A quantum computer would exploit the phenomena of quantum superposition and entanglement in its functioning and with them offer pathways to solving problems that are too hard or complex to even the best classical computers built today. The implementation of a large-scale working quantum computer could bring about a change in our society rivaling the one started by the digital computer. However, the field is still in its infancy and there are many theoretical and practical issues needing to be solved before large-scale quantum computing can become reality. In digital computers, data is stored in bits. The quantum equivalent is called a qubit (quantum bit) and it is basically a quantum mechanical two-level system that can be in a superposition of its two basis states. There are many different proposals for implementing qubits, but one of the most promising ones is to encode the qubit using electron spins trapped in semiconductor quantum dots. Singlet-triplet qubits are spin qubits where the two-electron spin eigenstates are used as the qubit's basis. The required one and two-qubit operations have already been demonstrated experimentally by several research groups around the world in this qubit architecture. The most severe factor limiting the implementation of larger systems of qubits is decoherence. The qubits are not isolated systems, they interact with their environment, which can lead to the loss of quantum information. Few-electron systems can be simulated accurately using first principle methods that become too taxing when the particle number increases. The topic of this thesis is the simulation of quantum dot singlet-triplet qubit systems using accurate exact diagonalization based methods. The emphasis is on the realistic description of qubit operations, both single-qubit ones and those involving the interaction between neighboring qubits. The decoherence effects are also discussed alongside with certain proposals to alleviate their effects.
A quantum computer would exploit the phenomena of quantum superposition and entanglement in its functioning and with them offer pathways to solving problems that are too hard or complex to even the best classical computers built today. The implementation of a large-scale working quantum computer could bring about a change in our society rivaling the one started by the digital computer. However, the field is still in its infancy and there are many theoretical and practical issues needing to be solved before large-scale quantum computing can become reality. In digital computers, data is stored in bits. The quantum equivalent is called a qubit (quantum bit) and it is basically a quantum mechanical two-level system that can be in a superposition of its two basis states. There are many different proposals for implementing qubits, but one of the most promising ones is to encode the qubit using electron spins trapped in semiconductor quantum dots. Singlet-triplet qubits are spin qubits where the two-electron spin eigenstates are used as the qubit's basis. The required one and two-qubit operations have already been demonstrated experimentally by several research groups around the world in this qubit architecture. The most severe factor limiting the implementation of larger systems of qubits is decoherence. The qubits are not isolated systems, they interact with their environment, which can lead to the loss of quantum information. Few-electron systems can be simulated accurately using first principle methods that become too taxing when the particle number increases. The topic of this thesis is the simulation of quantum dot singlet-triplet qubit systems using accurate exact diagonalization based methods. The emphasis is on the realistic description of qubit operations, both single-qubit ones and those involving the interaction between neighboring qubits. The decoherence effects are also discussed alongside with certain proposals to alleviate their effects.
The computer industry has seen an immense development in the last decades. \nPersonal computers have become available for everybody living in industrialized \ncountries with rapidly increasing performance in terms of speed and storage capacities. \nHowever, the performance of nowadays' computers is fundamentally limited by the \nlaws of classical physics: a classical bit can only take on either of the two distinct \nvalues `0' or `1'. \nIn contrast, a quantum computer could, in principle, make direct use \nof quantum phenomena, such as state superpositions \n-- a quantum bit can be in both states `0' and `1' simultaneously --, \nto perform complex computational tasks much faster than any classical computer. \n \nThe idea of building computers that work according to the laws of quantum physics \nhas opened various fields of research, one of which is the search for the best \nphysical system to use as a quantum bit (qubit). One important criterion for determining \nthe optimal qubit system is the lifetime of state superpositions. \nTypically, once initialized, such superpositions are destroyed on remarkably short \ntimescales due to interactions with the environment -- a process which is referred to \nas decoherence --, posing the question which physical qubit candidate system might show a high-enough robustness against the \ninfluence of the `outside world' to allow for viable quantum