Keimpema, A. (2004) Toys for quantum computation. Master's Thesis / Essay, Computing Science.

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Abstract
A formal definition of a quantum computer would be that it is a system whose quantum mechanical time evolution is used to do computation. A quantum computer just like an dassical computer has the bit as unit of information, in the context of quantum computation called a qubit. Because a quantum computer operates in the quantum mechanical domain, a collection of qubits can hold not one value but a superposition of many values. This "quantum parallelism" is a fundamental advantage that a quantum computer has over a classical computer and algorithms that exploit this feature can be significantly faster than any classical algorithm. That said, the number of algorithm available today is small and their usefulness is limited. Several hardware implementations of quantum computers have been made. Unfortunately existing quantum hardware technology limits the size of these experimental quantum computers to just a few qubits. In chapter 1 we review the field of quantum computation and introduce an interesting new hardware candidate, dubbed the "quantum toy" by its creators. This quantum toy is a metallic ring with three embedded ferromagnets. In certain configurations these ferromagnets can induce a current in the ring. In fact the configurations that will or will not induce a current can be identified with an CNOT(XOR) gate. This motivates us to explore the viability of this system using computer simulations. The quantum toy system is related to a long standing problem in condensed matter physics, that of the persistent currents. When a small metal ring is placed in a static magnetic field, the magnetic field will induce a current in the ring. Persistent currents have been observed experimentally and a great discrepancy between theory and experiment was found. The qualitative features of the phenomenon are explained well by theory, but the size of the current measured was one to two orders of magnitude larger then that predicted by experiment. As a second object of this master thesis we will study persistent current numerically using a simple fightbinding model and compare this to existing theory and experiment. In chapter two we review the field of persistent currents and we give an overview of existing theory. To make a simulation tool of a quantum mechanical system basically means that we have to solve the Schrodinger equation numerically. In chapter 3 we briefly discuss two such method, namely the Cranck Nicholson method and the SuzukiTrotter method. We will use the latter in our simulation tool. The actual simulation is described in chapter 4. In this chapter we detail the development of the numerical tool and show its correctness at each step. We find that the simulations qualitatively show the same features as we would expect theoretically and experimentally. The size of current is a different matter. When compared to experiment we find that although our results are of the same order of magnitude we cannot make an accurate fit to the experimental data using our simple model. In the last chapter we investigate the properties of the quantum toy system by computer simulations. We will explore its application for quantum computation and its use as a measurement device. Upon investigating how a quantum toy system can be used in a quantum circuit, we find that the quantum toy is not useful for quantum computation. We then focus on the second application1 that of a measurement device. The idea being that we can exploit the dependence of the induced current to the direction of the embedded ferromagnets. To verify if such a scheme will work, we need to investigate if we can reproduce the theoretical values of the induced current. We found major discrepancies between theory and our simulations. We found for instance that the induced current has an erratic dependence on the electron density. We then split our investigation between the single electron case and the halffilled case found in most metals. The single electron case only matches the theory qualitatively. We find that indeed such a single electron system can be build and be used as a measurement device. In the halffilled case we find that the dependence of the current on the various parameters is so erratic that we condude that according to our simulations, a metallic quantum toy could not be used as a measurement device. An explanation for the difference between theory and our simulations could unfortunately not be found. To investigate this we would have to repeat the original calculations, which is outside the scope of this master thesis
Item Type:  Thesis (Master's Thesis / Essay) 

Degree programme:  Computing Science 
Thesis type:  Master's Thesis / Essay 
Language:  English 
Date Deposited:  15 Feb 2018 07:30 
Last Modified:  15 Feb 2018 07:30 
URI:  http://fse.studenttheses.ub.rug.nl/id/eprint/8944 
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