scholarly journals Stability of point spectrum for three-state quantum walks on a line

2014 ◽  
Vol 14 (13&14) ◽  
pp. 1219-1226
Author(s):  
Martin Štefaňák ◽  
Iva Bezděková ◽  
Igor Jex ◽  
Stephen M. Barnett
Keyword(s):  
Point Spectrum ◽  
Quantum Walks ◽  
Evolution Operators ◽  
Continuous Part ◽  
Time Quantum ◽  
The Rich ◽  
The Stability ◽  
Rich Spectrum ◽  
Partial Trapping ◽  
Shed Light

Evolution operators of certain quantum walks possess, apart from the continuous part, also a 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.

scholarly journals MIXING OF QUANTUM WALKS ON GENERALIZED HYPERCUBES

2008 ◽  
Vol 06 (06) ◽  
pp. 1135-1148 ◽  
Author(s):  
ANA BEST ◽  
MARKUS KLIEGL ◽  
SHAWN MEAD-GLUCHACKI ◽  
CHRISTINO TAMON
Keyword(s):  
Mixing Time ◽  
Quantum Walk ◽  
Circulant Matrix ◽  
Quantum Walks ◽  
Spectral Structure ◽  
The Status ◽  
Time Quantum ◽  
Hamming Graphs ◽  
The Rich ◽  
The Fourier Transform

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: • The graph obtained by augmenting the hypercube with an additive matching x ↦ x ⊕ η is instantaneous uniform mixing whenever |η| is even, but with a slower mixing time. This strictly includes the result of Moore and Russell1 on the hypercube. • The class of Hamming graphs H(n,q) is not uniform mixing if and only if q ≥ 5. This is a tight characterization of quantum uniform mixing on Hamming graphs; previously, only the status of H(n,q) with q < 5 was known. • The bunkbed graph [Formula: see text] whose adjacency matrix is I ⊗ Qn + X ⊗ Af, where Af is a [Formula: see text]-circulant matrix defined by a Boolean function f, is not uniform mixing if the Fourier transform of f has support of size smaller than 2n-1. This explains why the hypercube is uniform mixing and why the join of two hypercubes is not. Our work exploits the rich spectral structure of the generalized hypercubes and relies heavily on Fourier analysis of group-circulants.

scholarly journals CONTINUOUS-TIME QUANTUM WALKS AND TRAPPING

2010 ◽  
Vol 20 (02) ◽  
pp. 271-279 ◽  
Author(s):  
ELENA AGLIARI ◽  
OLIVER MÜLKEN ◽  
ALEXANDER BLUMEN
Keyword(s):  
Energy Transfer ◽  
Continuous Time ◽  
Tight Binding ◽  
Quantum Walks ◽  
Random Environments ◽  
Quantum Mechanical ◽  
Evolution Operators ◽  
Time Quantum ◽  
Photosynthetic Antenna ◽  
Binding Type

Recent findings suggest that processes such as the excitonic energy transfer through the photosynthetic antenna display quantal features, aspects known from the dynamics of charge carriers along polymer backbones. Hence, in modeling energy transfer one has to leave the classical, master-equation-type formalism and advance towards an increasingly quantum-mechanical picture, while still retaining a local description of the complex network of molecules involved in the transport, say through a tight-binding approach. Interestingly, the continuous time random walk (CTRW) picture, widely employed in describing transport in random environments, can be mathematically reformulated to yield a quantum-mechanical Hamiltonian of tight-binding type; the procedure uses the mathematical analogies between time-evolution operators in statistical and in quantum mechanics: The result are continuous-time quantum walks (CTQWs). However, beyond these formal analogies, CTRWs and CTQWs display vastly different physical properties. In particular, here we focus on trapping processes on a ring and show, both analytically and numerically, that distinct configurations of traps (ranging from periodical to random) yield strongly different behaviors for the quantal mean survival probability, while classically (under ordered conditions) we always find an exponential decay at long times.

scholarly journals Impact of Lattice Vibrations on the Dynamics of a Spinor Atom-Optics Kicked Rotor

2019 ◽  
Vol 4 (1) ◽  
pp. 10 ◽  
Author(s):  
Caspar Groiseau ◽  
Alexander Wagner ◽  
Gil Summy ◽  
Sandro Wimberger
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Atom Optics ◽  
Lattice Vibrations ◽  
Noise Sources ◽  
The Road ◽  
Spin Degree ◽  
Kicked Rotor ◽  
Time Quantum ◽  
The Stability

