Long-range interactions, weak magnetic fields amplification, and end states for quantum computing

It was Richard Feynman who first proposed, in 1982, the far-reaching concept of a ”quantum computer”—a device more powerful than classical computers. The idea of a quantum computer is to employ the fascinating and often counterintuitive laws of quantum mechanics to process information. It is far from obvious that the proposed concept of a quantum computer is more powerful than its classical counterpart, it was only in 1994 when Peter Shor theoretically demonstrated the existence of a quantum algorithm for factorizing integers into prime factors that runs in polynomial time unlike its classical counterpart which works in sub-exponential time. The factorization of integers into prime factors is the basis of asymmetric cryptography. These early theoretical results lunched an immense interest of the scientific community. Already during ’90s, the first proposals for the physical implementation of quantum computation emerged. Ever since, many experimental groups around the world pursued different physical implementations of quantum bits (qubits). The first decade of the new century saw a steady improvement in the control and decoherence time (the time over which the information carried by the qubit is lost) for various qubits by many orders of magnitude. The natural next step in this context is to answer the question of how to scale the system up to include many qubits and thus build a quantum computer? One of the main parts of this thesis addresses exactly this question, namely the question of architecture and scalability of future quantum computer. Among various different physical realizations of qubits, the idea of using electron spins trapped in electrostatic semiconductor quantum dots as the building blocks of a quantum computer (the so-called spin qubits), put forward by Daniel Loss and David DiVincenzo in 1997, triggered tremendous interest in scientific community. Nevertheless, the implementation of the original Loss-DiVincenzo proposal posed a considerable technical challenge. It used quantum tunneling between qubits to enable their communication with each other, and thus required that the qubits to be placed very close to each other. This requirement not only leaves little space for the placement of the vast amount of gates and wirings needed to define the electrostatic quantum dots, but also makes it challenging to control the local magnetic field needed for single-qubit operations. For these reasons, no system with more than a couple of spin qubits has been successfully implemented thus far. In the first part of this thesis, we leap over this long-standing problem with an entirely different strategy of using metallic floating gates or ferromagnets to couple together qubits that are separated over a long distance. Our scheme works for any type of spin qubits, including the qubits based on nitrogen-vacancy center (NV-center) in diamond and technologically very important silicon qubits. The main topic of this thesis is related to quantum computer. Still, quite unexpectedly, some of the ideas we employed in order to tackle the problem of quantum computer scalability can be utilized in a completely different field of research, namely, in the field of magnetic field sensing. Qubit are not only a necessary ingredient of quantum computer but they also provide a way to measure very accurately magnetic fields. The magnetometer build upon the qubit based on NV-center, so-called NV-magnetometer, emerged in recent years as most sensitive magnetic moment sensor. These magnetometers are able to detect about hundred nuclear spins within a minute of acquisition time. In the second part of this thesis, we propose an entirely novel experimental realization of NV-magnetometers which increases present NV-center sensitivities by four orders of magnitude at room temperature. This unprecedented amplification of sensitivity will render magnetometers capable of detecting individual nuclear spins. This amplification is achieved by introducing a ferromagnetic particle between the nuclear spin that needs to be detected and the NV-magnetometer. Our setup, in contrast to existing schemes, is particularly advantageous because, due to the large amplification of sensitivity, the nuclear spin need not lie within a few nanometers of the surface but rather can be detectable at a distance of 30 nm. With these novelties, our scheme provides chemically sensitive spin detection under ambient conditions allowing nanoscale resolution of nuclear magnetic moments in biological systems—the holy grail of nuclear magneticresonance. In the last part of the thesis we focus our attention to a new direction in quantum computer implementation that deals with topological quantum computer introduced by Alexei Kitaev in 1997; in this approach the idea is to use quasiparticles with ”fractional” statistics and to perform the single- and two-qubit gates by merely exchanging these quasiparticles. Additionally, information in this system is stored non-locally thus it mitigates the problem of decoherence caused by local noise from the environment. Majorana fermions are one of the most well known examples of such excitations. We analyze transport signatures of different topological states in one-dimensional systems, like Majorana fermions and fractionally charged states. We envision an Aharonov-Bohm setup wherein conductance measurement provides a clear signature of presence of fractionally charged fermionic states, since oscillations with double period emerge in this case. Additionally, we propose a very simple setup that enables existence of degenerate mid-gap states, so-called Tamm-Shockley states that are characterized by fractional charge and discuss possible ways of detecting these states experimentally.