computation. \n \nIn this thesis, we will consider three particular realizations of one specific and \nvery promising type of qubit candidate system: \nan electron (or hole) confined to a quantum dot -- a nanoscale structure \nwithin a (typically semiconducting) material --, where the spin states `down' \nand `up' of our electron (or hole) encode the logical states `0' and `1'. \nOur task will be to study the decay of spin-state superpositions in such quantum-dot \nsystems. The main objective of this thesis is to understand the most important physical \nprocesses that lead to spin decoherence and to show ways to suppress this undesirable effect. \nIt turns out that at low temperatures, the main source of decoherence \nis the coupling of the electron (hole) to the surrounding nuclear spins. \n \nThis thesis is divided into three logical parts, corresponding to the three qubit candidate \nsystems under consideration. First we will study electron-spin qubits in III-V semiconductor \nquantum dots, where the electron spin interacts with the nuclear spins of the semiconducting \nhost material \nvia the isotropic Fermi contact hyperfine interaction. Second we consider quantum-dot-confined \nheavy holes and the decoherence of their (pseudo-)spin states due to anisotropic interactions with the \nnuclear spins. Third and last, we study electron-spin qubits made from carbon-nanotube and \ngraphene quantum dots. Quantum dots made of carbon have the advantage of a low abundance \nof spin-carrying nuclear isotopes, therefore reducing decoherence effects significantly. \n \nFor each of the systems under consideration, we will carry out analytical calculations on \nthe nuclear-spin interactions and the spin dynamics of the qubit. Although one main goal of \nthis thesis is to show ways to extend spin decoherence times, we will also focus on physically more \nfundamental questions. Not only the timescale of the decay is relevant for \nthe system's applicability as a qubit, but also the form of the decay which can vary \nsignificantly from system to system. For example, the decay of spin-state superpositions \ncan follow an exponential, super-exponential or power-law decay, and can even pass through \nvarious stages. This is not only of academic interest, but also important for practical \npurposes, such as the implementation of quantum error-correction schemes in a potential \nquantum computer. \n
A two-level quantum system is the building block of a quantum computer. This pair of quantum states defines the computational unit of a quantum computer, and it is called a quantum bit or qubit in analogy to the bit that is the binary unit of a classical computer. The spin of a single electron naturally defines such a two-level quantum system. A quantum dot can be tuned to the single-electron regime, and an array of singly occupied quantum dots provides a multi-qubit register. This thesis describes the multielectron encoding of a spin qubit using a pair or a trio of quantum dots, and it analyzes the coherence properties and manipulation protocols of this system. The singlet-triplet qubit, coded using the singlet state and the spinless triplet state of a pair of singly occupied quantum dots, can be controlled all electrically when applying voltages to gates close to the quantum dot structure. Rapid, subnanosecond modifications of the double quantum dot's charge configuration have been realized successfully. A small magnetic field gradient across the double quantum dot enables universal qubit control. Two modifications of the normal singlet-triplet encoding are introduced. (1) The six-electron configuration of a double quantum dot encodes a singlet-triplet qubit in the same way as for the two-electron double quantum dot. Two electrons at each quantum dot are irrelevant for the qubit manipulations because they are paired in a singlet state; the remaining two electrons encode the qubit. The qubit's wave function is immune to charge noise at moderate out-of-plane magnetic fields. (2) A singlet-triplet qubit encoded using two quantum dots of different sizes has an orbital state degeneracy of the singlet state and the spinless triplet state at finite out-of-plane magnetic fields. Spin-orbit interactions lift this state degeneracy, while the magnitude of the state coupling is determined by the size difference of the QDs. This setup enables the manipulations of singlet-triplet qubits without the need for magnetic field gradients. Finally, two-qubit gates between singlet-triplet qubits are proposed that use mediated exchange interactions via one quantum state. These operations are well controlled and highly noise insensitive. Orbital interactions alone can control spin qubits coded in a three electron Hilbert space. For example, the exchange-only qubit is encoded using three singly occupied quantum dots. The exchange interactions of two quantum dot pairs need to be modified to manipulate this qubit. The noise sensitivity of the exchange-only qubit is discussed. Alternatively, a three-electron qubit at a double quantum dot can be operated when single electrons are transferred between the quantum dots. Fast, subnanosecond manipulations of the double quantum dot's charge configuration are required to realize single-qubit gates. A novel two-qubit pulse gate for the three-electron double quantum dot qubit is proposed.
