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D e p a r t me nto fA p p l i e dP h y s i c s T uukka H il t une n Quant um c o mput at io nw it ht w o e l e c t ro n spins in se mic o nduc t o r quant um do t s Quant um c o mput at io nw it h t w o e l e c t ro n spins in se mic o nduc t o r quant um do t s T uuk k aH i l t une n A a l t oU ni v e r s i t y D O C T O R A L D I S S E R T A T I O N S Aalto University publication series DOCTORAL DISSERTATIONS 66/2015 Quantum computation with twoelectron spins in semi-conductor quantum dots Tuukka Hiltunen A doctoral dissertation completed for the degree of Doctor of Science (Technology) to be defended, with the permission of the Aalto University School of Science, at a public examination held at the lecture hall M1 of the school on 5th June 2015 at 12. Aalto University School of Science Department of Applied Physics Quantum Many-body Physics Group Supervising professor Aalto Distinguished Professor Risto Nieminen Thesis advisor Adjunct Professor Ari Harju Preliminary examiners Professor Esa Räsänen, Tampere University of Technology, Finland Adjunct Professor Jyrki Piilo, University of Turku, Finland Opponent Professor Guido Burkard, University of Konstanz, Germany Aalto University publication series DOCTORAL DISSERTATIONS 66/2015 © Tuukka Hiltunen ISBN 978-952-60-6199-3 (printed) ISBN 978-952-60-6200-6 (pdf) ISSN-L 1799-4934 ISSN 1799-4934 (printed) ISSN 1799-4942 (pdf) http://urn.fi/URN:ISBN:978-952-60-6200-6 Unigrafia Oy Helsinki 2015 Finland Abstract Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto.fi Author Tuukka Hiltunen Name of the doctoral dissertation Quantum computation with two-electron spins in semi-conductor quantum dots Publisher School of Science Unit Department of Applied Physics Series Aalto University publication series DOCTORAL DISSERTATIONS 66/2015 Field of research Theoretical and Computational Physics Manuscript submitted 6 March 2015 Permission to publish granted (date) 23 April 2015 Monograph Date of the defence 5 June 2015 Language English Article dissertation (summary + original articles) Abstract 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. Keywords Quantum computer, Spin qubit, Singlet-triplet qubit, entanglement, decoherence, configuration interaction, Hubbard model ISBN (printed) 978-952-60-6199-3 ISBN (pdf) 978-952-60-6200-6 ISSN-L 1799-4934 ISSN (printed) 1799-4934 ISSN (pdf) 1799-4942 Location of publisher Helsinki Pages 117 Location of printing Helsinki Year 2015 urn http://urn.fi/URN:ISBN:978-952-60-6200-6 Tiivistelmä Aalto-yliopisto, PL 11000, 00076 Aalto www.aalto.fi Tekijä Tuukka Hiltunen Väitöskirjan nimi Kvanttilaskenta käyttäen kahden elektronin spini-tiloja puolijohdekvanttipisteissä Julkaisija Perustieteiden korkeakoulu Yksikkö Teknillisen fysiikan laitos Sarja Aalto University publication series DOCTORAL DISSERTATIONS 66/2015 Tutkimusala Teoreettinen ja laskennallinen fysiikka Käsikirjoituksen pvm 6.3.2015 Julkaisuluvan myöntämispäivä 23.4.2015 Monografia Väitöspäivä 5.6.2015 Kieli Englanti Yhdistelmäväitöskirja (yhteenveto-osa + erillisartikkelit) Tiivistelmä Kvanttitietokone hyödyntäisi kvantti-ilmiöitä, kuten superpositiota ja lomittumista ja mahdollistaisi tiettyjen nykyisille klassisille tietokoneille vaikeiden ongelmien ratkaisemisen. Suuren mittakaavan kvanttilaskenta saattaisi muuttaa yhteiskuntaamme yhtä perustavalla tavalla kuin digitaalisen tietokoneen keksiminen 1900-luvulla. Toimivaan ison mittakaavan kvanttitietokoneeseen on kuitenkin yhä matkaa, ja on useita teoreettisia ja käytännöllisiä ongelmia, jotka täytyy ratkaista ennen kuin tästä tulee todellisuutta. Klassisessa tietokoneessa data tallennetaan bitteihin. Bitin kvanttivastinetta kutsutaan kubitiksi. Se on kvanttimekaaninen kaksitasosysteemi, joka voi olla ominaistilojensa superpositiossa. Kubitti voidaan realisoida monessa erilaisessa systeemissä. Yksi lupaavimmista ehdotuksista on käyttää puolijohdekvanttipisteisiin vangittujen elektronien spinejä kubitin pohjana. Jos kubitin tilat muodostuvat kahden elektronin spin-ominaistiloista, kubittia kutsutaan singletti-tripletti kubitiksi. Universaaliin kvanttilaskentaan tarvittavat yksi- ja kaksikubittioperaatiot on jo toteutettu kokeellisesti singletti-tripletti kubiteilla. Suuren mittakaavan kubittisysteemien implementoimista estävät tällä hetkellä ennen kaikkea dekoherenssi-ilmiöt. Kubitit eivät ole eristettyjä systeemejä; ne vuorovaikuttavat ympäristönsä kanssa, mikä johtaa kvantti-informaation katoamiseen. Muutaman elektronin systeemejä voidaan mallintaa tarkoilla yksittäisten elektronien tasolla toimivilla malleilla. Monet tällaiset mallit ovat liian kompleksisia laskennallisesti suurempien systeemien simuloimiseen, mutta soveltuvat hyvin esimerkiksi muutaman singletti-tripletti kubitin tapaukseen. Tämän väitöskirjan aihe kvanttipiste singletti-tripletti kubittien mallintaminen käyttäen tarkkoja ns. exact diagonalization-metodeita. Painopiste on yksi- ja monikubittioperaatioiden tarkassa kuvaamisessa. Myös dekoherenssi-ilmiöitä ja tiettyjä ratkaisuja niiden helpottamiseksi käsitellään. Avainsanat Kvanttilaskenta, spin-kubitti, lomittuminen, dekoherenssi, CI-metodi, Hubbardin malli ISBN (painettu) 978-952-60-6199-3 ISBN (pdf) 978-952-60-6200-6 ISSN-L 1799-4934 ISSN (painettu) 1799-4934 ISSN (pdf) 1799-4942 Julkaisupaikka Helsinki Sivumäärä 117 Painopaikka Helsinki Vuosi 2015 urn http://urn.fi/URN:ISBN:978-952-60-6200-6 Preface The research in this thesis has been carried in the Department of Applied Physics in the Aalto University School of Science and Technology during the years 2012-2015. The author has been a member of the Quantum Many-body Physics (QMP) group led by Doctor Ari Harju. The QMPgroup is a part of the COMP Centre of Excellence which is funded by the Academy of Finland. I am grateful to my supervisor Aalto Distinguished Professor Risto Nieminen for his participation in the process of making this thesis. Furthermore, I would like to thank Risto and Professor Päivi Törmä, the current head of COMP, for providing excellent working facilities and an opportunity to perform graduate studies in the COMP Centre of Excellence. I thank my instructor Doctor Ari Harju for all of his immense help during these years. Ari’s contributions to this thesis span from the actual simulation codes and computational issues to providing ideas for new research topics. Most importantly though, he has always been available when I have needed advice. For all of this, I am deeply grateful. I would also like to thank my former and current co-workers Mikko Ervasti, Juha Ritala, Topi Siro, and Doctor Zheyong Fan for the many useful discussions related to the topics and articles of this thesis. The atmosphere in the QMP-group has always been informal and relaxed while simultaneously being serious towards doing science; it has been an excellent environment for graduate studies. Despite this, everything does not always go as in the gardening program Strömsö. There have been those days when the computers do not work, the code refuses to compile, and the referees disagree with everything. The off-topic discussions and peer-support from my co-workers Andreas Uppstu, Mari Ijäs, and Ilja Makkonen (and those already mentioned) have helped in getting me through these occasional dark moments. Special thanks goes to Mikko 1 Preface for the TV-show related analyses and to Andreas for all the cookies I have stolen. Finally, I thank my family and friends for everything else. After all, some people say there is more to life than just quantum mechanics. Espoo, April 28, 2015, Tuukka Hiltunen 2 Contents Preface 1 Contents 3 List of Publications 5 Author’s Contribution 7 1. Introduction 9 2. Quantum dots 11 2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Lateral gate-defined semiconductor quantum dots . . . . . . 12 3. Quantum computation with spin qubits 15 3.1 Quantum computation . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Spin qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4 Singlet-triplet qubits . . . . . . . . . . . . . . . . . . . . . . . 21 3.4.1 Two-electron spin states . . . . . . . . . . . . . . . . . 21 3.4.2 Single-qubit operations . . . . . . . . . . . . . . . . . . 22 3.4.3 Two-qubit operations . . . . . . . . . . . . . . . . . . . 26 3.4.4 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . 29 4. Simulation methods 35 4.1 Continuum model . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.1.1 Continuum Hamiltonian . . . . . . . . . . . . . . . . . 35 4.1.2 Single-particle states . . . . . . . . . . . . . . . . . . . 37 4.1.3 Exact diagonalization . . . . . . . . . . . . . . . . . . . 39 4.2 Extended Hubbard model . . . . . . . . . . . . . . . . . . . . 40 3 Contents 5. Discussion 43 Bibliography 47 Publications 53 4 List of Publications This thesis consists of an overview and of the following publications which are referred to in the text by their Roman numerals. I Tuukka Hiltunen and Ari Harju. Y-junction splitting spin states of moving quantum dot. Physical Review B (Rapid Communications), 86, 12, 12301(R), September 2012. II Tuukka Hiltunen, Juha Ritala, Topi Siro, and Ari Harju. Non-adiabatic charge state transitions in singlet-triplet qubits. New Journal of Physics, 15, 10, 103015, October 2013. III Tuukka Hiltunen and Ari Harju. Maximal tripartite entanglement between singlet-triplet qubits in quantum dots. Physical Review B, 89, 11, 115322, March 2014. IV Tuukka Hiltunen and Ari Harju. Capacitative coupling of singlettriplet qubits in different interqubit geometries. Physical Review B, 90, 12, 125303, September 2014. V Tuukka Hiltunen, Hendrik Bluhm, Sebastian Mehl, and Ari Harju. Charge-noise tolerant exchange gates of singlet-triplet qubits in asymmetric double quantum dots. Physical Review B, 91, 7, 075301, February 2015. VI Michiel A. Bakker, Sebastian Mehl, Tuukka Hiltunen, Ari Harju, David P. DiVincenzo. Validity of the single-particle description and charge 5 List of Publications noise resilience for multi-electron quantum dots. Physical Review B, 91, 15, 155425, April 2015. 