3].…”

Section: Sufficient Conditions For Uniform Mixing In Hamming Schemesmentioning

confidence: 99%

“…Uniform mixing on graphs is rare, and almost all the known examples are Cayley graphs. The current list of Cayley graphs contains the complete graphs K 2 , K 3 and K 4 [1], the Hamming graphs H(d, 2), H(d, 3) and H(d, 4) [2,8], the Paley graph of order nine [6], some strongly regular graphs from regular symmetric Hadamard matrices [6], some linear Cayley graphs over Z d 2 , Z d 3 and Z d 4 [3,9], and the Cartesian product of graphs which admit uniform mixing at the same time. These graphs share the following features.…”

Section: Introductionmentioning

confidence: 99%

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Uniform Mixing on Cayley Graphs

Godsil¹,

Zhan²

2017

Electron. J. Combin.

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We provide new examples of Cayley graphs on which the quantum walks reach uniform mixing. Our first result is a complete characterization of all 2(d+2)-regular Cayley graphs over Z d 3 that admit uniform mixing at time 2π/9. Our second result shows that for every integer k ≥ 3, we can construct Cayley graphs over Z d q that admit uniform mixing at time 2π/q k , where q = 3, 4.We also find the first family of irregular graphs, the Cartesian powers of the star K 1,3 , that admit uniform mixing.

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Section: Introductionsupporting

confidence: 88%

Section: Hamming Schemesmentioning

confidence: 99%

Section: Sufficient Conditions For Uniform Mixing In Hamming Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Uniform Mixing on Cayley Graphs

Godsil¹,

Zhan²

2017

Electron. J. Combin.

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“…Another area for future work is finding additional examples of graphs in association schemes that admit uniform mixing. Recent work by Chan shows that such graphs exist in certain schemes derived from n-cubes [8].…”

Section: Future Workmentioning

confidence: 99%

Uniform Mixing and Association Schemes

Godsil

Mullin²,

Roy³

2017

Electron. J. Combin.

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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 ≥ 3. Using a result of Haagerup, we show that uniform mixing does not occur on C p for any prime p such that p ≥ 5. In contrast to this result, we see that ǫ-uniform mixing occurs on C p for all primes p.

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Discretization of continuous-time quantum walks via the staggered model with Hamiltonians

Coutinho

Portugal²

2018

Nat Comput

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We characterize a close connection between the continuous-time quantum-walk model and a discrete-time quantum-walk version, based on the staggered model with Hamiltonians in a class of Cayley graphs, which can be considered as a discretization of continuous-time quantum walks. This connection provides examples of perfect state transfer and instantaneous uniform mixing in the staggered model. On the other hand, we provide some more examples of perfect state transfer and instantaneous uniform mixing in the staggered model that cannot be reproduced by the continuous-time model. * Acknowledges the support of grants FAPESP 15/16339-2 and FAPESP 03447-6. † Acknowledges the support of CNPq grant 474143/2013-9. arXiv:1701.03423v2 [quant-ph]

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Complex Hadamard Matrices, Instantaneous Uniform Mixing and Cubes

Cited by 11 publications

References 13 publications

Uniform Mixing on Cayley Graphs

Uniform Mixing on Cayley Graphs

Uniform Mixing and Association Schemes

Discretization of continuous-time quantum walks via the staggered model with Hamiltonians

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