This thesis addresses the concept of quantum computing with semiconductor quantum dots. The basic unit of a quantum computer is a quantum mechanical two-level system, the so-called quantum bit (qubit). The qubit can be defined as the spin of an electron confined in a quantum dot or as a two-dimensional subspace of the Hilbert space for several spins. Some semiconductors have several minima in their conduction band, socalled valleys. A two-dimensional valley degree of freedom can also be considered as a qubit. In this thesis, quantum registers storing spin as well as valley qubits are described theoretically. Virtual hopping yields an exchange interaction between neighboring quantum dots, which reduces the energy of antisymmetric spin-valley states according to the Pauli principle. Considering the spin and the valley degrees of freedom of two electrons in neighboring quantum dots as qubits, the exchange interaction is a four-qubit interaction. In this thesis, it will be shown that it is, nevertheless, possible to generate a universal two-qubit gate for spin qubits or for valley qubits by combining the exchange interaction and individual single-qubit gates. The exchange interaction in this two-electron double quantum dot directly provides a universal two-qubit gate for one spin and one valley qubit as well. For this, the qubits are defined as suitable subspaces, spanned by the singlet and one triplet state of the spin and the valley states, respectively, Using those gates, arbitrary quantum operations can be performed for single-spin and single-valley qubits in the same quantum register, where in one double quantum dot the quantum states are restricted to the singlet-triplet subspace. While in this register single-spin and single-valley rotations are required, these gates are dispensable in a quantum register storing spin and valley singlet-triplet qubits. The single-qubit operations, in this case, can be provided by the exchange interaction and a gradient in the energy splitting of spin and valley states when the spin qubit is stored in a double quantum dot with spin degrees of freedom only and when the spin states in the double quantum dot which stores the valley qubit are polarized. For providing the two-qubit gate, the spin and the valley qubits need to be stored in the same double quantum dot. The crucial task for this register is to interchange the spin of a spin-only and a spin-valley quantum dot in order to switch between the single-qubit and the two-qubit operation mode. This interchanging of the spin can be achieved by a valley-depending virtual hopping between the quantum dots. Furthermore, echo sequences for a so-called exchange-only qubit are considered. An exchange-only qubit is defined as a subspace in a system of three electron spins in a triple quantum dot. An inhomogeneous magnetic field, which may be caused by nuclear spins in the semiconductor, can lead to decoherence and leakage, i.e., the state might leave the qubit subspace. Suppressing this effect by an echo sequence which consists of pulses interchanging neighboring spin states, so-called SWAP gates, was proposed. Controllability over these quantum operations is provided via the exchange interaction. In this thesis, the performance of different such sequences is investigated. The analysis shows that optimization strategies developed for single spin echoes can be adapted for the three spin system. The echo sequences can be different in their order of applied SWAP gates interchanging the spins in the first and second dots or in the second and third dots. The noise suppression performance depends slightly on this order of SWAP gates. For two spins in a double quantum dot a scheme for quantum state tomography is presented. The corresponding measurement operators are constructed from a set of mutually unbiased bases of the four-dimensional two-qubit Hilbert space. These measurements are represented by short sequences of no more than three elementary quantum gates and projective measurements by the well-established spin-to-charge conversion.
The recent discovery of efficient quantum algorithms for factoring and database search has shown that quantum computing would allow to solve important problems which are intractable with conventional computers. In contrast to the very demanding task of building a large-scale quantum computer, there are quantum communication protocols, e.g. quantum key distribution for cryptography, which—though still difficult—require much less effort and can be implemented with current technology. Apart from the technological motivation, the study of quantum information offers (at least) two additional benefits. First, new insight into fundamental questions on quantum mechanics, e.g. concerning non-locality and entanglement, are gained from an information-theoretical approach. And second, investigating a particular physical implementation of quantum information can give rise to independent physical results. Spintronics, the use of spin as opposed to charge in (classical) electronics is a new field for which some results presented here could be relevant. In this dissertation we investigate several theoretical aspects of the physical implementation of quantum computation and communication in which the fundamental unit of quantum information, the qubit, is represented by the spin of electrons in semiconductor quantum dots. The required coupling between the spins can be obtained by allowing for tunneling of electrons between adjacent dots, leading to a Heisenberg exchange coupling J S1 · S2 between the spins, a scenario which we study for laterally coupled quantum dots in a two-dimensional electron system, and for a three-dimensional setup with vertically coupled quantum dots. Furthermore, an alternative scheme to couple the spins via the interaction with