6 Author’s Contribution Publication I: “Y-junction splitting spin states of moving quantum dot” The author contributed significantly to the writing of the computation codes, did all the computations, and wrote the manuscript. Publication II: “Non-adiabatic charge state transitions in singlet-triplet qubits” The author suggested the research topic, contributed significantly to the writing of the computation codes, performed all the computations, and wrote the manuscript. Publication III: “Maximal tripartite entanglement between singlet-triplet qubits in quantum dots” The author suggested the research topic, wrote the computation code, performed the computations, and wrote the manuscript. Publication IV: “Capacitative coupling of singlet-triplet qubits in different interqubit geometries” The author suggested the research topic, wrote most of the computation code, performed the computations, and wrote the manuscript. 7 Author’s Contribution Publication V: “Charge-noise tolerant exchange gates of singlet-triplet qubits in asymmetric double quantum dots” The author contributed significantly to the writing of the computation codes, did all the computations, and wrote most of the manuscript. Publication VI: “Validity of the single-particle description and charge noise resilience for multi-electron quantum dots” The author contributed significantly to the writing of the computation codes for the configuration interaction calculations and perfomed these computations. 8 1. Introduction The invention of the digital computer and the subsequent rise of the information technology have propelled our society and culture to an unprecedented new era. The technological advances and innovations done in the last twenty years have been so enormous that even the wildest visions imagined in the preceding decades cannot compete with the reality today. This technological shift has altered all aspects of our every day lives in profound ways. The ever-accelerating progress in the computer age has been fueled by the so-called Moore’s law, predicting an exponential rise in the capacities of computers. It has held thus far, but there might be a limit to this exponential growth. Moore’s law is based on the ever-diminishing size of computers’ microchips. The microchips cannot, however, shrink forever. Today, we are approaching the point when the chips have become so small that quantum mechanical phenomena can inhibit their functioning. Instead of considering the quantum effects as a mere error source, one could also try to harness them. A fundamentally new kind of a computer, a quantum computer, has indeed been a subject of intensive theoretical and experimental research in the last couple of decades. 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- 9 Introduction 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. These systems, the confinement of the electrons in the quantum dots, can be defined and controlled accurately using electric pulses. There are many ways how the spin degree of freedom can be used to encode qubits, but the one most relevant to this thesis is the proposition to use the two-electron spin eigenstates as the qubits’ basis states. Such qubits are often called singlet-triplet qubits and they offer many advantages over the superficially simpler one-spin qubit. The required one and two-qubit operations, including the initiation and measurement of the qubit’s states, have already been demonstrated experimentally by several research groups around the world with singlet-triplet qubits. The basic pieces of the puzzle are in place, but there is still a lot of what we do not understand, many problems to be solved. 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. Solid theoretical understanding of the underlying phenomena is a requisite for the successful experimental implementation of these qubit systems. 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 with accurate exact diagonalization based methods. The emphasis is on the realistic description of qubit operations, both singlequbit ones and those involving the interaction between neighboring qubits. The decoherence effects are also discussed alongside with certain proposals to alleviate their effects. This thesis is organized as follows. Chapter 2 provides an overview to the concept of a quantum dot, with emphasis being on the lateral gate defined semiconductor dots that for the basis of in the singlet-triplet qubit scheme. In Chapter 3, the topic is the fundamentals of quantum computing using electron spins. The singlet-triplet qubit architecture is discussed with more detail, including the proposals and advances done in this thesis. In Chapter 4, the computational and theoretical methods used are reviewed. Chapter 5 summarizes the contents of this thesis. 10 2. Quantum dots 2.1 Basics A quantum dot (QD) is an artificial structure that can be filled with either electrons or holes. The dot can exchange charges with reservoirs coupled to it through tunnel barriers, and its electronic properties can be measured using current and voltage probes [1, 2]. Having discrete spectra of eigenstates, QDs are often described as artificial atoms. They allow the study and application of few electron level physics in a controlled manner [3]. The term “quantum dot” is quite loosely defined; there is a multitude of different physical systems often called as quantum dots. QDs can be realized using for example trapped molecules, different kinds of nanoparticles, semiconductor systems, and carbon nanotubes [2]. All these different realizations, however, share the same essential properties. The energy levels in a QD are discrete as the system is confined in all three dimensions. The stronger the confinement is, the larger the separations between energy levels are. The three-dimensional confinement also results in making the Coulomb interactions between electrons stronger. It has been proposed that the strong correlations may allow the study of many exotic physical phenomena, such as the fractional quantum Hall effect [4–6], using QD systems. QDs have several promising applications, for example in light emitting devices [7, 8], solar-cells [9], and imaging in biological and medical sciences [10, 11]. However, the one application at the heart of this thesis is the use of QDs in quantum information. More specifically, the main focus is on two-electron spins in semiconductor double QD devices being used as qubits [12, 13]. The next sections concentrate on the definition and 11 Quantum dots Figure 2.1. The formation of the two-dimensional electron gas in the heterostructure of GaAs and n-doped Alx Ga1−x As. When brought in contact, the differing Fermi-levels of the two materials are matched, resulting in their energy bands bending and a potential well forming at the interface. Electrons occupying the bound states of the well are strongly confined in one direction and the 2DEG is formed (figure adapted from Ref. [14] by Davies). fabrication of the lateral gate defined dots and the experimental progress made in this field. 2.2 Lateral gate-defined semiconductor quantum dots In this thesis, the QDs discussed are all of the lateral semiconductor kind. The name ’lateral quantum dot’ derives from the lateral gate geometry of these electrostatically confined QD systems. In them, the movement of electrons is first restricted to two dimensions, and a two-dimensional electron gas (2DEG) is formed. The actual QD is then created electrostatically by gate voltages. A potential landscape is formed as different voltages are applied to the lateral gates. Individual electrons can be trapped in the minima of the electric potential and a few-electron QD is created. The number of trapped electrons can be controlled reliably using suitable gate geometries [2]. The 2DEG essential to this QD realization is formed at the interface of certain semiconductor materials. The most typical choice of material is the heterostructure of epitaxially grown layers of GaAs and AlGaAs, where the latter is an alloy of AlAs and GaAs, with the chemical formula Alx Ga1−x As. The AlGaAs-layer is doped with Si, creating free electrons in the structure. When the two materials are connected, their Fermi-levels are matched, resulting in their energy bands bending, and the dopants in AlGaAs transferring to GaAs so that an electrical potential minimum is formed at the interface. The excess electrons from the Si-dopants occupy the bound states of the well. The electrons are strongly confined in the zdirection (see Fig. 2.1), so strongly that their movement in that direction can often be disregarded, and the 2DEG is formed. 