scholarly journals Quantum walking in curved spacetime: (3+1) dimensions, and beyond

2017 ◽  
Vol 17 (9&10) ◽  
pp. 810-824 ◽  
Author(s):  
Pablo Arrighi ◽  
Stefno Facchini
Keyword(s):  
Dirac Equation ◽  
Continuum Limit ◽  
Space Dimension ◽  
Quantum Walk ◽  
Internal Degree ◽  
Curved Spacetime ◽  
Arbitrary Space ◽  
Time Quantum ◽  
General Continuum ◽  
Internal Degree Of Freedom

A discrete-time Quantum Walk (QW) is essentially an operator driving the evolution of a single particle on the lattice, through local unitaries. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). Recently it was discovered that prior grouping and encoding allows for more general continuum limit equations (e.g. the Dirac equation in (1+ 1) curved spacetime). In this paper, we extend these results to arbitrary space dimension and internal degree of freedom. We recover an entire class of PDEs encompassing the massive Dirac equation in (3 + 1) curved spacetime. This means that the metric field can be represented by a field of local unitaries over a lattice.

scholarly journals Quantum walking in curved spacetime: discrete metric

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 84 ◽  
Author(s):  
Pablo Arrighi ◽  
Giuseppe Di Molfetta ◽  
Stefano Facchini
Keyword(s):  
Continuum Limit ◽  
Quantum Walk ◽  
Scalar Transport ◽  
Curved Spacetime ◽  
Continuous Parameter ◽  
Slight Generalization ◽  
Time Quantum ◽  
Quantum Coin ◽  
Speed Of Propagation ◽  
The One

A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs have familiar physics PDEs as their continuum limit. Some slight generalization of them (allowing for prior encoding and larger neighbourhoods) even have the curved spacetime Dirac equation, as their continuum limit. In the(1+1)−dimensional massless case, this equation decouples as scalar transport equations with tunable speeds. We characterise and construct all those QWs that lead to scalar transport with tunable speeds. The local coin operator dictates that speed; we provide concrete techniques to tune the speed of propagation, by making use only of a finite number of coin operators-differently from previous models, in which the speed of propagation depends upon a continuous parameter of the quantum coin. The interest of such a discretization is twofold : to allow for easier experimental implementations on the one hand, and to evaluate ways of quantizing the metric field, on the other.

scholarly journals Massless Dirac equation from Fibonacci discrete-time quantum walk

2015 ◽  
Vol 2 (3) ◽  
pp. 243-252 ◽  
Author(s):  
Giuseppe Di Molfetta ◽  
Lauchlan Honter ◽  
Ben B. Luo ◽  
Tatsuaki Wada ◽  
Yutaka Shikano
Keyword(s):  
Dirac Equation ◽  
Discrete Time ◽  
Quantum Walk ◽  
Time Quantum

scholarly journals Erratum to: Massless Dirac equation from Fibonacci discrete-time quantum walk

2015 ◽  
Vol 2 (3) ◽  
pp. 253-254
Author(s):  
Giuseppe Di Molfetta ◽  
Lauchlan Honter ◽  
Ben B. Luo ◽  
Tatsuaki Wada ◽  
Yutaka Shikano
Keyword(s):  
Dirac Equation ◽  
Discrete Time ◽  
Quantum Walk ◽  
Time Quantum

scholarly journals A Discontinuity of the Energy of Quantum Walk in Impurities

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1134
Author(s):  
Kenta Higuchi ◽  
Takashi Komatsu ◽  
Norio Konno ◽  
Hisashi Morioka ◽  
Etsuo Segawa
Keyword(s):  
Unit Circle ◽  
Stationary State ◽  
Discrete Time ◽  
Quantum Walk ◽  
Time Limit ◽  
The Other ◽  
Unitary Matrix ◽  
Initial State ◽  
Time Quantum ◽  
Long Time

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

scholarly journals Depinning of domain walls with an internal degree of freedom

2009 ◽  
Vol 80 (5) ◽  
Author(s):  
V. Lecomte ◽  
S. E. Barnes ◽  
J.-P. Eckmann ◽  
T. Giamarchi
Keyword(s):  
Domain Walls ◽  
Internal Degree ◽  
Degree Of Freedom ◽  
Internal Degree Of Freedom

scholarly journals Marking Vertices to Find Graph Isomorphism Mapping Based on Continuous-Time Quantum Walk

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 586 ◽  
Author(s):  
Xin Wang ◽  
Yi Zhang ◽  
Kai Lu ◽  
Xiaoping Wang ◽  
Kai Liu
Keyword(s):  
Polynomial Time ◽  
Continuous Time ◽  
Graph Isomorphism ◽  
Quantum Walk ◽  
Isomorphism Problem ◽  
Quantum Walks ◽  
Regular Graphs ◽  
Structure Preserving ◽  
Topological Structures ◽  
Time Quantum

The isomorphism problem involves judging whether two graphs are topologically the same and producing structure-preserving isomorphism mapping. It is widely used in various areas. Diverse algorithms have been proposed to solve this problem in polynomial time, with the help of quantum walks. Some of these algorithms, however, fail to find the isomorphism mapping. Moreover, most algorithms have very limited performance on regular graphs which are generally difficult to deal with due to their symmetry. We propose IsoMarking to discover an isomorphism mapping effectively, based on the quantum walk which is sensitive to topological structures. Firstly, IsoMarking marks vertices so that it can reduce the harmful influence of symmetry. Secondly, IsoMarking can ascertain whether the current candidate bijection is consistent with existing bijections and eventually obtains qualified mapping. Thirdly, our experiments on 1585 pairs of graphs demonstrate that our algorithm performs significantly better on both ordinary graphs and regular graphs.

scholarly journals INFLUENCE OF GRAVITY ON NONCOMMUTATIVE DIRAC EQUATION

2005 ◽  
Vol 20 (26) ◽  
pp. 1997-2005 ◽  
Author(s):  
SOFIANE BOUROUAINE ◽  
ACHOUR BENSLAMA
Keyword(s):  
Dirac Equation ◽  
Electric Field ◽  
Covariant Derivative ◽  
Strong Electric Field ◽  
Curved Spacetime ◽  
Tetrad Formalism ◽  
Dirac Particles ◽  
Modified Dirac Equation ◽  
New Form

In this paper, we investigate the influence of gravity and noncommutativity on Dirac particles. By adopting the tetrad formalism, we show that the modified Dirac equation keeps the same form. The only modification is in the expression of the covariant derivative. The new form of this derivative is the product of its counterpart given in curved spacetime with an operator which depends on the noncommutative θ-parameter. As an application, we have computed the density number of the created particles in the presence of constant strong electric field in an anisotropic Bianchi universe.

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