Discrete-Time Quantum Walk - Dynamics and Applications

cycle highlighting the interplay between noise and topology. \n \nUsing the properties and behavior of quantum walk discussed in part two, in part three we propose the application of quantum walk to realize quantum phase transition in optical lattice, that is to efficiently control and redistribute ultracold atoms in optical lattice. We also discuss the implementation scheme. Another application we consider is creation of spatial entanglement using quantum walk on a quantum many body system.

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C. M. Chandrashekar

This dissertation presents investigations on dynamics of discrete-time quantum walk and some of its applications. Quantum walks has been exploited as an useful tool for quantum algorithms in quantum computing. Beyond quantum computational purposes, it has been used to explain and control the dynamics in various physical systems. In order to use the quantum walk to its fullest potential, it is important to know and optimize the properties purely due to quantum dynamics and in presence of noise. Various studies of its dynamics in the absence and presence of noise have been reported. We propose new approaches to optimize the dynamics, discuss symmetries and effect of noise on the quantum walk. Making use of its properties, we propose the use of quantum walk as an efficient new tool for various applications in physical systems and quantum information processing. In the first and second part of this dissertation, we discuss evolution process of the quantum walks, propose and demonstrate the optimization of discrete-time quantum walk using quantum coin operation from SU(2) group and discuss some of its properties. We investigate symmetry operations and environmental effects on dynamics of the walk on a line and an n-cycle highlighting the interplay between noise and topology. Using the properties and behavior of quantum walk discussed in part two, in part three we propose the application of quantum walk to realize quantum phase transition in optical lattice, that is to efficiently control and redistribute ultracold atoms in optical lattice. We also discuss the implementation scheme. Another application we consider is creation of spatial entanglement using quantum walk on a quantum many body system.This dissertation presents investigations on dynamics of discrete-time quantum walk and some of its applications. Quantum walks has been exploited as an useful tool for quantum algorithms in quantum computing. Beyond quantum computational purposes, it has been used to explain and control the dynamics in various physical systems. In order to use the quantum walk to its fullest potential, it is important to know and optimize the properties purely due to quantum dynamics and in presence of noise. Various studies of its dynamics in the absence and presence of noise have been reported. We propose new approaches to optimize the dynamics, discuss symmetries and effect of noise on the quantum walk. Making use of its properties, we propose the use of quantum walk as an efficient new tool for various applications in physical systems and quantum information processing. In the first and second part of this dissertation, we discuss evolution process of the quantum walks, propose and demonstrate the optimization of discrete-time quantum walk using quantum coin operation from SU(2) group and discuss some of its properties. We investigate symmetry operations and environmental effects on dynamics of the walk on a line and an n-cycle highlighting the interplay between noise and topology. Using the properties and behavior of quantum walk discussed in part two, in part three we propose the application of quantum walk to realize quantum phase transition in optical lattice, that is to efficiently control and redistribute ultracold atoms in optical lattice. We also discuss the implementation scheme. Another application we consider is creation of spatial entanglement using quantum walk on a quantum many body system.

