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arXiv:1309.3911 · Scientific Reports

Two-component Dirac-like Hamiltonian for generating quantum walk on one-, two- and three-dimensional lattices

C. M. Chandrashekar

Published Oct 3, 2013• 64 citations
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Abstract

From the unitary operator used for implementing two-state discrete-time quantum walk on one-, two- and three- dimensional lattice we obtain a two-component Dirac-like Hamiltonian. In particular, using different pairs of Pauli basis as position translation states we obtain three different form of Hamiltonians for evolution on one-dimensional lattice. We extend this to two- and three-dimensional lattices using different Pauli basis states as position translation states for each dimension and show that the external coin operation, which is necessary for one-dimensional walk is not a necessary requirement for a walk on higher dimensions but can serve as an additional resource to control the dynamics. The two-component Hamiltonian we present here for quantum walk on different lattices can serve as a general framework to simulate, control, and study the dynamics of quantum systems governed by Dirac-like Hamiltonian.

DOI
10.1038/srep02829

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From the unitary operator used for implementing two-state discrete-time quantum walk on one-, two- and three- dimensional lattice we obtain a two-component Dirac-like Hamiltonian. In particular, using different pairs of Pauli basis as position translation states we obtain three different form of Hamiltonians for evolution on one-dimensional lattice. We extend this to two- and three-dimensional lattices using different Pauli basis states as position translation states for each dimension and show that the external coin operation, which is necessary for one-dimensional walk is not a necessary requirement for a walk on higher dimensions but can serve as an additional resource to control the dynamics. The two-component Hamiltonian we present here for quantum walk on different lattices can serve as a general framework to simulate, control, and study the dynamics of quantum systems governed by Dirac-like Hamiltonian.
We present a new scheme for a discrete-time quantum walk on two- and three-dimensional lattices using a two-state particle. We use different Pauli basis as translational eigestates for different axis and show that the coin operation, which is necessary for one-dimensional walk is not a necessary requirement for two- and three- dimensional walk but can serve as an additional resource. Using this scheme, the probability distribution from Grover walk using four-state particle and other equivalent schemes on a square lattice using coin operation is reproduced. We also present the Hamiltonian form of evolution which can serve as a general framework to simulate, control, and study the dynamics in different physical systems.
We investigate a discrete-time quantum walk of a three-component quantum particle on a one-dimensional lattice. As coin operators, we employ parameterized rotations generated by the Gell-Mann matrices, which enable systematic tuning of the couplings between the internal components. We analyze how these couplings influence and control the dynamics by systematically exploring the position-space probability distribution across a broad region of the parameter space. To quantify the impact of different inter-component couplings, we further examine the ratio of the mean position to the variance in each half of the lattice. Our results indicates that the system supports a rich variety of transport regimes, ranging from nearly symmetric, rapidly spreading walks to strongly anisotropic dynamics with partial localization. This framework thus provides a new avenue for engineering targeted spreading and trapping behavior in multicomponent discrete-time quantum walks.
Dirac particle represents a fundamental constituent of our nature. Simulation of Dirac particle dynamics by a controllable quantum system using quantum walks will allow us to investigate the non-classical nature of dynamics in its discrete form. In this work, starting from a modified version of one-spatial dimensional general inhomogeneous split-step discrete quantum walk we derive an effective Hamiltonian which mimics a single massive Dirac particle dynamics in curved (1 + 1) space-time dimension coupled to U (1) gauge potential—which is a forward step towards the simulation of the unification of electromagnetic and gravitational forces in lower dimension and at the single particle level. Implementation of this simulation scheme in simple qubit-system has been demonstrated. We show that the same Hamiltonian can represent (2 + 1) space-time dimensional Dirac particle dynamics when one of the spatial momenta remains fixed. We also discuss how we can include U ( N ) gauge potential in our scheme, in order to capture other fundamental force effects on the Dirac particle. The emergence of curvature in the two-particle split-step quantum walk has also been investigated while the particles are interacting through their entangled coin operation.
It has been observed that quantum walks on regular lattices can give rise to wave equations for relativistic particles in the continuum limit. In this paper, we define the three-dimensional discrete-time walk as a product of three coined one-dimensional walks. The factor corresponding to each one-dimensional walk involves two projection operators that act on an internal coin space; each projector is associated with either the ``forward'' or ``backward'' direction in that physical dimension. We show that the simple requirement that there is no preferred axis or direction along an axis---that is, that the walk be symmetric under parity transformations and steps along different axes of the cubic lattice be uncorrelated---leads, in the case of the simplest solution, to the requirement that the continuum limit of the walk is fully Lorentz-invariant. We show further that, in the case of a massive particle, this symmetry requirement necessitates the use of a four-dimensional internal space (as in the Dirac equation). The ``coin flip'' operation is generated by the parity transformation on the internal coin space, while the differences of the projection operators associated with each dimension must all anticommute. Finally, we discuss the leading correction to the continuum limit, and the possibility of distinguishing through experiment between the discrete random walk and the continuum-based Dirac equation as a description of fermion dynamics.
A coinless, discrete-time quantum walk possesses a Hilbert space whose dimension is smaller compared to the widely studied coined walk. Coined walks require the direct product of the site basis with the coin space; coinless walks operate purely in the site basis, which is clearly minimal. These coinless quantum walks have received considerable attention recently because they have evolution operators that can be obtained by a graphical method based on lattice tessellations and they have been shown to be as efficient as the best known coined walks when used as a quantum search algorithm. We argue that both formulations in their most general form are equivalent. In particular, we demonstrate how to transform the one-dimensional version of the coinless quantum walk into an equivalent extended coined version for a specific family of evolution operators. We present some of its basic, asymptotic features for the one-dimensional lattice with some examples of tessellations, and analyze the mixing time and limiting probability distributions on cycles.
An explicit formula of the Hamiltonians generating one-dimensional discrete-time quantum walks is given. The formula is deduced by using the algebraic structure introduced before. The square of the Hamiltonian turns out to be an operator without, essentially, the ``coin register,'' and hence it can be compared with the one-dimensional continuous-time quantum walk. It is shown that, under a limit with respect to a parameter, which expresses the magnitude of the diagonal components of the unitary matrix defining the discrete-time quantum walks, the one-dimensional continuous-time quantum walk is obtained from operators defined through the Hamiltonians of the one-dimensional discrete-time quantum walks. Thus, this result can be regarded, in one-dimension, as a partial answer to a problem proposed by Ambainis.