We investigate the effect of amplitude and phase noise on the dynamics of a discrete-time quantum walk and its related evolution. Our findings underline the robustness of the motion with respect to these noise sources, and can explain the stability of quantum walks that has recently been observed experimentally. This opens the road to measure topological properties of an atom-optics double kicked rotor with an additional internal spin degree of freedom.

scholarly journals Forward Rate Bias in Developed and Developing Countries: More Risky Not Less Rational

Econometrics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 43
Author(s):  
Michael D. Goldberg ◽  
Olesia Kozlova ◽  
Deniz Ozabaci
Keyword(s):  
Developing Countries ◽  
Forward Rate ◽  
Currency Markets ◽  
Developed Country ◽  
Slope Coefficient ◽  
Break Points ◽  
Time Periods ◽  
The Stability ◽  
Widespread View ◽  
Shed Light

This paper examines the stability of the Bilson–Fama regression for a panel of 55 developed and developing countries. We find multiple break points for nearly every country in our panel. Subperiod estimates of the slope coefficient show a negative bias during some time periods and a positive bias during other time periods in nearly every country. The subperiod biases display two key patterns that shed light on the literature’s linear regression findings. The results point toward the importance of risk in currency markets. We find that risk is greater for developed country markets. The evidence undercuts the widespread view that currency returns are predictable or that developed country markets are less rational.

scholarly journals Critical slowing down in circuit quantum electrodynamics

2021 ◽  
Vol 7 (21) ◽  
pp. eabe9492
Author(s):  
Paul Brookes ◽  
Giovanna Tancredi ◽  
Andrew D. Patterson ◽  
Joseph Rahamim ◽  
Martina Esposito ◽  
...  
Keyword(s):  
Quantum Electrodynamics ◽  
Quantum Circuit ◽  
Critical Slowing Down ◽  
Microwave Cavity ◽  
Many Body ◽  
Circuit Quantum Electrodynamics ◽  
Slowing Down ◽  
The Rich ◽  
First Order Phase ◽  
Shed Light

Critical slowing down of the time it takes a system to reach equilibrium is a key signature of bistability in dissipative first-order phase transitions. Understanding and characterizing this process can shed light on the underlying many-body dynamics that occur close to such a transition. Here, we explore the rich quantum activation dynamics and the appearance of critical slowing down in an engineered superconducting quantum circuit. Specifically, we investigate the intermediate bistable regime of the generalized Jaynes-Cummings Hamiltonian (GJC), realized by a circuit quantum electrodynamics (cQED) system consisting of a transmon qubit coupled to a microwave cavity. We find a previously unidentified regime of quantum activation in which the critical slowing down reaches saturation and, by comparing our experimental results with a range of models, we shed light on the fundamental role played by the qubit in this regime.

scholarly journals Discrete-time quantum walks generated by aperiodic fractal sequence of space coin operators

2018 ◽  
Vol 29 (10) ◽  
pp. 1850098 ◽  
Author(s):  
R. F. S. Andrade ◽  
A. M. C. Souza
Keyword(s):  
Wave Function ◽  
Probability Distribution ◽  
Discrete Time ◽  
Numerical Study ◽  
Entanglement Entropy ◽  
Quantum Effects ◽  
Quantum Walks ◽  
Cantor Set ◽  
One Dimensional ◽  
Time Quantum

Properties of one-dimensional discrete-time quantum walks (DTQWs) are sensitive to the presence of inhomogeneities in the substrate, which can be generated by defining position-dependent coin operators. Deterministic aperiodic sequences of two or more symbols provide ideal environments where these properties can be explored in a controlled way. Based on an exhaustive numerical study, this work discusses a two-coin model resulting from the construction rules that lead to the usual fractal Cantor set. Although the fraction of the less frequent coin [Formula: see text] as the size of the chain is increased, it leaves peculiar properties in the walker dynamics. They are characterized by the wave function, from which results for the probability distribution and its variance, as well as the entanglement entropy, were obtained. A number of results for different choices of the two coins are presented. The entanglement entropy has shown to be very sensitive to uncovering subtle quantum effects present in the model.

scholarly journals Dynamics and energy spectra of aperiodic discrete-time quantum walks

2017 ◽  
Vol 96 (1) ◽  
Author(s):  
N. Lo Gullo ◽  
C. V. Ambarish ◽  
Th. Busch ◽  
L. Dell'Anna ◽  
C. M. Chandrashekar
Keyword(s):  
Discrete Time ◽  
Quantum Walks ◽  
Energy Spectra ◽  
Time Quantum
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