arXiv:1001.5326Jan 29, 201014 citations

Toward quantum control in discrete-time quantum walks

S. Panahiyan

Discrete-time quantum walks are among the branches of quantum information and computation. They are platforms for developing quantum algorithms for quantum computers. In addition, due to their universal primitive nature, discrete-time quantum walks have been used to simulate other quantum systems and phenomena that are observed in physics and chemistry. To fully utilize the potentials that the discrete-time quantum walks hold in their applications, control over the discrete-time quantum walks and their properties becomes essential. In this dissertation, we propose two models for attaining a high level of control over the discrete-time quantum walks. In the first one, we incorporate a dynamical nature for the unitary operator performing the quantum walks. This enables us to readily control the properties of the walker and produce diverse behaviors for it. We show that with our proposal, the important properties of the discrete-time quantum walks such as variance would indeed improve. To explore the potential of this proposal, we apply it in the simulations of topological phases in condensed matter physics. With our proposal, we can control the simulations and determine the type of topological phenomena that should be simulated. In addition, we confirm simulations of topological phases and boundary states that can be observed in one-, two- and three-dimensional systems. Finally, we report the emergence of exotic phase structures in form of cell-like structures that contain all types of topological phases and boundary states of certain classes. In our second proposal, we take advantage of resources available in quantum mechanics, namely quantum entanglement and entangled qubits. In this proposal, we use entangled qubits in the structure of a quantum walk and show that by tuning the initial entanglement between these qubits and how these qubits are modified through the walk, one is able to produce diverse behaviors for the quantum walk and control its behavior.Discrete-time quantum walks are among the branches of quantum information and computation. They are platforms for developing quantum algorithms for quantum computers. In addition, due to their universal primitive nature, discrete-time quantum walks have been used to simulate other quantum systems and phenomena that are observed in physics and chemistry. To fully utilize the potentials that the discrete-time quantum walks hold in their applications, control over the discrete-time quantum walks and their properties becomes essential. In this dissertation, we propose two models for attaining a high level of control over the discrete-time quantum walks. In the first one, we incorporate a dynamical nature for the unitary operator performing the quantum walks. This enables us to readily control the properties of the walker and produce diverse behaviors for it. We show that with our proposal, the important properties of the discrete-time quantum walks such as variance would indeed improve. To explore the potential of this proposal, we apply it in the simulations of topological phases in condensed matter physics. With our proposal, we can control the simulations and determine the type of topological phenomena that should be simulated. In addition, we confirm simulations of topological phases and boundary states that can be observed in one-, two- and three-dimensional systems. Finally, we report the emergence of exotic phase structures in form of cell-like structures that contain all types of topological phases and boundary states of certain classes. In our second proposal, we take advantage of resources available in quantum mechanics, namely quantum entanglement and entangled qubits. In this proposal, we use entangled qubits in the structure of a quantum walk and show that by tuning the initial entanglement between these qubits and how these qubits are modified through the walk, one is able to produce diverse behaviors for the quantum walk and control its behavior.

Jan 1, 20210 citations

Quantum Walks: Theory, Application, and Implementation

Albert T. Schmitz

The quantum walk is a method for conceptualizing and designing quantum computing algorithms and it comes in two forms: the continuous-time and discrete-time quantum walk. The thesis is organized into three parts, each of which looks to develop the concept and uses of the quantum walk. The ﬁrst part is the theory of the quantum walk. This includes deﬁnitions and considerations for the various incarnations of the discrete-time quantum walk and a discussion on the general method for connecting the continuous-time and discrete-time versions. As a result, it is shown that most versions of the discrete-time quantum walk can be put into a general form and this can be used to simulate any continuous-time quantum walk. The second part uses these results for a hypothetical application. The application presented is a search algorithm that appears to scale in the time for completion independent of the size of the search space. This behavior is then elaborated upon and shown to have general qualitative agreement with simulations to within the approximations that are made. The third part introduces a method of implementation. Given a universal quantum computer, the method is discussed and shown to simulate an arbitrary discrete-time quantum walk. Some of the beneﬁts of this method are that half the unitary evolution can be achieved without the use of any gates and there may be some possibility for error detection. The three parts combined suggest a possible experiment, given a quantum computing scheme of suﬃcient robustness.The quantum walk is a method for conceptualizing and designing quantum computing algorithms and it comes in two forms: the continuous-time and discrete-time quantum walk. The thesis is organized into three parts, each of which looks to develop the concept and uses of the quantum walk. The ﬁrst part is the theory of the quantum walk. This includes deﬁnitions and considerations for the various incarnations of the discrete-time quantum walk and a discussion on the general method for connecting the continuous-time and discrete-time versions. As a result, it is shown that most versions of the discrete-time quantum walk can be put into a general form and this can be used to simulate any continuous-time quantum walk. The second part uses these results for a hypothetical application. The application presented is a search algorithm that appears to scale in the time for completion independent of the size of the search space. This behavior is then elaborated upon and shown to have general qualitative agreement with simulations to within the approximations that are made. The third part introduces a method of implementation. Given a universal quantum computer, the method is discussed and shown to simulate an arbitrary discrete-time quantum walk. Some of the beneﬁts of this method are that half the unitary evolution can be achieved without the use of any gates and there may be some possibility for error detection. The three parts combined suggest a possible experiment, given a quantum computing scheme of suﬃcient robustness.

