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arXiv:0904.4588 · Physical Review B

Quantum walk on the line with quantum rings

Orsolya Kálmán, T. Kiss, Péter Földi

Published Jul 29, 2009• 19 citations
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Abstract

We propose a scheme to implement the one-dimensional coined quantum walk with electrons transported through a two-dimensional network of spintronic semiconductor quantum rings. The coin degree of freedom is represented by the spin of the electron while the discrete position of the walker corresponds to the label of the rings in one of the spatial directions in the network. We assume that Rashba-type spin-orbit interaction is present in the rings, the strength of which can be tuned by an external electric field. The geometry of the device, together with the appropriate spin-orbit interaction strengths, ensure the realization of the coin toss (i.e., spin flip) and the step operator.

DOI
10.1103/physrevb.80.035327

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We propose a scheme to implement the one-dimensional coined quantum walk with electrons transported through a two-dimensional network of spintronic semiconductor quantum rings. The coin degree of freedom is represented by the spin of the electron while the discrete position of the walker corresponds to the label of the rings in one of the spatial directions in the network. We assume that Rashba-type spin-orbit interaction is present in the rings, the strength of which can be tuned by an external electric field. The geometry of the device, together with the appropriate spin-orbit interaction strengths, ensure the realization of the coin toss (i.e., spin flip) and the step operator.
We propose a different implementation of discrete-time quantum walks for a neutral atom in an array of optical microtraps or an optical lattice. We analyze a one-dimensional walk in position space, with the coin, the additional qubit degree of freedom that controls the displacement of the quantum walker, implemented as a spatially delocalized qubit, i.e., the coin is also encoded in position space. We analyze the dependence of the quantum walk on temperature and experimental imperfections such as shaking in the trap positions. Finally, combining a spatially delocalized qubit and a hyperfine qubit, we also give a scheme to realize a quantum walk on a two-dimensional square lattice with the possibility of implementing different coin operators.
We propose a physical implementation of the step operator of the discrete quantum walk for an electron in a one-dimensional chain of quantum dots. The operating principle of the step operator is based on locally enhanced Zeeman splitting and the role of the quantum coin is played by the spin of the electron. We calculate the probability of successful transfer of the electron in the presence of decoherence due to quantum charge fluctuations, modeled as a bosonic bath. We then analyze two mechanisms for creating locally enhanced Zeeman splitting based on, respectively, locally applied electric and magnetic fields and slanting magnetic fields. Our results imply that a success probability of >90>90%>90 is feasible under realistic experimental conditions.
Quantum walks are more than tools for building quantum algorithms. They have been used effectively to model and simulate quantum dynamics in many complex physical processes. Particularly, a variant of discrete-time quantum walk known as split-step quantum walk is closely related to Dirac cellular automata and topological insulators, whose realizations rely on position-dependent control of evolution operators. Owing to the ease of manipulating multiple degrees of freedom of photons, we provide an optical setup of split-step operators which, in combination with position-dependent coin (PDC) operation, can accomplish a table-top setup of generalized split-step walks. Also, we propose an optical implementation for PDC operation that allows, for instance, realizing electric quantum walks, control localization dynamics, and emulate space-time curvature effects. In addition, we propose a setup to realize any t -step split-step quantum walk involving 2 J -plates, 2 variable waveplates, a half-waveplate, an optical switch, and an optical delay line.
Spin transport typically relies on direct manipulation of the spin degree of freedom via magnetic fields, spin-orbit coupling, or engineered spin-dependent potentials. We show theoretically that directional spin currents can arise in a relatively simple setting - a one-dimensional interacting fermionic ring with static, spin-independent asymmetric barriers. By introducing asymmetric potential barrier geometry, spin-resolved circulating currents emerge on a closed chain even for symmetric initial configurations. The effect can be enhanced or reversed by appropriate initial state preparation and tuning the barrier asymmetry to resonant conditions.
Quantum state preparation in high-dimensional systems is an essential requirement for many quantum-technology applications. The engineering of an arbitrary quantum state is, however, typically strongly dependent on the experimental platform chosen for implementation, and a general framework is still missing. Here we show that coined quantum walks on a line, which represent a framework general enough to encompass a variety of different platforms, can be used for quantum state engineering of arbitrary superpositions of the walker's sites. We achieve this goal by identifying a set of conditions that fully characterize the reachable states in the space comprising walker and coin and providing a method to efficiently compute the corresponding set of coin parameters. We assess the feasibility of our proposal by identifying a linear optics experiment based on photonic orbital angular momentum technology.