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.