Jan 1, 20161 citation

Quantum walks and quantum computation

Katherine Elizabeth Barr

The field of quantum information applies the concepts of quantum physics to problems in computer science, and shows great potential for allowing efficient computation. \nIn this thesis I concentrate on a particular quantum information theoretic tool known as the quantum walk. There are two widely studied versions of the quantum walk, the continuous time walk, and the discrete time walk. The discrete time walk is particularly amenable to investigation using numerical methods, from which most of the results in this thesis are derived, and is the main focus of the work presented. Two aspects of the discrete time walk are investigated: their transport properties \nand their interpretation as quantum computers. I investigated the transport properties in two ways, by looking for a particular type of transport known as perfect \nstate transfer, and examining the transport properties of a new type of coin operator. \nThe search for perfect state transfer concentrated on modifications of small cycles. I found that perfect state transfer is rare for the choices of coin operators \ntested. The structures tested for perfect state transfer were based on cycles, and it appears that the type of modification has more of an effect than the size of the \ncycle. This makes intuitive sense, as the modifications found to lead to walks exhibiting perfect state transfer affected only the initial and target node of the cycle. \nI then investigated a new type of coin operator which does not allow amplitude to return to the node it has come from. This effectively simulates a dimer. Using the general form of this type of operator and random variables for each parameter, I found that the expected distance of the walker from the origin, and standard deviation, were independent of the initial condition. The second half of the thesis concentrates on computational applications of quantum \nwalks using the language acceptance model. I first note their equivalence to a type of quantum automaton known as the QFA-WOM, and this provides an intuitive understanding of the role of the WOM. I then use a more direct construction to show that they can accept a range of formal languages. Using this construction allows us to use superpositions of words as inputs, and the insights provided by investigating these suggest a new way of approaching the problem of quantum state discrimination.The field of quantum information applies the concepts of quantum physics to problems in computer science, and shows great potential for allowing efficient computation. \nIn this thesis I concentrate on a particular quantum information theoretic tool known as the quantum walk. There are two widely studied versions of the quantum walk, the continuous time walk, and the discrete time walk. The discrete time walk is particularly amenable to investigation using numerical methods, from which most of the results in this thesis are derived, and is the main focus of the work presented. Two aspects of the discrete time walk are investigated: their transport properties \nand their interpretation as quantum computers. I investigated the transport properties in two ways, by looking for a particular type of transport known as perfect \nstate transfer, and examining the transport properties of a new type of coin operator. \nThe search for perfect state transfer concentrated on modifications of small cycles. I found that perfect state transfer is rare for the choices of coin operators \ntested. The structures tested for perfect state transfer were based on cycles, and it appears that the type of modification has more of an effect than the size of the \ncycle. This makes intuitive sense, as the modifications found to lead to walks exhibiting perfect state transfer affected only the initial and target node of the cycle. \nI then investigated a new type of coin operator which does not allow amplitude to return to the node it has come from. This effectively simulates a dimer. Using the general form of this type of operator and random variables for each parameter, I found that the expected distance of the walker from the origin, and standard deviation, were independent of the initial condition. The second half of the thesis concentrates on computational applications of quantum \nwalks using the language acceptance model. I first note their equivalence to a type of quantum automaton known as the QFA-WOM, and this provides an intuitive understanding of the role of the WOM. I then use a more direct construction to show that they can accept a range of formal languages. Using this construction allows us to use superpositions of words as inputs, and the insights provided by investigating these suggest a new way of approaching the problem of quantum state discrimination.