12 Quantum dots Figure 2.2. Left: a schematic of a lateral gate-defined quantum dot device. The 2DEG (light gray) is formed between the AlGaAs and GaAs layers. Metal gate electrodes (dark gray) define the potential of the dot and voltages applied to them can be used to control the number of electrons in it (figure taken from Ref. [2] by Hanson et al.). Right: A scanning electron micrograph of a lateral double dot device by Elzerman et al. [17]. The gate electrodes are shown in light gray, and the two white circles denote the locations of the dots. The electron occupancy of the dots can be measured with the quantum point contacts. The QD is created when the electrons are further confined in the directions of the 2DEG using gate electrodes (see Fig. 2.2). A suitable design of the gate structure allows small islands of electrons to be separated from the 2DEG. The dot is connected to electron reservoirs via low-resistance leads and the current due to the transport of the electrons through the dot can be used to measure its electrical properties. A nearby electrometer, such as a quantum point contact (QPC) [15], can also be used to detect the charge inside the dot, allowing a non-invasive probing of the properties of the QD system. It should be noted that in order to access the quantum phenomena discussed in this thesis, the GaAs/AlGaAs-system has to be cooled below 1K [2]. The first experiments demonstrating a successful implementation of a lateral few-electron QD device were done by Ciorga et al. in 2000 [16]. Systems of multiple coupled QDs are often called QD molecules. The first demonstration of a lateral double quantum dot (DQD) system was done by Elzerman et al. in 2003 [17] (see Fig. 2.2). In their design, QPCs were used to measure the electron occupancy of the dots. This particular design has later become the standard in the field and is used by several different research groups around the world in fabrication of DQDs. Systems larger than two dots have also been implemented. A triple dot was first demonstrated by Gaudreau et al. in 2006 [18], and systems of two DQDs have been used by several groups [19, 20]. Single electrons can also be trapped from the 2DEG into the minima of a surface acoustic wave (SAW). In piezoelectric materials, such as GaAs, SAWs create a moving potential modulation that can then be used to transport electrons between two QDs separated in space. This is a fea- 13 Quantum dots ture that is by no means trivial as for distances larger than a few hundred nanometers neither free propagation nor tunneling are viable transfer methods [21]. SAW induced single-electron transport between two QDs via a linear channel has been demonstrated experimentally [21, 22]. SAW transport can also be done through X or Y-shaped junctions [23–25]. 14 3. Quantum computation with spin qubits 3.1 Quantum computation Despite the enormous leaps done in information science in the last few centuries, many computational problems still greatly exceed the capabilities of even the best supercomputers. A classical example of such an immensely taxing problem is the prime factorization of very large integer numbers that becomes relevant in the context of many cryptographic protocols. One possible way to solve some of these hard problems is to build a completely new kind of a computer that exploits quantum mechanical phenomena to enhance its computational power. The invention of the concept of a quantum computer and the field of quantum information science is often attributed to Richard Feynman and his 1982 paper [26]. The idea was further developed by Deutsch in 1985 [27], when he presented the first quantum algorithm showing that a quantum computer can indeed perform certain tasks faster than its classical counterpart. The interest in the field exploded with the famous Shor algorithm [28] which could be used to solve the aforementioned prime factorization problem in polynomial time, phenomenally faster than classical computers. Classical computers encode data into bits; each bit has a value of either 0 or 1. The main idea behind a quantum computer is that the bits, called qubits (quantum bit) in this context, are quantum mechanical two-level systems spanned by the states |0 and |1. Due to the principles of quantum mechanics, a quantum bit can then also be in a superposition of 0 and 1. This fact makes the quantum computer fundamentally different from the classical one, as each qubit can be simultaneously 0 and 1 and everything in between. A qubit can either be a true two-level system (like the polarization angles of a photon) or an approximate one in which case the 15 Quantum computation with spin qubits low lying subspace of the states |0 and |1 should be separated in energy from the rest of the eigenstates of the system or otherwise decoupled from them. In addition to the spin qubits [12, 13] that are the topic of this thesis, there are many other promising realizations for a qubit, for example atoms and ions [29] and superconducting qubits [30]. A general state |ψ of a qubit can be written as |ψ = α0 |0 + α1 |1, where α0 and α1 are complex numbers satisfying the normalization condition, |α0 |2 +|α1 |2 = 1. The possible states can thus be represented as the surface of a sphere. In this so called Bloch-sphere representation, a general state of a qubit can be written as (up to a global phase that has no physical meaning in this context), |ψ = cos θ θ |0 + eiφ sin |1, 2 2 (3.1) where θ ∈ [0, π] and φ ∈ [0, 2π] are the polar and azimuthal angles in the spherical coordinates, respectively. The state |ψ is thus equated with a point defined by the angles θ and φ on the sphere’s surface and the basis states |0 and |1 are represented by the north and south poles. In the ’equator’ of the sphere (called also the xy-plane), both qubit states 0 and 1 have equal occupations. The most studied and thus best known model of quantum computation is the so-called circuit model [31], although there are also other schemes such as the one-way quantum computer [32] and adiabatic quantum computation [33]. In the circuit model, the quantum computer works in principle quite similarly to its classical equivalent. A single computation is a specific sequence of quantum gates, reversible transformations on an n-qubit register. An important requirement that a functioning quantum computer should fulfill is universality, i.e. that given any classical reversible circuit, a quantum circuit consisting of certain basic building blocks of gates can simulate it. There exists a theorem saying that a quantum circuit consisting just of single-qubit rotations (see Sec. 3.4.2) and a two-qubit gate, such as the complex phase (CPHASE) gate (see Sec. 3.4.3) can meet the requirements of universality [34]. The universality is actually one of the five so-called Loss-DiVincenzo criteria, requirements that have to be fulfilled in a physical implementation of a quantum computer [35]. The criteria are: 1) a scalable physical system with well defined qubits, 2) reliable state preparation, 3) much longer decoherence times than the gate operation times, 4) universal set of quantum gates, and 5) a qubit specific measurement capacity. 16 Quantum computation with spin qubits The field of quantum information and computing is still in its infancy. Although the principles are clear and a lot of progress has been made in the last few centuries, there is still a long way to go until a functioning large-scale quantum computer becomes reality. The biggest obstacles to be overcome are issues related to scalability and decoherence. The former means that any useful quantum circuit consists of an enormous number of qubits and gates, while thus far researchers have been able to make only very small number of qubits working together in a controlled and reliable manner. Decoherence, on the other hand, is a very serious problem stopping much of the progress that would otherwise be made. The interaction of the qubits with their environment results in the quantum superpositions being short-lived and the qubit systems basically leaking information out (see Sec. 3.4.4 for more on quantum decoherence). There are some proposed remedies to the decoherence problem, but it’s still far from being solved. 3.2 Entanglement Entanglement is a quantum phenomenon that has no analogue in classical physics. Pairs or groups of particles can interact in ways that the state of each particle or subsystem can no longer be described individually. Instead, the whole system behaves like single entity. Entanglement was first ’discovered’ as an argument against the plausibility of the whole quantum mechanics is the famous paper by Einstein, Podolsky, and Rosen in 1935 [36]. The phenomenon was thought to be so peculiar and alien to human understanding that Einstein et al. reasoned that the laws of quantum mechanics predicting it must be wrong. Later on, it was discovered that entanglement indeed is just a part of how the nature works in the quantum scale. Entanglement is at the heart of the whole field of quantum information, having numerous applications from quantum cryptography and teleportation to actual quantum computing. For example, the many-qubit gates required for universality in the circuit model of quantum computation are entangling in nature [37]. An example describing the effects of entanglement involves a state |Ψ+ = √1 (|01 2 + |10) of a system of two qubits, A and B. Here |01 = |0A ⊗ |1B , i.e. a state with qubit A in |0 and qubit B in |1. Now, suppose the qubits are separated by a large distance, and the state of the qubit A is measured. If the result is 0, |ψ has collapsed to |01 and the state of the qubit 17 Quantum computation with spin qubits B must be 1. Similarly, if A’s measurement yields 1, B will be in the state 0. Here, the measurement of the state of A seems to affect the state of B, instantaneously regardless of where B is located in space. Einstein called this phenomenon ’a spooky action in distance’. The state |Ψ+ is indeed an entangled state, one of the so called four Bell-states. These are defined as 1 1 |Ψ± = √ (|01 ± |10), |Φ± = √ (|00 ± |11). 