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arXiv:0903.1236 · Physical Review B

Implementation of the quantum-walk step operator in lateral quantum dots

Kevin A. van Hoogdalem, M. Blaauboer

Published Sep 14, 2009• 6 citations
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Abstract

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 $>90%$ is feasible under realistic experimental conditions.

DOI
10.1103/physrevb.80.125309

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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.
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 show that a many-body system of single-electron quantum dots, whose orbital states are dressed by a global magnetic field, can be described by an effective Hamiltonian with an anisotropic XZ\mathit{XZ}XZ spin-spin interaction which is proportional to the Zeeman splitting. We show that these interaction potentials give rise to spin-dependent Hubbard models with tunable nearest neighbor two-body and three-body interactions. The two-body interactions can even be switched off via the external electric field, and hence the three-body interaction plays a dominant role. The derivation of these effective interaction potentials follows from a well-controlled and systematic expansion into many-body interaction terms. Models of this type have appeared in the recent discussion of exotic quantum phases, in particular in the context of topological quantum phases and quantum computing, and we show that quantum dots can be regarded as a realistic experimental route which provides the basic building blocks and techniques toward the study of these phenomena. The main application of this derived model is to develop topological quantum computation.
Quantum random walks are shown to have non-intuitive dynamics, which makes them an attractive area of study for devising quantum algorithms for well-known classical problems as well as those arising in the field of quantum computing. In this work we propose a novel scheme for the physical implementation of a discrete-time quantum random walk using laser excitations of the electronic states of an array of quantum dots. These dots represent the discrete nodes of the walk, while transitions between the energy levels inside each dot correspond to the required coin operation and stimulated Raman adiabatic passage (STIRAP) processes are employed to induce the steps of the walk. The quantum dot design is tailored in such a way as to enable selective coupling of the energy levels. Our simulation results show a close agreement with the ideal quantum walk distribution as well as modest robustness towards noise disturbance.
We consider a single electron in a 1D quantum dot with a static slanting Zeeman field. By combining the spin and orbital degrees of freedom of the electron, an effective quantum two-level (qubit) system is defined. This pseudospin can be coherently manipulated by the voltage applied to the gate electrodes, without the need for an external time-dependent magnetic field or spin-orbit coupling. Single-qubit rotations and the controlled-NOT operation can be realized. We estimated the relaxation (T1) and coherence (T2) times and the (tunable) quality factor. This scheme implies important experimental advantages for single electron spin control.
The discrete-time quantum walk (DTQW) with step-dependent (SD) scattering operator is implemented on quantum computer. The probabilities of different states, with their respective fidelities, are calculated. This is done by generalizing the coin with a rotation gate using the quantum gate model. The CNOT gates in the shift operator are replaced with the alternative to CNOT gates Rx( π). They are applied on a quantum device and a quantum simulator (QS). The fidelities varied around 50% and the probability distribution of SD-DTQW for the angle π/4 spread symmetrically, while the step-independent DTQW (SI-DTQW) tended to peak at one side. The symmetric probability distribution of SD-DTQW can help in better control of the walk on QS. In the case of angle π/2, the SI-DTQW spread equally across the states with four peaks, while the SD-DTQW spread with two peaks to one side.