Universal quantum computation using the discrete-time quantum walk

Lovett, Neil B.; Cooper, S.; Everitt, Mark S.; Trevers, Matthew; Kendon, Viv

doi:10.1103/physreva.81.042330

Cited by 492 publications

(418 citation statements)

References 30 publications

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“…Their hallmark property is that on a regular lattice they spread quadratically more rapidly than random walks, i.e., ballistically rather than diffusively. This makes them valuable in quantum search algorithms [2] or even for general quantum computing [3]. Experiments on quantum walks range from realizations on trapped ions [4][5][6], to cold atoms in optical lattices [7][8][9], to light on an optical table [10][11][12][13][14][15], but there have been many other experimental proposals [16,17].…”

Section: Introductionmentioning

confidence: 99%

Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk

Rakovszky

Asbóth

2015

Phys. Rev. A

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Quantum walks are promising for information processing tasks because in regular graphs they spread quadratically more rapidly than random walks. Static disorder, however, can turn the tables: unlike random walks, quantum walks can suffer Anderson localization, with their wave function staying within a finite region even in the infinite time limit, with a probability exponentially close to 1. It is thus important to understand when a quantum walk will be Anderson localized and when we can expect it to spread to infinity even in the presence of disorder. In this work we analyze the response of a one-dimensional quantum walk-the split-step walk-to different forms of static disorder. We find that introducing static, symmetry-preserving disorder in the parameters of the walk leads to Anderson localization. In the completely disordered limit, however, a delocalization transition occurs, and the walk spreads subdiffusively to infinity. Using an efficient numerical algorithm, we calculate the bulk topological invariants of the disordered walk and find that the disorder-induced Anderson localization and delocalization transitions are governed by the topological phases of the quantum walk.

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Section: Introductionmentioning

confidence: 99%

Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk

Rakovszky

Asbóth

2015

Phys. Rev. A

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“…Lovett et al [33] later showed that the discrete time quantum walk can implement the universal quantum gate set, following Childs' [34] earlier proof of the equivalent result for the continuous time quantum walk. Ryan et al [35] experimentally demonstrated eight steps of a discrete time quantum walk over three points using an NMR-based quantum computer.…”

Section: Discrete Time Quantum Walksmentioning

confidence: 96%

Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms

Dellar¹

2015

ESAIM: Proc.

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Abstract. Quantum lattice algorithms originated with the Feynman checkerboard model for the onedimensional Dirac equation. They offer discrete models of quantum mechanics in which the complex numbers representing wavefunction values on a discrete spatial lattice evolve through discrete unitary operations. This paper draws together some of the identical, or at least unitarily equivalent, algorithms that have appeared in three largely disconnected strands of research. Treated as conventional numerical algorithms, they are all only first order accurate under refinement of the discrete space/time grid, but may be raised to second order by a unitary change of variables. Much more efficient implementations arise from replacing the evolution through a sequence of unitary intermediate steps with a short path integral formulation that expresses the wavefunction at each spatial point on the most recent time level as a linear combination of values at immediately preceding time levels and neighbouring spatial points. In one dimension, a particularly elegant reformulation replaces two variables at two time levels with a single variable over three time levels. The resulting algorithm is a variational integrator arising from a discrete action principle, and coincides with the Ablowitz-Kruskal-Ladik finite difference scheme for the Klein-Gordon equation.

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“…They have been intensively investigated, especially in connection with quantum information science [3,4,5,6,7,8,9]. As in the classical case, QWs have been proposed as elements to design quantum algorithms [10,11,12,13] and more recently it has been shown that they can be used as a universal model for quantum computation [14,15].…”

Section: Introductionmentioning

confidence: 99%

Quantum walk, entanglement and thermodynamic laws

Romanelli

2015

Physica A: Statistical Mechanics and its Applications

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We consider an special dynamics of a quantum walk (QW) on a line. Initially, the walker localized at the origin of the line with arbitrary chirality, evolves to an asymptotic stationary state. In this stationary state a measurement is performed and the state resulting from this measurement is used to start a second QW evolution to achieve a second asymptotic stationary state. In previous works, we developed the thermodynamics associated with the entanglement between the coin and position degrees of freedom in the QW. Here we study the application of the first and second laws of thermodynamics to the process between the two stationary states mentioned above. We show that: i) the entropy change has upper and lower bounds that are obtained analytically as a function of the initial conditions. ii) the energy change is associated to a heat-transfer process.

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Universal quantum computation using the discrete-time quantum walk

Cited by 492 publications

References 30 publications

Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk

Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk

Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms

Quantum walk, entanglement and thermodynamic laws

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