2 2 (3.2) The Bell states are maximally entangled two-qubit states. Entanglement can be thought of as a resource; generating it for a given task requires a lot of effort, and in general it diminishes in the process of that task. The amount of how much a given state is entangled can be determined using an entanglement measure. There are many different kinds of such measures, but the ones relevant to this thesis are concurrence for twoqubit systems and its generalizations for a system of three qubits [37–39], and these will be introduced in the next paragraphs. First, define a general state |ΨNq of Nq qubits as |ΨNq = αk1 k2 ...kNq |k1 k2 ...kNq . (3.3) k1 ,k2 ,...,kNq =0,1 Here, |k1 k2 ...kNq = |k1 1 ⊗ |k2 2 ⊗ ... ⊗ |kNq Nq and the αs are complex numbers satisfying the normalization of the wave function |ΨNq . In the case of two qubits, Eq. (3.3) becomes |Ψ2 = α00 |00 + α01 |01 + α10 |10 + α11 |11. The concurrence can then be computed as C = 2|α00 α11 − α01 α10 |. (3.4) Concurrence assumes values between 0 and 1, and the higher the value, the more entangled the state. C = 1 corresponds to a maximally entangled Bell state and C = 0 to an non-entangled state or a product state [37]. Entanglement in a three-qubit system is a more complex phenomenon than the above two-qubit case, exhibiting several inequivalent classes [37]. Any two qubits of the system can be entangled pairwisely, but there exists also genuine three-qubit entanglement as well. Tn fact, there are two different classes of it, the Greenberg-Horne-Zeilinger (GHZ) states [40] and the W-states [37, 38]. The entanglement properties of a threequbit state can be characterized by using pairwise concurrences and the three-tangle [38]. Given a general tripartite wave function of the form as in Eq. (3.3), the three-tangle can be computed as τ123 = 2 αijk αi j m αnpk αn p m ηii ηjj ηkk ηmm ηnn ηpp , 18 (3.5) Quantum computation with spin qubits where the sum goes from 0 to 1 for all indices, with η01 = 1, η10 = −1, and ηii = 0. The value of τ123 is between 0 and 1, and it is maximal for the GHZ-states. Conversely, the W-class states have zero tangle [38, 39]. The entanglement in a subsystem of two qubits (out of total three) can be measured using pairwise concurrences. For example the pairwise con√ √ √ currence C12 of qubits 1 and 2 is obtained as C12 = max{ λ4 − λ3 − λ2 − √ λ1 , 0}, where the λs are the eigenvalues of the matrix R12 = ρ12 (Y ⊗ Y)ρ̄12 (Y ⊗ Y) (3.6) in an ascending order. Here, ρ12 is the reduced density matrix of the pair 1 and 2 obtained by tracing out the qubit 3 from the full 8 × 8 density matrix of the system, Y is the y-Pauli matrix, and ρ̄12 denotes complex conjugation. The W-states maximize all three pairwise concurrences, so 2 + C 2 + C 2 = 4/3. In the GHZ-states, the pairwise concurrences that C12 13 23 are zero [38]. In the context of a general many-qubit system, the classification of different kinds of entanglement becomes more and more complex as the qubit number is increased. The two-body Bell state and three-body GHZstate are actually special cases of a general class of highly entangled many-qubit states called cluster states [41, 42]. 3.3 Spin qubits Electron spin (and the spin of other particles such as protons and neutrons) is a relativistic property that arises as the fourth quantum number describing the electron’s state (the other three correspond to the three spatial dimensions) when solving the electron’s Dirac equation. The spin of an electron is a quantum mechanical observable represented by the operator Ŝ 2 = Ŝx2 + Ŝy2 + Ŝz2 . Here, Ŝx , Ŝy , and Ŝz are the projections of the spin onto the three Cartesian coordinates. For all fermions (e.g. electrons), Ŝ has the value h̄/2 = sh̄, with s = 1/2 being the spin quantum number. The spin state of the electron usually means the eigenstates of the Ŝz2 -operator, corresponding to its two eigenvalues −h̄/2 and h̄/2 (or sz h̄, with sz = ±1). These eigenstates are often called the spin down and up states, and denoted with | ↓ and | ↑. The existence of this kind of a non-spatial degree of freedom was first observed experimentally in the famous Stern-Gerlach experiment in 1922 [43]. In the experiment, a beam of silver atoms was split in two as it was led through a magnetic field. The reason for this is 19 Quantum computation with spin qubits that the magnetic momentum of an electron is defined by its spin state. Electron spins trapped in lateral gate defined QDs were proposed as qubits by Loss et al. in 1998 [12]. In this qubit architecture, the |0 and |1 states of the bits would be the | ↓ and | ↑-states of the spin of a trapped electron. The spin state of the electron can be changed using a local magnetic field that couples to its magnetic momentum, and this procedure can be used to create the one-qubit gates required for universality. In this scheme, the coupling of two adjacent qubits is achieved by exploiting the exchange interaction (more on this in Sections 3.4.1, 3.4.2, and 3.4.3) arising from the overlap of the wave functions of the qubits’ electrons. Coherent oscillations of a single electron spin in a GaAs/AlGaAs-quantum dot were demonstrated by Koppens et al. in 2006 [44]. Quantum dot spin qubits can also be realized using certain eigenstates of the many-body spin operators Ŝ 2 and Ŝz as the qubit’s basis states. For N 2 N spin-1/2 fermions, these operators are defined as Ŝ 2 = i=1 Ŝi and N Ŝz = i=1 Ŝz,i , where Ŝi2 and Ŝz,i are the one-particle spin operators corresponding to particle i. The many-body Ŝ 2 and Ŝz have the eigenvalues s2 h̄ and sz h̄, respectively, with the quantum numbers s and sz satisfying s ∈ {|σ1 /2 + σ2 /2 + ... + σN /2| : σi = ±1 ∀i}, sz ∈ {−s, −s + 1, ..., s}. (3.7) If the qubit states |0 and |1 are encoded into the many-body spin, the qubit consists of multiple electrons in several QDs. The most common and widely studied case is the singlet-triplet qubit, i.e. two electrons in a DQD, a qubit architecture first proposed by Levy in 2002 [13]. This qubit realization is also the main topic of this thesis and it is discussed in detail in the following sections. Three-electron qubits in triple QDs are another possibility and one-qubit operations in them have been demonstrated experimentally [45–47]. There has also been proposals on many-electron qubits in double dots [48, 49]. The main advantage of the many-electron realizations over the singlespin qubits is that the many-particle spin eigenstates sometimes have natural protection against certain kinds of decoherence. This is true especially in the case of the nuclear spins of the host material that cause a severe decoherence channel in GaAs-QD qubits [50, 51]. Both Ga and As in GaAs/AlGaAs-heterostructures have non-zero nuclear spins. These couple to the spins confined in the QDs via the hyperfine interaction and result in an effective random magnetic field across the material. In case of the single spin qubits, the effective field affects the energies of the qubit states via the Zeemann-interaction and can lead to spin-flip errors in the 20 Quantum computation with spin qubits qubit’s operation. The many-body spin states are often more weakly coupled to the nuclear spins and the effects due to them are thus less severe. Another possibility to mitigate the effects of the hyperfine coupling is to move to host material that has zero nuclear spin (or a greatly smaller number of non-zero spin host atoms). In silicon, the concentrations of non-zero nuclear spin isotopes are low and silicon QD based qubits have showed a lot of promise in the recent years [52–54]. 3.4 3.4.1 Singlet-triplet qubits Two-electron spin states The two-electron spin operator Ŝ 2 = Ŝ12 + Ŝ22 has four eigenstates, the so-called singlet state (the spin part of the singlet’s wave function), 1 1 |S = √ (| ↑1 ⊗ | ↓2 − | ↓1 ⊗ | ↑2 ) = √ (| ↑↓ − | ↓↑), 2 2 (3.8) that has sz = s = 0 and three triplet states, |T− = | ↓↓, 1 |T0 = √ (| ↑↓ + | ↓↑), 2 |T+ = | ↑↑, (3.9) all having s = 1. Here, |T± has sz = ±1, while in |T0 , sz = 0. The spin part of the singlet’s wave function is antisymmetric with respect to the change of the particle indices, while those of the triplets are symmetric. If there is no coupling between the spin and spatial degrees of freedom in the Hamiltonian of the system, the total wave function of two particles can be written as a product of the spin and spatial parts, Ψ = χΨS , where χ is the spin part (e.g. one of the eigenstates in Eqs (3.8) and (3.9)) and ΨS is the spatial part. The total wave function of fermions is always antisymmetric with respect to a change of particle indices. Hence, if the system is in a triplet state (s = 1), ΨS must be antisymmetric and if it is in the singlet (s = 0), ΨS is symmetric. The different symmetry properties of the spatial wave functions affect the behavior of the electrons. Electrons in the singlet, having a symmetrical spatial wave function, behave effectively as bosons. The so-called exchange force, a geometrical consequence of the symmetrization of the spatial wave function, causes an attraction between 21 Quantum computation with spin qubits the electrons in the singlet state whenever their wave functions overlap. For the triplet state, the effect is the opposite, the overlap of the wave functions results in an effective repulsive force between the electrons. The difference in the energy of the singlet and the triplet |T0 is called the exchange energy. A singlet-triplet qubit is defined by two electrons trapped in a lateral DQD system. Logical basis states of the qubit are the singlet and the sz = 0 triplet, i.e. |0 = |S and |1 = |T0 . In zero magnetic field, the three triplet states are degenerate in energy. Adding a magnetic field Bz in the z-direction (e.g. a homogeneous one) lifts this degeneracy as the states |T± have non-zero values of sz and couple to the magnetic field via the Zeemann interaction, so that the energies of the states |T± with respect to |T0 are given as ±g ∗ μB Bz (g ∗ is the gyromagnetic ratio and μB the Bohr magneton). If the field Bz is sufficiently strong (g ∗ μB Bz >> J), the sz = 0 subspace is separated from the other two triplets and the system can be regarded as two-level, spanned by the states |S and |T0 . The magnetic field Bz also affects the energy splitting of the |S and |T0 -states. In zero fields, the singlet is always the ground state. Increasing the magnetic field will eventually cause the triplet to cross the singlet in energy and become the lowest eigenstate [55]. 