Nature of Quantum Ultra Walks: Localization and Delocalization Open Access

Sharma, Richa (Summer 2021)

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Classical Random walks have proven to be a useful tool for various computational problems. Similarly, its quantum mechanical counterpart quantum walks have applications in the development of quantum algorithms specifically quantum search and element distinctness algorithm[8]. Classical random walk spreads diffusively across the geometric structure which scales proportionally to the square root of time. Homogeneous Quantum Walks, however, is ‘ballistic’, i.e., its spread scales linearly with time. This property contributes to its algorithmic applications, as it enhances quantum energy transfer on certain graphs. However, the static disorder can make quantum walks suffer Anderson-like localization which can confine the walker within a finite region for an indefinite time[11, 12, 13], taking away the quantum advantage. Therefore, the exploration of the heterogeneity in disordered systems is important for the successful realization of quantum computers. To this end, we study for a discrete-time quantum walk the effect of a quantum coin that varies randomly in space but only on a hierarchy of lattice points. Such a hierarchical system, but with a regular sequence of coins representing reflective barriers, was introduced by Boettcher et al.[3, 4, 5] who found sub-ballistic spread but no localization of the quantum particle, irrespective of barrier strength. We find that any strength of randomness on the hierarchy of lattice points by itself, without barriers, also does not localize the walk. However, applying non-trivial combinations of randomness and barrier strength on that hierarchy appears to induce transition into a localized state. This behavior of the system is determined numerically using the walk dimension d_w we extract from the mean-square displacement, averaged over repeated realizations of the randomness.

Table of Contents

Introduction ... 1

1.1 Motivation

1.2 Background

1.2.1 Fundamental Postulates of Quantum Mechanics

1.2.2 Geometrical Representation of Qubits

1.2.3 Classical Random Walks

1.2.4 Discrete-Time Quantum Walks

1.2.5 Localization: Absence of Diffusion

1.3 Methods

1.3.1 Simulation of Heterogenous Quantum Walks

1.3.2 Uncorrelated phase disorders in the lattice

2.Results ... 19

2.1 Behaviours of homogeneous and heterogeneous Quantum Walks

2.2 Localization for regular phase disorders

2.2.1 Homogeneity affected by phase disorders

2.2.2 Phase disorders in Heterogeneous systems

2.3 System affected by phase disorders correlated with Hierarchical indices

2.3.1 Novel Disorder introduced with ε = 1

2.4 Increasing Reflectivity of the system

2.4.1 Heterogeneity and hierarchical phase disorders effects 

2.4.2 Indicators of Localized, sub-ballistic and ballistic behavior

3.Discussion ... 27

3.1 Heterogeneity in Quantum Walks

3.2 Disorder Strength and localization length

3.3 Re-normalization and Mathematical modeling of the Quantum System

4.Future Studies ... 31

4.1 Novel Phase Disorders

4.2 Quantum Advantage of Quantum Walks modeling

5.Summary ... 33

5.1 Results

5.2 Discussion

5.3 Conclusion

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