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Mixing of Quantum Walks on Generalized Hypercubes | Ana Best
; Markus Kliegl
; Shawn Mead-Gluchacki
; Christino Tamon
; | Date: |
18 Aug 2008 | Abstract: | We study continuous-time quantum walks on graphs which generalize the
hypercube. The only known family of graphs whose quantum walk instantaneously
mixes to uniform is the Hamming graphs with small arities. We show that quantum
uniform mixing on the hypercube is robust under the addition of perfect
matchings but not much else. Our specific results include:
(1) The graph obtained by augmenting the hypercube with an additive matching
is instantaneous uniform mixing whenever the parity of the matching is even,
but with a slower mixing time. This strictly includes Moore-Russell’s result on
the hypercube.
(2) The class of Hamming graphs is not uniform mixing if and only if its
arity is greater than 5. This is a tight characterization of quantum uniform
mixing on Hamming graphs; previously, only the status of arity less than 5 was
known.
(3) The bunkbed graph B(A[f]), defined by a hypercube-circulant matrix A and
a Boolean function f, is not uniform mixing if the Fourier transform of f has
small support. This explains why the hypercube is uniform mixing and why the
join of two hypercubes is not. | Source: | arXiv, 0808.2382 | Services: | Forum | Review | PDF | Favorites |
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