3.4.2 Single-qubit operations The requirements of universality in quantum computing can be met if the qubit architecture contains a gate for a general one-qubit rotation and an entangling two-qubit gate [12]. A general rotation means here that a rotation around an arbitrary axis in the Bloch sphere can be achieved by gate operation. Note that a general rotation can always be generated using rotations around two different (non-parallel) axes. In DQD singlet-triplet qubits, rotations around the z-axis (axis that goes through poles, the |0 and |1 states) can be created by inducing an energy splitting J = ET0 − ES between the basis states |T0 and |S [13, 56]. This scheme was first demonstrated experimentally by Petta et al. in 2005 [57]. A singlet-triplet qubit is typically defined so that J ≈ 0 when there is no potential energy difference between the minima of the two dots of the qubit. This is achieved by making the tunneling barrier between the dots is high enough that the overlap of the wave functions of electrons in different dots is insignificant. The exchange J can then be turned on electrostatically, by applying volt- 22 Quantum computation with spin qubits Figure 3.1. The energies of the |S and |T0 as functions of the potential energy difference of the two dots in the DQD system. As the detuning is increased and it overcomes the Coulomb repulsion of two electrons in a single dot, the singlet will tunnel to the doubly occupied state S(0, 2). The exchange splitting J = J() driving z-rotations in the Bloch sphere is denoted in the figure in gray. ages to gates defining the lateral DQD system, so that one of the dots becomes preferable energetically. This procedure is called detuning the qubit (See Fig. 4.1 in Section 4.1). As the detuning is increased, the singlet starts to localize fully in the low-detuned dot, i.e. to tunnel into the doubly occupied singlet state S(0, 2) (see Fig. 3.1), as the detuning overcomes the Coulomb repulsion. Due to the anti-symmetric spatial wave function of the triplet, it is energetically less preferable for it to tunnel to T0 (0, 2) and it will stay split between the two dots (until the detuning is further increased significantly). This creates a splitting between the energies of the qubit states. When the qubit system is then evolved under exchange, the energy difference J causes a complex phase difference between the states |0 and |1, i.e. increases the angle φ in Eq. (3.1). This amounts to rotating the qubit’s state in the Bloch sphere around the z-axis. The value of the exchange generally increases with increasing confinement of the electrons and the value is also affected by the magnetic field strength Bz . The so-called exchange gate then functions as follows. At first the qubit is in the (1, 1)-charge state, one electron in both dots. A detuning pulse 23 Quantum computation with spin qubits is then applied, turning the exchange splitting J on. The qubit is let to evolve for a time T under exchange so that a phase difference of φ = JT /h̄ is accumulated between the singlet and triplet states (for example the commonly used π-rotation around the z-axis is achieved with T = h̄π/J). When the desired rotation angle is achieved, the detuning pulse ends and the qubit returns to (1, 1), but now with the accumulated phase difference. The speed of the detuning pulses should be adiabatic in the time scales on the inter-dot tunneling, or Landau-Zener type leakage errors may occur [34, 58]. The charge state leakage in one and two-qubit systems is simulated in Publication II of thesis. The detuning pulses should optimally be small and induce only small charge differences between the singlet and triplet states. The larger the charge differences, the more susceptible the qubit will be to charge based noise and decoherence [34, 59] (more on this in Section 3.4.4). Rotations around x-axis in singlet-triplet qubits can be created by using local non-homogeneous magnetic fields across the two dots [34, 56]. When the two spins are subject to different magnetic fields in the z-direction, they obtain different phases in the time-evolution, which leads to the qubit’s state precessing between |S and |T0 . The rate of the precession is determined by the magnetic field difference ΔBz between the dots. The magnetic field gradients required for the operation can be created using nearby micromagnets or by exploiting the nuclear magnetic field of the host material. In the latter method, the nuclear spins in the two dots are aligned using a pumping scheme that transfers magnetic momentum from the electronic system to the nuclear spins. Foletti et al. demonstrated this method in 2009 experimentally [60]. They were able to achieve magnetic field differences of over 200 mT between the dots and, along with the aforementioned exchange rotations, universal single-spin control. The unitary operators (2 × 2 matrices) giving the rotations of the angle α around the two axes are given as Ux,α = e−iαg ∗μ B ΔBz X/h̄ , Uz,α = e−iαJZ/h̄ . (3.10) The z- and x-rotations can be combined to produce a rotation around a general axis. In the logical basis {|S, |T0 }, the Hamiltonian describing the qubit’s state H= J Z + g ∗ μB ΔBz X, 2 (3.11) where X and Z are the x- and z-Pauli matrices. The axis of a general 24 Quantum computation with spin qubits Figure 3.2. Single-qubit rotations on the Bloch sphere. Left: Pure x rotation with J ≈ 0 and a tilted rotation with J > 0. Right: A non-trivially time-dependent J(t) (obtained with an oscillating detuning (t) = 0 sin(2πf t)) produces complex contour on the Bloch sphere. rotation in the Bloch sphere is then n = J êz + μB g ∗ ΔBz êx , (3.12) and the frequency is f= J 2 + (μB g ∗ ΔBz )2 /h, (3.13) where êi are Cartesian coordinate vectors and h is the Planck’s constant. The single-qubit rotations on the Bloch sphere are demonstrated in Fig. 3.2. The plots are obtained using the continuum exact diagonalization model (see Section 4.1) by projecting the wave function of two-electron DQD-system onto the qubit’s basis states and plotting the evolution of the state according to Eq. (3.1). The left figure shows simple rotations with constant exchange and magnetic fields. Without detuning, i.e. exchange set to (approximately) zero, the magnetic field gradient produces a pure x-rotation, a contour that goes through the |S and |T0 -states at poles and √ the | ↑↓ = 1/ 2(|S + |T0 ) state in the xy-plane. With detuning turned on, the rotation axis becomes tilted as given by Eq. (3.12). The right figure shows a more complex pulse sequence. Here, the detuning strength is oscillating, (t) = 0 sin(2πf t), producing a non-trivially evolving J(t) and a complicated contour on the Bloch sphere. The qubit’s state can be measured using the same phenomenon responsible for the z-rotations [56, 57]. Detuning the dots of the qubit results in the singlet tunneling to the doubly occupied state S(0, 2) while the triplet stays in T0 (1, 1). Now, placing a quantum point contact near one of the dots allows the measurement of the singlet content in the qubit’s state. This projective measurement could also be used in SAW based transport, 25 Quantum computation with spin qubits Figure 3.3. SAW induced transfer of electrons through an Y-shaped junction. The singlet state (denoted by S) is transferred to the Y-branch that is energetically preferable due to a detuning potential, while the triplet state (denoted by two spin-up arrows) is split into the two Y-branches. as proposed in Publication I. If a two-electron system is led through an Y-shaped junction in which one of the Y-branches is in lower potential, the singlet will localize into that branch while the two electrons of the triplet stay split (see Fig. 3.3). 3.4.3 Two-qubit operations In the original paper by Levy on the singlet-triplet qubit architecture, the two-qubit coupling was achieved using the exchange interaction [13]. In this scheme, two singlet-triplet qubits are placed close to each other, separated by a tunneling barrier. Controlling the barrier height (or by applying detuning pulses), the wave functions of the neighboring electrons can be made to overlap, turning on the exchange interaction between them. These electrons start to entangle, which then results in the actual qubit states to become entangled as well (the qubits are first initiated in a suitable product state), producing a two-qubit operation that meets the requirements of universality. A drawback in this scheme is however that, due to the non-zero tunneling between the qubits, leakage out from the two-qubit computational space {|00, |01, |10, |11} is possible. Namely, the states |T+ T− and |T− T+ are coupled to the computational basis states via the inter-qubit tunneling. These states are in the four-electron sz = 0 subspace, so they cannot be decoupled from the qubit states by a uniform magnetic field. However, local micromagnets could be used to achieve this decoupling [61]. The efficiency of the exchange coupling scheme could be improved by introducing 26 Quantum computation with spin qubits Figure 3.4. The singlet and triplet states have differing charge densities under detuning and the Coulomb repulsion between two neighboring qubits thus depends on their states. Left: The computed charge densities of two qubits A and B in the state |SA ⊗ |SB . Right: the density in the state |T0 A ⊗ |T0 B . an intermediate quantum state via which the coupling is meditated [62]. Two-qubit exchange coupling has not yet been achieved experimentally. The capacitative coupling based on the charge distribution differences of the singlet and triplet states is the other proposed way of achieving