Quantum walk with a time-dependent coin

We revisit the one dimensional discrete time quantum walk with 3-states and the Grover coin, the simplest model that exhibits localization in a quantum walk. We derive analytic expressions for the localization and a long-time approximation for the entire probability density function (PDF). We also connect the time-averaged approximation of the PDF found by Inui et. al. to a spatial average of the walk. We show that this smoothed approximation predicts moments of the real PDF accurately.

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“…Before we can apply the method, we introduce the parameter v, see Eq. (17). This allows us to write the integrals in the form 1 2π…”

Section: Appendix B: Approximation For Long Timesmentioning

confidence: 99%

Weak limit of the three-state quantum walk on the line

Falkner

Boettcher

2014

Phys. Rev. A

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“…. }, opens a much richer array of phenomena including localization and quasiperiodicity [7,8]. Here we demonstrate a time-dependent QW and use this technique to demonstrate a revival of the walker's position distribution.…”

mentioning

confidence: 92%

Experimental Quantum-Walk Revival with a Time-Dependent Coin

et al. 2015

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We demonstrate a quantum walk with time-dependent coin bias. With this technique we realize an experimental single-photon one-dimensional quantum walk with a linearly-ramped time-dependent coin flip operation and thereby demonstrate two periodic revivals of the walker distribution. In our beam-displacer interferometer, the walk corresponds to movement between discretely separated transverse modes of the field serving as lattice sites, and the time-dependent coin flip is effected by implementing a different angle between the optical axis of half-wave plate and the light propagation at each step. Each of the quantum-walk steps required to realize a revival comprises two sequential orthogonal coin-flip operators, with one coin having constant bias and the other coin having a time-dependent ramped coin bias, followed by a conditional translation of the walker. [4][5][6] plus the fundamental interest of being a natural quantized version of the ubiquitous random walk that appears in statistics, computer science, finance, physics, and chemistry. QW research has focused on evolution due to repeated applications of a time-independent unitary step operator U , but a QW with time-dependent unitary steps U (t), with discrete time t ∈ N := {0, 1, 2, . . . }, opens a much richer array of phenomena including localization and quasiperiodicity [7,8]. Here we demonstrate a time-dependent QW and use this technique to demonstrate a revival of the walker's position distribution.Rather than employing direct time-dependent control, we simulate time-dependent coin control by setting different coin parameters for different steps, which are effected in different locations along the longitudinal axis within our photonic beam-displacer interferometer (BDI) [9]. The quantum walker within the BDI is a single heralded photon produced by spontaneous parametric down conversion, and its walking degree of freedom is the set of discretely spaced transverse beam modes. The coin flip is effected by employing quarter-and half-wave plates.Our method for realizing the first time-dependent QW demonstrates the phenomenon of revivals and also opens the door to realizing a multitude of time-dependent QWs experimentally. Compared to prior work employing position-dependent control [10][11][12], our new technique decreases experimental complexity by relaxing the requirement of optical compensation. Our QW revival displays a different characteristic than typical QW properties such as ballistic spreading and localization of the walker distribution.The QW with a coin proceeds as a sequence of coin flips and then walker-coin entangling operations whereby the walker's position is displaced according to the coin state. We explain the QW now in full generality so the coin operator admits both spatial and temporal dependence. Spatially-dependent coin operations have dramatically demonstrated the realization of topological phases by QWs [4][5][6], but the time-dependent QW is, until now, only a theoretical construct and not yet explored experimentally.We employ a two...

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“…From the bath Hamiltonian (36) we obtain e iHBτ b k e −iHBτ = e −iω k τ b k and the hermitian conjugate, respectively. We will consider the limit of a large bath with a continuous spacing of oscillator frequencies.…”

Section: Bath Correlation Functionsmentioning

confidence: 99%

Preservation of positivity by dynamical coarse graining

2008

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We compare different quantum Master equations for the time evolution of the reduced density matrix. The widely applied secular approximation (rotating wave approximation) applied in combination with the Born-Markov approximation generates a Lindblad type master equation ensuring for completely positive and stable evolution and is typically well applicable for optical baths. For phonon baths however, the secular approximation is expected to be invalid. The usual Markovian master equation does not generally preserve positivity of the density matrix. As a solution we propose a coarse-graining approach with a dynamically adapted coarse graining time scale. For some simple examples we demonstrate that this preserves the accuracy of the integro-differential Born equation. For large times we analytically show that the secular approximation master equation is recovered. The method can in principle be extended to systems with a dynamically changing system Hamiltonian, which is of special interest for adiabatic quantum computation. We give some numerical examples for the spin-boson model of cases where a spin system thermalizes rapidly, and other examples where thermalization is not reached.

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Quantum walk with a time-dependent coin

Cited by 100 publications

References 32 publications

Weak limit of the three-state quantum walk on the line

Weak limit of the three-state quantum walk on the line

Experimental Quantum-Walk Revival with a Time-Dependent Coin

Preservation of positivity by dynamical coarse graining

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