an optical cavity mode is presented. Quantum error correction represents one of the important ingredients for the physical implementation of a quantum computer by protecting \nit from the e�ects of a noisy environment. As a �rst test for errorcorrection \nin a solid-state device using spins, we propose an optimized \nimplementation of the most primitive error correction scheme (the threebit \ncode). In this context, we introduce parallel switching, allowing \nto operate a quantum computer more e�ciently than the usual serial \nswitching. \nCoupling spins with the exchange interaction J S1 �S2 is not su�cient \nfor quantum computation; the spins also have to be addressed individually \nusing controllable local magnetic �elds or g-factors giBi �Si in order \nto allow for single-qubit operations. On the one hand, we discuss several \nschemes for overcoming the di�culty of applying local magnetic �elds \n(requiring large gradients), e.g. g-factor engineering, which allows for \nall-electric operation of the device. On the other hand, we show that \nat the expense of additional devices (spins) and switching operations, \nsingle-spin rotations can be dispensed with completely. \nAddressing the feasibility of quantum communication with entangled \nelectrons in mesoscopic wires, i.e. interacting many-body environments, \nwe propose an interference experiment using a scattering set-up with an \nentangler and a beam splitter. The current noise for electronic singlet \nstates turns out to be enhanced (bunching), while it is reduced for triplets \n(antibunching). Due to interactions, the �delity of the entangled singlet \nand triplet states is reduced by z4F \nin a conductor described by Fermi \nliquid theory, zF being the quasiparticle weight factor. \nFinally, we study the related but more general problem of the noise of \nthe cotunneling current through one or several tunnel-coupled quantum \ndots in the Coulomb blockade regime. The various regimes of weak and \nstrong, elastic and inelastic cotunneling are analyzed for quantum-dot \nsystems (QDS) with few-level, nearly-degenerate, and continuous electronic \nspectra. In contrast to sequential tunneling, the noise in inelastic \ncotunneling can be super-Poissonian. In order to investigate strong cotunneling \nwe develop a microscopic theory of cotunneling based on the \ndensity-operator formalism and using the projection operator technique. \nWe have derived the master equation for the QDS and the current and \nnoise in cotunneling in terms of the stationary state of the QDS. These \nresults are then applied to QDS with a nearly degenerate and continuous \nspectrum.
Quantum information processing in semiconductor quantum dots using single and multi-spin qubitsThe idea of a quantum computer, a device capable of processing information using quantum bits (qubit) instead of classical bits that outperforms classical computer in terms of computation speed, is still science ction. However, recent advances in quantum information processing showed that this ction may come true faster than anticipated. A promising platform for the physical realization of a qubit are spin qubits where the information is encoded in the quantum mechanical spin of an electron con ned in a semiconductor quantum dot. There are a variety of di erent implementations how the logical qubit is physically encoded, the use of a single electron spin or the use of the spins of multi electrons. This thesis focuses on the implementation of the spin- 1/2 qubit using the spin of a single electron in a single quantum dot, the exchangeonly (EO) qubit using three electron spins distributed in three quantum dots, and the novel quadrupolar exchange-only (QUEX) qubit using four electron spins distributed in three quantum dots. In this thesis we investigate several theoretical aspects of the physical implementation of the aforementioned qubit encodings. Since all three qubit encodings are allelectric controlled these implementations su er decoherence, the loss of information, from electric uctuations of the environment commonly called charge noise. Typically, the use of more quantum dots increases the decoherence due to charge noise while it decreases the decoherence due to magnetic noise. The rst main aspect investigated in this thesis is the protection of the quantum information encoded in the di erent qubit implementations against charge noise. Special attention is paid to the multi-electron spin implementations where high-symmetry points of operation are discovered which are less sensitive to the charge noise. Protection against charge noise is even better by increasing the number of electrons while keeping the number of quantum dots making the QUEX qubit superior to the EO qubit. So far the main direction in this thesis is the reduction of the qubit sensitivity to electric uctuations. However, coherent electromagnetic elds in form of microwaves v can be used to interconnect qubits. In this thesis also a high-performance coupling between two spin-1/2 is investigated. The use of a resonator, subsequently, allows for a coupling of qubits over macroscopic distances as well as to extract valuable information about the physical system. Because of the larger susceptibility of multi-spin qubits to the electric eld of the resonator, this thesis focuses on the coupling between an EO or a QUEX qubit and a microwave resonator. Points of operations are discovered for both qubit implementations which o er a strong coupling towards a microwave resonator while maintaining their protection against charge noise. Subsequently, a series of measurements are proposed to extract information about the microscopic structure of the quantum dots. Finally, a promising high-performance protocol for coupling two qubits remotely via the resonator is investigated in detail.