a universal two-qubit gate, namely the CPHASE gate [34, 56]. The scheme works via the same principle as the projective measurement of the qubits state, i.e. the doubly occupied singlet state becomes energetically preferable when one of the qubit’s dots is detuned to low potential energy. As the singlet and triplet states have different charge distributions, the Coulomb repulsion between two neighboring qubits depends on their states (see Fig. 3.4). This conditioning results in an entangling operation when two nearby qubits are evolved under exchange. The capacitative coupling of singlet-triplet qubits has been done experimentally by several research groups [19, 20]. As an example of this scheme, consider the CPHASE gate operation simulated in this thesis [58, 63, 64], i.e. two singlet-triplet qubits coupled capacitatively. Both qubits are initiated in the xy-plane of the Bloch sphere, √1 (|0+|1). So that the state of the two-qubit system is 2 |ψ1 ⊗ |ψ2 = 12 1k1 =0 1k2 =0 |k1 k2 , i.e. all coefficients α in Eq. (3.3) in the state | ↑↓ = |Ψ2 = are 1/2. The detunings of the qubits are then turned on, lifting the degeneracy in the computational space {|00, |01, |10, |11}. As the qubits are now evolved, keeping the detunings constant, each term in the two-qubit wave function obtains a phase factor so that αk1 k2 = 1 −iEk1 k2 t/h̄ , 2e where Ek1 k2 is the energy of the computational basis state |k1 k2 . Inserting these coefficients into Eq. (3.4) gives the time evolution of the concurrence, C(t) = 1 2 − 2 cos (Ecc t/h̄). 2 (3.14) 27 Quantum computation with spin qubits 1 τ 123 C2 +C2 +C2 12 0.8 13 23 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 t (ns) Figure 3.5. The evolution of the pairwise concurrences and the three-tangle in a system of three qubits in a triangular geometry. The thick blue line shows the threetangle and the red line the squared sum of the pairwise concurrences. At t = 0, all qubits are initiated in the xy-plane of the Bloch sphere with their detunings on. The qubits are then let to evolve, and the pairwise concurrences and the three-tangle are computed at each time step. Here, Ecc is the differential cross capacitance energy between the two double-dot systems, Ecc = |E00 + E11 − E01 − E10 |. (3.15) Ecc determines the speed of the gate operation and the frequency of the entanglement oscillations. At time t when tEcc /h̄ is an odd multiple of π, C(t) = 1 and the maximal Bell-state entanglement is achieved. The cross capacitance energy Ecc , determined by the difference in the Coulomb repulsion in the computational basis, is a measure of the coupling strength of the qubits. If the charge distributions of the computational basis states are equal, Ecc = 0. Ecc assumes its maximum when the singlet is fully in S(0, 2) but the triplet is still in (1, 1). As the detuning is then further increased, the triplet as well starts to undergo the same charge transition, and the value of Ecc starts to decrease. The detuning dependence of Ecc is, however, problematic regarding actual implementation of the gates. The operational regions with high charge distribution differences, i.e. with fast operation, are also the ones that are more susceptible to charge noise based decoherence [59] (see Section 3.4.4 for more 28 Quantum computation with spin qubits on this). The value of Ecc is also affected by the geometry of the two qubit system, as analyzed in Publication IV of this thesis. The closer the qubits are, the more there is repulsion and the stronger the coupling is. The orientation of the qubits also plays a role, for example there are certain dead angles in the orientations that result in the coupling disappearing completely, regardless of the charge differences. The largest values of Ecc are obtained when the four QDs of the qubit are in a rectangular formation [64]. As a part of this thesis, in Publication III and Publication IV, also threequbit capacitative coupling is simulated. Coupling two qubits this way allows the creation of Bell-states. Similarly, three-qubit coupling enables the creation of maximally entangled GHZ-states. As an example, consider three singlet triplet qubits placed in the corners of an equilateral triangle [63]. The qubits are all initiated in the xy-plane of the Bloch sphere, 1 so that the full three-qubit state of the system is |Ψ3 = 2√ ijk=0,1 |ijk 2 and the detunings are switched on. The qubits start to interact capacitatively and entangle with each other. The evolution of the three-tangle and the pairwise concurrences are shown in Fig. 3.5. As seen in the figure, the system starts to oscillate between pairwise entanglement (the Wstate is not achieved though) and GHZ-type tri-body entanglement. The maximally entangled GHZ-state is obtained, as evident from the tangle obtaining the value τ123 = 1. Generally, coupling Nq qubits via CPHASE gates enables the creation of cluster states, a class of highly entangled states that the Bell-states and GHZ-states are examples of [41]. 3.4.4 Decoherence Large scale quantum computation requires long-lasting quantum states that can be stored and used in qubit operations. The interaction of the qubits with their environment poses a serious problem in this regard. It leads to the qubits becoming entangled with the environment in complex ways and their original states thus becoming destroyed in the process. This phenomenon is called quantum decoherence. Mathematically speaking, it means the decay of the off-diagonal elements in the density matrix of the system. For quantum error correction schemes to be applicable, the relevant gate-operation times must be considerably shorter than the lifetimes of the states under decoherence. There are three important time scales in this context. The relaxation time T1 means the average time for the transition |0 → |1 to happen due to decoherence. The decoher- 29 Quantum computation with spin qubits √ ence time T2 means the time it takes for the superposition 1/ 2(|0 + |1) to decay to |0 or |1. The dephasing time T2∗ is the decoherence time T2 obtained experimentally from an ensemble of measurements. It can be shown that T2 ≤ 2T1 , and T2∗ is usually much shorter than T2 due to destructive interference and damping caused by averaging over the measurement ensemble [65]. The quantum decoherence and its measurement was at the heart of the life’s work of the 2012 Nobel laureates Serge Haroche and David Wineland (see e.g. Refs. [66, 67]). For example in Ref. [66], Haroche et al. directly measured the decay of the off-diagonal elements of the density matrix associated with quantum decoherence in a cavity photon system. In singlet-triplet qubits, the hyperfine interaction with the host material’s nuclear spins causes precession around the x axis in the Bloch sphere, similarly as in the x-rotation gate. An electron in a lateral GaAs QD interacts with around 106 nuclear spins. The orientation of these spins is basically random, and also fluctuates during an experiment, resulting in randomness in the measurement’s outcome. The local differences in in the hyperfine field give rise to an effective magnetic field gradient between the two dots of the qubits. This difference is typically in the order of 1 mT, and the z-directional gradient is the most important regarding the decoherence effects. The random hyperfine fields cause random phase differences in the qubit’s states between different measurements and lead to relaxation and dephasing. The hyperfine dephasing time T2∗ is typically in the order of 10 ns [34, 50, 68, 69]. The hyperfine decoherence is mainly of concern in the (1, 1)-charge configuration. With large values of the exchange, its effects are suppressed as the exchange term will then dominate over the small magnetic field differences ΔBz in Eq. (3.11). Spin orbit coupling (SOI) is a relativistic effect that couples an electron’s spin with its orbital motion. In materials with bulk inversion asymmetry, the so-called Dresselhaus term of the SOI is present [70]. Structural inversion asymmetry, caused by asymmetric confining potentials in the material, cause the so-called Rashba term of SOI [65]. The 2DEG in GaAs/AlGaAs-heterostructures is subject to both of these effects. In singlet-triplet qubits, the SOI effects cause decoherence by coupling the singlet and triplet states. In GaAs, the hyperfine induced relaxation is dominant though in the sub-ms time-scales [34]. In singlet-triplet qubits implemented in materials without nuclear spin, the SOI can be the dominant interaction channel with an electron spin trapped in a QD and its 30 Quantum computation with spin qubits environment. The electric environment of the QD presents another important decoherence source. There are several effects contributing in the electric decoherence, including fluctuations in the gate voltages and impurities and charge traps in the QDs vicinity. In singlet-triplet qubits this charge based decoherence is found to manifest itself dominantly as -noise, i.e. as effective fluctuations in the qubits detuning [59]. The random differences caused by the qubit’s electrical environment lead to fluctuations in the exchange and thus also to dephasing. Pure dephasing by -noise is measured in a free induction decay (FID) experiment, in which the qubit is initiated in the state |ψ = √1 (|S + |T0 ) 2 and it evolves under the exchange J = J(). The charge environment of the qubit causes an effective fluctuation δ in the detuning that in turn results in a fluctuating of J: J( + δ) ≈ J() + dJ d δ. This fluctuation leads to decoherence in the form of a random phase accumulation between the T evolution of the singlet state and the triplet state: Δφ(T ) = h̄1 0 dt dJ d δ(t) as the qubit is evolved (for a time T ). It can be shown that, as a consequence, the coherence between the |S