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Register a new userInvariance in quantum walks with time-dependent coin operators
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Authors
Miquel Montero
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In this paper we unveil some features of a discrete-time quantum walk on the line whose coin depends on the temporal variable. After considering the most general form of the unitary coin operator, we focus on the role played by the two phase factors that one can incorporate there, and show how both terms influence the evolution of the system. A closer analysis reveals that the probabilistic properties of the motion of the walker remain unaltered when the update rule of these phases is chosen adequately. This invariance is based on a symmetry with consequences not yet fully explored.
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Although quantum walks exhibit peculiar properties that distinguish them from random walks, classical behavior can be recovered in the asymptotic limit by destroying the coherence of the pure state associated to the quantum system. Here I show that this is not the only way: I introduce a quantum walk driven by an inhomogeneous, time-dependent coin operator, which mimics the statistical properties of a random walk at all time scales. The quantum particle undergoes unitary evolution and, in fact, the high correlation evidenced by the components of the wave function can be used to revert the outcome of an accidental measurement of its chirality.
We provide an explanation of recent experimental results of Xue et al., where full revivals in a time-dependent quantum walk model with a periodically changing coin are found. Using methods originally developed for electric walks with a space-dependent, rather than a time-dependent coin, we provide a full explanation of the observations of Xue et al. We extend the analysis from periodic time-dependence to quasi-periodic behaviour with periods incommensurate to the step size. Spectral analysis, one of the principal tools for the study of electric walks, fails for time-dependent systems, but we find qualitative propagation behaviour of the time-dependent system in close analogy to the electric case.
We introduce quantum walks with a time-dependent coin, and show how they include, as a particular case, the generalized quantum walk recently studied by Wojcik et al. {[}Phys. Rev. Lett. textbf{93}, 180601(2004){]} which exhibits interesting dynamical localization and quasiperiodic dynamics. Our proposal allows for a much easier implementation of this particular rich dynamics than the original one. Moreover, it allows for an additional control on the walk, which can be used to compensate for phases appearing due to external interactions. To illustrate its feasibility, we discuss an example using an optical cavity. We also derive an approximated solution in the continuous limit (long--wavelength approximation) which provides physical insight about the process.
Quantum walk (QW) is the quantum analog of the random walk. QW is an integral part of the development of numerous quantum algorithms. Hence, an in-depth understanding of QW helps us to grasp the quantum algorithms. We revisit the one-dimensional discrete-time QW and discuss basic steps in detail by incorporating the most general coin operator. We investigate the impact of each parameter of the general coin operator on the probability distribution of the quantum walker. We show that by tuning the parameters of the general coin, one can regulate the probability distribution of the walker. We provide an algorithm for the one-dimensional quantum walk driven by the general coin operator. The study conducted on general coin operator also includes the popular coins -- Hadamard, Grover, and Fourier coins.
The control of quantum walk is made particularly transparent when the initial state is expressed in terms of the eigenstates of the coin operator. We show that the group-velocity density acquires a much simpler form when expressed in this basis. This allows us to obtain a much deeper understanding of the role of the initial coin state on the dynamics of quantum walks and control it. We find that the eigenvectors of the coin result in an extremal regime of a quantum walk. The approach is illustrated on two examples of quantum walks on a line.
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