May 1, 20130 citations

Spatial entanglement using a quantum walk on a many-body system

Sandeep K. Goyal, C. M. Chandrashekar

The evolution of a many-particle system on a one-dimensional lattice, subjected to a quantum walk can cause spatial entanglement in the lattice position, which can be exploited for quantum information/communication purposes. We demonstrate the evolution of spatial entanglement and its dependence on the quantum coin operation parameters, the number of particles present in the lattice and the number of steps of the quantum walk on the system. Thus, spatial entanglement can be controlled and optimized using a many-particle discrete-time quantum walk.The evolution of a many-particle system on a one-dimensional lattice, subjected to a quantum walk can cause spatial entanglement in the lattice position, which can be exploited for quantum information/communication purposes. We demonstrate the evolution of spatial entanglement and its dependence on the quantum coin operation parameters, the number of particles present in the lattice and the number of steps of the quantum walk on the system. Thus, spatial entanglement can be controlled and optimized using a many-particle discrete-time quantum walk.

arXiv:0901.0671Journal of Physics A Mathematical and TheoreticalMay 20, 201032 citations

Discrete Time Quantum Lattice Systems

Holger Vogts

Discrete time quantum lattice systems recently have come into the focus of quantum computation because they provide a versatile tool for many different applications and they are potentially implementable in current experimental realizations. In this thesis we study the fundamental structures of such quantum lattice systems as well as consequences of experimental imperfections. Essentially, there are two models of discrete time quantum lattice systems, namely quantum cellular automata and quantum walks, which are quantum versions of their classical counterparts, i.e., cellular automata and random walks. In both cases, the dynamics acts locally on the lattice and is usually also translationally invariant. The main difference between these structures is that quantum cellular automata can describe the dynamics of many interacting particles, where quantum walks describe the evolution of a single particle. The first part of this thesis is devoted to quantum cellular automata. We characterize one-dimensional quantum cellular automata in terms of an index theory up to local deformations. Further, we characterize in detail a subclass of quantum cellular automata by requiring that Pauli operators are mapped to Pauli operators. This structure can be understood in terms of certain classical cellular automata. The second part of this thesis is concerned with quantum walks. We identify a quantum walk with the one-particle sector of a quantum cellular automaton. We also establish an index theory for quantum walks and we discuss decoherent quantum walks, i.e., the behavior of quantum walks with experimental imperfections.Discrete time quantum lattice systems recently have come into the focus of quantum computation because they provide a versatile tool for many different applications and they are potentially implementable in current experimental realizations. In this thesis we study the fundamental structures of such quantum lattice systems as well as consequences of experimental imperfections. Essentially, there are two models of discrete time quantum lattice systems, namely quantum cellular automata and quantum walks, which are quantum versions of their classical counterparts, i.e., cellular automata and random walks. In both cases, the dynamics acts locally on the lattice and is usually also translationally invariant. The main difference between these structures is that quantum cellular automata can describe the dynamics of many interacting particles, where quantum walks describe the evolution of a single particle. The first part of this thesis is devoted to quantum cellular automata. We characterize one-dimensional quantum cellular automata in terms of an index theory up to local deformations. Further, we characterize in detail a subclass of quantum cellular automata by requiring that Pauli operators are mapped to Pauli operators. This structure can be understood in terms of certain classical cellular automata. The second part of this thesis is concerned with quantum walks. We identify a quantum walk with the one-particle sector of a quantum cellular automaton. We also establish an index theory for quantum walks and we discuss decoherent quantum walks, i.e., the behavior of quantum walks with experimental imperfections.

Jun 25, 20096 citations

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