and |T0 states in a FID evolution (of time T ) decays as sin2 (ωT /2) dJ 2 ∞ dωS(ω) f (T ) = exp − , d (ω/2)2 −∞ (3.16) where S(ω) is the spectral density of the detuning fluctuations [71, 72]. Assuming specific noise properties [i.e. the spectral density S(ω)], the time dependence of the decoherence can be further analyzed. For example, in the experimentally relevant case of low-frequency voltage noise (quasistatic noise), the coherence decays as exp(−(T /T2∗ )2 ), with the coherence time √ T2∗ = 2h̄ dJ d rms , (3.17) where rms is the root-mean-squared fluctuation of the detuning [59, 73]. The qubit’s sensitivity to charge noise is directly related to the differences in the charge densities of the singlet and triplet states. The larger the differences are, the more severe the dephasing is. In case of the noise, it is evident from Eq. (3.16) that the effects of the noise are mitigated if one can minimize the derivative |dJ/d|. In exchange gate operations, such operational regions that have high noise resistance are called sweet-spots [34, 49, 59, 74]. One such region is found in the high detuning regime where both the singlet and triplet are in the doubly occupied 31 Quantum computation with spin qubits Figure 3.6. The asymmetric double quantum dot scheme. The energies of the singlet (blue lines) and triplet (dashed red lines) are shown in the two dots [(0, 2) and (2, 0)-configurations] as functions of the detuning of the dots . The exchange gate operations are done in the larger right dot so that both the singlet and triplet are in the (0, 2)-configuration. The projective measurement is conducted using the smaller left dot, with the singlet in (2, 0) and the triplet in (1, 1). states. In this region, the detuning and thus also -noise has in principle no effect on the qubit’s operation. In Publication V, an exchange gate implemented in the doubly occupied region of an symmetric double dot system is proposed. With magnetic fields, one can set the exchange J = ET0 (0,2) − ES(0,2) to a value smaller than the intra-qubit tunneling in the larger of the dots of the system, enabling good performance in the gate operation. The smaller dot can then be used in the projective measurement, as the J-splitting stays large enough there to make it possible to distinguish the S and T0 -states via charge-sensing techniques [75] (see Fig. 3.6). It should be noted that the densities of the qubits states still differ even though they are both located in one QD. These differences will also contribute to charge based decoherence, but they could be mitigated by moving to multi-electron singlet-triplet qubits [49, 76]. In Publication VI, the charge-noise resilience properties of two and four-electron QDs are compared. It is found that the charge noise has considerably smaller effects in the four-electron dots. The charge noise is an especially severe decoherence source in the two-qubit capacitative coupling, as there the coupling 32 Quantum computation with spin qubits strength is directly related to charge differences. If the fluctuations in the noise decohering the qubit are slow compared to gate operation times (quasi-static noise), their effects can be greatly mitigated by dynamically decoupling echo techniques [34,57]. These techniques are based on the fact that if the noise does not fluctuate during an experiment, one can cancel its effect by performing a π-rotation in the middle of the experiment. The phases accumulated before this echo-pulse and the ones after that are equal, but have opposite signs and add up to zero. More echo pulses can be added to the pulse sequence, dividing the phase accumulation to even shorter intervals. In the presence of quasistatic noise, the echo techniques can greatly increase the dephasing times. Times up to T2echo = 200 μs have been demonstrated experimentally in singlet-triplet qubits [77]. 33 Quantum computation with spin qubits 34 4. Simulation methods 4.1 4.1.1 Continuum model Continuum Hamiltonian The effective mass approximation can be used to describe electrons moving in the two-dimensional electron gas. In the approximation, only the electrons localized into the lateral QDs are described. The surrounding semiconductor material is taken into account by substituting the electron mass me in the Schrödinger equation with the material dependent effective mass m∗ and the same for the Lande factor g ∗ . The Hamiltonian for N interacting electrons confined in two dimensions (in the effective mass approximation) is N (p̂j + eA(rj ))2 e2 ∗ Ĥ = + V (r ) + g μ B(r ) · S /h̄ + . (4.1) j j j B ∗ 2m 4πrij j=1 i<j Here, rj is the coordinate of the j:th electron, V the external electric potential, Sj = h̄ 2 (Xj , Yj , ¸Zj ) the spin operator (with Xj , Yj and Zj being the Pauli-matrices corresponding to the electron j), and A the magnetic vector potential. In this thesis, the main interest is in magnetic fields perpendicular to the xy-plane where the electrons are confined. In particular, an uniform magnetic field in the positive z-direction and with the field strength Bext is given by the vector potential A(r) = Bext 2 (−y, x, 0) as Bext = ∇ × A. Typical GaAs-values for the parameters in Eq. (4.1) are: the effective electron mass m∗ = 0.067me , the effective electron gyro magnetic ratio, g ∗ = −0.44, and the dielectric permittivity = 12.70 . It is important to note that the electron-electron Coulomb interactions are taken into account exactly in Eq. (4.1). This allows for an accurate description of the 35 Simulation methods effects arising from the electron-electron interactions and correlations. In simulating QD systems, the individual QDs are usually simulated as minima of the electric potential V (r). Quadratic potentials are a common choice for the form of the confinement. The actual form depends on the gating of the QDs, but quadratic potentials are found to be a good approximation to the exact solution to the Poisson’s equation in realistic setups [74]. A confinement potential of n quadratic wells (located at the coordinates {Rm }nm=1 ) can be written as 1 Vc (r) = m∗ ω02 min {|r − Rm |2 }, 1≤m≤n 2 (4.2) where ω0 is the confinement strength of the wells. In the above potential, the derivative is discontinuous in the ridges between different QDs. This is, however, found to cause no qualitative effects compared to cases where the ridges are smoothed to obtain a continuous derivative. In Eq. (4.2), the confinement strength is the same in all dots and the the dots are circular. Inhomogeneous confinement strengths and elliptical dots are also possible. In DQD singlet-triplet qubits, the electric detuning of the dots of the DQD system plays an important role in for example exchange rotations around the z-axis in the Bloch sphere. The detuning of the dots, a potential difference between the dot minima, can be simulated with a step function Vd that assumes constant values at each dot, so that the total external potential in Eq. (4.1) is V (r) = Vc (r) + Vd (r), with the detuning between dots k and l given by kl = V (Rk ) − V (Rl ). It should be noted that the step function causes a discontinuity in electric potential, that can be smoothed out, if needed, although this is again found to have very little effect in the physics of the system [58, 64]. The confinement potential of a DQD system as in Eq. (4.2) and the detuning are illustrated in Fig. 4.1. The hyperfine interaction arising from the coupling of the electron and the host semiconductor material’s nuclear spins acts as a decoherence channel [50, 68, 78] but also as a means of qubit control [34, 60]. The effect of the nuclear spin coupling can be taken into account as an effective magnetic field Bnuc so that the total magnetic field in the system is written as B = Bext + Bnuc [34]. In the context of singlet-triplet qubits, the difference in the magnetic field between the two dots plays a more important role than the actual values. The magnetic field difference drives the rotations around the x-axis in the Bloch sphere, coupling the singlet 36 Simulation methods V(meV) 12 10 V, ε=0 Vd, ε=6 meV 8 V, ε=6 meV 6 4 2 0 −2 −4 −50 0 x(nm) 50 Figure 4.1. A two-dot potential and the detuning. The potential of Eq. (4.2), with two minima at R1 = (−40 nm, 0) and R2 = (40 nm,0) and with no detuning is shown in the x-axis (the thick blue line). The confinement strength is h̄ω0 = 4 meV. A detuning potential Vd of = 6 meV is shown as the thin red line. The detuning is a step function that is made continuous with a linear ramp. The detuned potential is shown with the dashed black line. state to the triplet, and can be simulated similarly to the detuning, as a step function that assumes constant values at each dot. 4.1.2 Single-particle states The single-particle eigenstates of the potential in Eq. (4.2) provide a convenient basis to be used in the diagonalization of the many-body Hamiltonian. The single particle part of the Hamiltonian in Eq. (4.1) (excluding the Zeemann term) is Ĥ0 = (p̂ + eA(r))2 + V (r). 