It was Richard Feynman who first proposed, in 1982, the far-reaching concept of \na ”quantum computer”—a device more powerful than classical computers. The idea \nof a quantum computer is to employ the fascinating and often counterintuitive \nlaws of quantum mechanics to process information. It is far from obvious that \nthe proposed concept of a quantum computer is more powerful than its classical \ncounterpart, it was only in 1994 when Peter Shor theoretically demonstrated the \nexistence of a quantum algorithm for factorizing integers into prime factors \nthat runs in polynomial time unlike its classical counterpart which works in \nsub-exponential time. The factorization of integers into prime factors is the \nbasis of asymmetric cryptography. These early theoretical results lunched an \nimmense interest of the scientific community. Already during ’90s, the first \nproposals for the physical implementation of quantum computation emerged. Ever \nsince, many experimental groups around the world pursued different physical \nimplementations of quantum bits (qubits). The first decade of the new century \nsaw a steady improvement in the control and decoherence time (the time over \nwhich the information carried by the qubit is lost) for various qubits by many \norders of magnitude. The natural next step in this context is to answer the \nquestion of how to scale the system up to include many qubits and thus build a \nquantum computer? One of the main parts of this thesis addresses exactly this \nquestion, namely the question of architecture and scalability of future quantum \ncomputer. \n \nAmong various different physical realizations of qubits, the idea of using \nelectron spins trapped in electrostatic semiconductor quantum dots as the \nbuilding blocks of a quantum computer (the so-called spin qubits), put forward \nby Daniel Loss and David DiVincenzo in 1997, triggered tremendous interest in \nscientific community. Nevertheless, the implementation of the original \nLoss-DiVincenzo proposal posed a considerable technical challenge. It used \nquantum tunneling between qubits to enable their communication with each other, \nand thus required that the qubits to be placed very close to each other. This \nrequirement not only leaves little \nspace for the placement of the vast amount of gates and wirings needed to define \nthe electrostatic quantum dots, but also makes it challenging to control the \nlocal magnetic field needed for single-qubit operations. For these reasons, no \nsystem with more than a couple of spin qubits has been \nsuccessfully implemented thus far. In the first part of this thesis, we leap \nover this long-standing problem with an entirely different strategy of using \nmetallic floating gates or ferromagnets to couple together qubits that are \nseparated over a long distance. Our scheme works for any type of \nspin qubits, including the qubits based on nitrogen-vacancy center (NV-center) \nin diamond and technologically very important silicon qubits. \n \nThe main topic of this thesis is related to quantum computer. Still, quite \nunexpectedly, some of the ideas we employed in order to tackle the problem of \nquantum computer scalability can be utilized in a completely different field of \nresearch, namely, in the field of magnetic field sensing. Qubit are not only a \nnecessary ingredient of quantum computer but they also provide a way to measure \nvery accurately magnetic fields. The magnetometer build upon the qubit based on \nNV-center, so-called NV-magnetometer, emerged in recent years as most sensitive \nmagnetic moment sensor. These magnetometers are able to detect about hundred \nnuclear spins within a minute of acquisition time. In the second part of this \nthesis, we propose an entirely novel experimental realization of \nNV-magnetometers which increases present NV-center sensitivities by four orders \nof magnitude at room temperature. This unprecedented amplification of \nsensitivity will render magnetometers capable of detecting individual nuclear \nspins. This amplification is achieved by introducing a ferromagnetic particle \nbetween the nuclear spin that needs to be detected and the NV-magnetometer. Our \nsetup, in contrast to existing schemes, is particularly advantageous because, \ndue to the large amplification of sensitivity, the nuclear spin need not lie \nwithin a few nanometers of the surface but rather can be detectable at a \ndistance of 30 nm. With these novelties, our scheme provides chemically \nsensitive spin detection under ambient conditions allowing nanoscale resolution \nof nuclear magnetic moments in biological systems—the holy grail of nuclear \nmagneticresonance. \n \nIn the last part of the thesis we focus our attention to a new direction in \nquantum computer implementation that deals with topological quantum computer \nintroduced by Alexei Kitaev in 1997; in this approach the idea is to use \nquasiparticles with ”fractional” statistics and to perform the single- and \ntwo-qubit gates by merely exchanging these quasiparticles. Additionally, \ninformation in this system is stored non-locally thus it mitigates the problem \nof decoherence caused by local noise from the environment. Majorana fermions are \none of the most well known examples of such excitations. We analyze transport \nsignatures of different topological states in one-dimensional systems, like \nMajorana fermions and fractionally charged states. We envision an Aharonov-Bohm \nsetup wherein conductance measurement provides a clear signature of presence of \nfractionally charged fermionic states, since oscillations with double period \nemerge in this case. Additionally, we propose a very simple setup that enables \nexistence of degenerate mid-gap states, so-called Tamm-Shockley states that are \ncharacterized by fractional charge and discuss possible ways of detecting these \nstates experimentally. \n