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Register a new userStability of point spectrum for three-state quantum walks on a line
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Evolution operators of certain quantum walks possess, apart from the continuous part, also point spectrum. The existence of eigenvalues and the corresponding stationary states lead to partial trapping of the walker in the vicinity of the origin. We analyze the stability of this feature for three-state quantum walks on a line subject to homogenous coin deformations. We find two classes of coin operators that preserve the point spectrum. These new classes of coins are generalizations of coins found previously by different methods and shed light on the rich spectrum of coins that can drive discrete-time quantum walks.
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We introduce an analytically treatable spin decoherence model for quantum walk on a line that yields the exact position probability distribution of an unbiased classical random walk at all-time scales. This spin decoherence model depicts a quantum channel in which simultaneous bit and phase flip operator is applied at random on the coin state. Based on this result we claim that there exist certain quantum channels that can produce exact classical statistical properties for a given one-dimensional quantum walk. Moreover, from the perspective of quantum computing, decoherence model introduced in this study may have useful algorithmic applications when it is applied on quantum walks with non-local initial states.
We perform a finite-time scaling analysis over the detrapping point of a three-state quantum walk on the line. The coin operator is parameterized by $rho$ that controls the wavepacket spreading velocity. The input state prepared at the origin is set as symmetric linear combination of two eigenstates of the coin operator with a characteristic mixing angle $theta$, one of them being the component that results in full spreading occurring at $theta_c(rho)$ for which no fraction of the wavepacket remains trapped near the initial position. We show that relevant quantities such as the survival probability and the participation ratio assume single parameter scaling forms at the vicinity of the detrapping angle $theta_c$. In particular, we show that the participation ratio grows linearly in time with a logarithmic correction, thus shedding light on previous reports of sublinear behavior.
We analyze the asymptotic scaling of persistence of unvisited sites for quantum walks on a line. In contrast to the classical random walk there is no connection between the behaviour of persistence and the scaling of variance. In particular, we find that for a two-state quantum walks persistence follows an inverse power-law where the exponent is determined solely by the coin parameter. Moreover, for a one-parameter family of three-state quantum walks containing the Grover walk the scaling of persistence is given by two contributions. The first is the inverse power-law. The second contribution to the asymptotic behaviour of persistence is an exponential decay coming from the trapping nature of the studied family of quantum walks. In contrast to the two-state walks both the exponent of the inverse power-law and the decay constant of the exponential decay depend also on the initial coin state and its coherence. Hence, one can achieve various regimes of persistence by altering the initial condition, ranging from purely exponential decay to purely inverse power-law behaviour.
We analyze two families of three-state quantum walks which show the localization effect. We focus on the role of the initial coin state and its coherence in controlling the properties of the quantum walk. In particular, we show that the description of the walk simplifies considerably when the initial coin state is decomposed in the basis formed by the eigenvectors of the coin operator. This allows us to express the limit distributions in a much more convenient form. Consequently, striking features which are hidden in the standard basis description are easily identified. Moreover, the dependence of moments of the position distribution on the initial coin state can be analyzed in full detail. In particular, we find that in the eigenvector basis the even moments and the localization probability at the origin depend only on incoherent combination of probabilities. In contrast, odd moments and localization outside the origin are affected by the coherence of the initial coin state.
In this note, we consider a physical system given by a two-dimensional quantum walk in an external magnetic field. In this setup, we show that both the topological structure as well as its type depend sensitively on the value of the magnetic flux $Phi$: while for $Phi/(2{pi})$ rational the spectrum is known to consist of bands, we show that for $Phi/(2{pi})$ irrational the spectrum is a zero-measure Cantor set and the spectral measures have no pure point part.
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