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arXiv:1506.06457

Spectral mapping theorem of an abstract quantum walk

Yusuke Higuchi, Etsuo Segawa, Akito Suzuki

Published Jun 22, 2015• 9 citations
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Abstract

Given two Hilbert spaces, $\mathcal{H}$ and $\mathcal{K}$, we introduce an abstract unitary operator $U$ on $\mathcal{H}$ and its discriminant $T$ on $\mathcal{K}$ induced by a coisometry from $\mathcal{H}$ to $\mathcal{K}$ and a unitary involution on $\mathcal{H}$. In a particular case, these operators $U$ and $T$ become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between $U$ and $T$ via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpiński lattice, which is pure point and has a Cantor-like structure.

DOI
10.48550/arxiv.1506.06457

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Given two Hilbert spaces, H\mathcal{H}H and K\mathcal{K}K, we introduce an abstract unitary operator UUU on H\mathcal{H}H and its discriminant TTT on K\mathcal{K}K induced by a coisometry from H\mathcal{H}H to K\mathcal{K}K and a unitary involution on H\mathcal{H}H. In a particular case, these operators UUU and TTT become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between UUU and TTT via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpiński lattice, which is pure point and has a Cantor-like structure.
We consider an abstract quantum walk defined by a unitary evolution operator UUU, which acts on a Hilbert space decomposed into a direct sum of Hilbert spaces {Hv}v∈V\{\mathcal{H}_v \}_{v \in V}{Hv​}v∈V​. We show that such UUU naturally defines a directed graph GUG_UGU​ and the probability of finding a quantum walker on GUG_UGU​. The asymptotic property of an abstract quantum walker is governed by the generator HHH of UUU such that Un=einHU^n = e^{inH}Un=einH. We derive the generator of an evolution of the form U=S(2dA∗dA−1)U = S(2d_A^* d_A -1)U=S(2dA∗​dA​−1), a generalization of the Szegedy evolution operator. Here dAd_AdA​ is a boundary operator and SSS a shift operator.
We propose a twisted Szegedy walk for estimating the limit behavior of a discrete-time quantum walk on a crystal lattice, an infinite abelian covering graph, whose notion was introduced by [14]. First, we show that the spectrum of the twisted Szegedy walk on the quotient graph can be expressed by mapping the spectrum of a twisted random walk onto the unit circle. Secondly, we show that the spatial Fourier transform of the twisted Szegedy walk on a finite graph with appropriate parameters becomes the Grover walk on its infinite abelian covering graph. Finally, as an application, we show that if the Betti number of the quotient graph is strictly greater than one, then localization is ensured with some appropriated initial state. We also compute the limit density function for the Grover walk on Zd\mathbb{Z}^dZd with flip flop shift, which implies the coexistence of linear spreading and localization. We partially obtain the abstractive shape of the limit density function: the support is within the ddd-dimensional sphere of radius 1/d1/\sqrt{d}1/d​, and 2d2^d2d singular points reside on the sphere's surface.
For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in the work of Higuchi et al. (e-print arXiv:1506.06457) [see also E. Segawa and A. Suzuki, Quantum Stud.: Math. Found. 3, 11 (2016)]. In this paper, as an application of the spectral mapping theorem, we investigate the spectrum of a one-dimensional split-step quantum walk. We give a criterion for when there are no eigenvalues around ±1 in terms of a discriminant operator. We also provide a criterion for when eigenvalues ±1 exist in terms of birth eigenspaces. Moreover, we prove that eigenvectors from the birth eigenspaces decay exponentially at spatial infinity and that the birth eigenspaces are robust against perturbations.
We make and generalize the observation that summing of probability amplitudes of a discrete-time quantum walk over partitions of the walking graph consistent with the step operator results in a unitary evolution on the reduced graph which is also a quantum walk. Since the effective walking graph of the projected walk is not necessarily simpler than the original, this may bring new insights into the dynamics of some kinds of quantum walks using known results from thoroughly studied cases like Euclidean lattices. We use abstract treatment of the walking space and walker displacements in aim for a generality of the presented statements. Using this approach we also identify some pathological cases in which the projection mapping breaks down. For walks on lattices, the operation typically results in quantum walks with hyper-dimensional coin spaces. Such walks can, conversely, be viewed as projections of walks on inaccessible, larger spaces, and their properties can be inferred from the parental walk. We show that this is is the case for a lazy quantum walk, a walk with large coherent jumps and a walk on a circle with a twisted boundary condition. We also discuss the relation of this theory to the time-multiplexing optical implementations of quantum walks. Moreover, this manifestly irreversible operation can, in some cases and with a minor adjustment, be undone, and a quantum walk can be reconstructed from a set of its projections.
We define coined Quantum Walks on the infinite rooted binary tree given by unitary operators U(C)U(C)U(C) on an associated infinite dimensional Hilbert space, depending on a unitary coin matrix C∈U(3)C\in U(3)C∈U(3), and study their spectral properties. For circulant unitary coin matrices CCC, we derive an equation for the Carathéodory function associated to the spectral measure of a cyclic vector for U(C)U(C)U(C). This allows us to show that for all circulant unitary coin matrices, the spectrum of the Quantum Walk has no singular continuous component. Furthermore, for coin matrices CCC which are orthogonal circulant matrices, we show that the spectrum of the Quantum Walk is absolutely continuous, except for four coin matrices for which the spectrum of U(C)U(C)U(C) is pure point.