2m∗ (4.3) The eigenstates of this operator can be solved numerically by writing Eq. N g (4.3) as a matrix in a given basis, {|φi }i=1 , so that the Hamiltonian and the overlap matrix elements are given as H0,ij = φi |Ĥ0 |φj , Sij = φi |φj . (4.4) Projecting the one-particle Schrödinger equation onto this basis thus yields a generalized Ng × Ng matrix eigenvalue equation H0 u = ESu. (4.5) 37 Simulation methods s Solving Eq. (4.5) gives the set of eigenvectors {un }N n=1 (the set size being (n) Ns ≤ Ng ) that contain the coefficients ui states, s {|ψn }N n=1 in the basis Ng {|φn }n=1 , of the single-particle eigenNg (n) i.e. |ψn = i=1 ui |φi . Increas- ing the basis size Ng results in better convergence of the eigenenergies and -states. N g In principle, the basis {|φi }i=1 can consist of any (complete) set of prefer- N g ably smooth functions. When choosing the basis {|φi }i=1 , one should con- sider whether the Hamiltonian elements H0,ij can be computed analytically, specifically whether the potential integrals Vij = φi |V |φj can be given in a closed form. However, these two-dimensional integrals can also be computed in a relatively short time with numerical methods as they are not that complex computationally. A more demanding requirement comes from the Coulomb interaction elements, 1 |φl |φk , (4.6) |r̂1 − r̂2 | as these integrals are four-dimensional and the size of the four-index maVijkl = φi |φj | trix Vijkl can thus be very large. The numerical computation of these elements can be too time-consuming. In choosing the basis, it is preferable that the elements in Eq. (4.6) can be computed (at least partially) analytically. In this thesis, two such bases are used. The first option is the Fock-Darwin states [48], ϕn,l (r, θ) = n! 2 −r2 /2 ilθ r|l| L|l| e , n (r )e π(n + |l|)! (4.7) |l| where Ln is an associated Laguerre polynomial, n (−1)k n + |l| k x . (x) = L|l| n k! n−k (4.8) k=0 Here, n and l are the radial and angular quantum numbers, respectively. The second one is the multi-centered gaussian basis [74], 2a −a|r−ri |2 φi (r) = e , π (4.9) where ri is the location of the gaussian and a the gaussian width. N g When the Vijkl elements are computed analytically in the basis {|φi }i=1 , s they can be transformed into the eigenbasis {|ψp }N p=1 as (p) ∗ (q) ∗ (r) (s) Ṽpqrs = uk ul Vijkl . ui uj (4.10) ijkl (p) In case of an orthogonal basis, i.e. if φi |φj = δij , it holds that ui = φi |ψp . The transformed elements Ṽpqrs are used in the many-body computations as discussed in the next section. The basis change of Eq. (4.10) 38 Simulation methods becomes very taxing computationally as the basis sizes Ng and Ns are increased (the complexity of computing just one element Ṽpqrs grows as Ng4 ). Fortunately, the computation of the above quadruple sum is easily parallelized. It can be done with a graphics processing unit and CUDA [79], a parallel programming model for Nvidia GPUs, greatly reducing its computation time. 4.1.3 Exact diagonalization In the basis of the single-particle eigenstates of Eq. (4.3), the secondquantized N -body Hamiltonian is written as Ĥ = p,σ + 1 2 Ep â†p,σ âp,σ + Vpq,σ â†p,σ âq,σ pq,σ Ṽpqrs â†p,σ â†q,σ âr,σ âs,σ , (4.11) pqrs,σ,σ where pqrs are the orbital indices (running from 1 to Ns ) and σσ are the spin indices. In Eq. (4.11), the first term corresponds to the single-particle Hamiltonian of Eq. (4.3), the second term contains any additional singleparticle operators V̂pert (such as the Zeemann term of Eq. (4.1)) so that Vpq,σ = ψp,σ |V̂pert |ψqσ , and the third is the electron-electron interaction. Eq. (4.11) is diagonalized in the basis of fully symmetrized Slater determinants, |Ψα = |nα1 , nα2 , ..., nα2Ns , where nαk denotes the number of electrons in the k:th single-particle state (here, states from 1 to Ns correspond to the spin σ = −1/2 and the states from Ns + 1 to 2Ns to σ = 1/2). The N -body Hamiltonian is cast into a matrix form by computing the matrix elements Hαβ = Ψα |Ĥ|Ψβ = nα1 , nα2 , ..., nα2Ns |Ĥ|nβ1 , nβ2 , ..., nβ2Ns (4.12) that correspond to two different occupations α and β. To reduce the size of the many-body basis, one can restrict the computation to for example the unpolarized spin subspace, Sz = 0 (with Sz being the z-projection of the N -body spin). Even so, the many-body basis size grows as Ns ! (N s−N )!N ! , i.e. faster than exponentially. As several single-particle states per electron are required for converged results, the continuum method is limited to computations involving just a few electrons. In this thesis, the method is used in two and four-electron computations. Systems larger than that are described with computationally less demanding models. 39 Simulation methods Even with small electron numbers, the size of the Hamiltonian matrix of Eq. (4.12) usually becomes so large that it cannot be fully diagonalized to obtain its eigenvalues and -states. Instead, a numerical method has to be used. One such method is the Lanczos diagonalization [80, 81]. In the method, given a Hermitian matrix H, the goal is to find a tridiagonal ∗ HV matrix Tm = Vm m in m steps. In practice, the Lanczos algorithm is just the Gram-Schmidt orthogonalization applied to the Krylov space, K = {v, Hv, H2 v, ..., HM v}, where v is a (random) starting vector and M is the dimension of H. The advantage of the Lanczos method is that the lowest eigenvalue of Tm is a good approximation to the lowest eigenvalue of H even when m is relatively small compared to M (when m = M , H and Tm are similar and have the same set of eigenvalues). The Lanczos method is accurate only for the lowest eigenvalue. However, the excited states can still be obtained with the method iteratively. The k:th eigenvector uk is obtained as the lowest eigenvector of the matrix Hk = H + δ k−1 ui u∗i . (4.13) i=1 Here, H is the original Hamiltonian matrix and δ > 0 is a penalizing factor that moves the lower eigenstates {ui }k−1 i=1 above the desired k:th one in energy. The dynamics in a QD system can be computed by propagation, as i ψ(t + Δt) = exp − Ĥ(t)Δt ψ(t), (4.14) h̄ where ψ(t) is the state of the system at time t (ψ(0) is the initial state), Δt the time step length, and Ĥ(t) the Hamiltonian at time t. In practice, computing the exponential in Eq. (4.14) involves the diagonalization of the Hamiltonian matrix H, which, as described above, often becomes too taxing computationally. Fortunately, the product exp − h̄i Ĥ(t)Δt ψ(t) can be computed using the Lanczos algorithm (without explicitly computing the actual matrix exponent). With small enough time steps Δt, the number of Lanczos steps needed for accurate description of the dynamics is much smaller than in the actual diagonalization, making the computation of the dynamics this way fast and efficient. 4.2 Extended Hubbard model The continuum model, although accurate, is limited to small particle numbers due to its high computational cost. In addition, scanning large sets of 40 Simulation methods different parameters of QD systems (e.g. dot distances and confinements) is cumbersome, as each different realization requires the solving of the single-particle Hamiltonian of Eq. (4.3) and the associated basis change for the interaction matrix elements. A more flexible method for studying systems of several S − T0 qubits is an extended Hubbard model with the inclusion of a long-range Coulomb interaction (the regular Hubbard model with just the on-site interactions will not usually suffice, as the long-range repulsions play quite an important role, e.g. in the qubit-qubit interactions). In this model, a system consisting of Nq singlet-triplet qubits (N = 2Nq electrons and QDs) can be described using the Hamiltonian, Ĥ = Eiσ â†iσ âiσ − iσ + U tijσ â†iσ âjσ ijσ n̂i↓ n̂i↑ + i Uij n̂iσ n̂jσ . (4.15) σσ i<j Here, i and j are the site indices, and σ and σ the spin indices. Eiσ is the on-site energies at site i with spin σ, tijσ the tunneling element between dots, n̂iσ = â†iσ âiσ the spin σ density operator at site i, and U the on-site Coulomb repulsion. Uij is the long-range Coulomb-interaction between sites i and j, and niσ the charge at site i with the spin σ. The long-range interaction is given as Uij = C , |ri − rj | − η (4.16) where C = e2 /4π is the Coulomb-strength, ri and rj are the locations of the dots i and j. η > 0 is an extra constant conveying the fact that in reality the wave functions have finite widths. The parameters of the Hubbard model (U , tij , and η) can be fitted to continuum model data in order to produce more realistic results. An example of such fit is shown in Fig. 4.2 with the fit done to continuum data corresponding to a two-qubit system (four QDs and electrons). The plot shows the lowest energies of the system as a function of the detuning of the qubits. A two-qubit fit like this contains information about both the intra- and inter-qubit effects, and the parameters obtained from it can thus be used to realistically describe larger systems of many qubits that are unreachable by the continuum model. 41 Simulation methods 21.8 ED Hubbard E(meV) 21.6 21.4 21.2 21 4.2 4.3 4.4 ε(meV) 4.5 Figure 4.2. The lowest energies of a two-qubit system as a function of the detunings of the qubits (both qubits in the same detuning ). The thick black dashed line shows the ED energies, and the red line the Hubbard energies with the parameters tij = 27.8 μeV, U = 3.472 meV, and η = 0.43 nm. 42 5. Discussion The main topic of this thesis is singlet-triplet qubits in lateral GaAs quantum dots. Specifically, the emphasis is on qubit control in these systems; the initiation, measurement and evolution of the qubits’ states. The single- and two-qubit gate operations that are required for universal quantum computing are discussed and simulated. Concerning the two-qubit gates, their entangling properties are also analyzed. Other important facets of quantum computing discussed here are decoherence and schemes that could be used to alleviate it. The thesis is theoretical and computational in nature. Small lateral QD systems are simulated using the accurate continuum model and configuration interaction methods that take the Coulomb-repulsion between electrons into account exactly. Larger systems of QDs can be simulated using a less taxing method, a generalized Hubbard model with long-range interactions. The parameters of the Hubbard model can be fitted to the data form the more accurate continuum model to produce realistic results. In Publication I, a scheme for measuring the state of a singlet-triplet qubit during SAW-based transport is proposed. The method is based on an electric potential splitting the qubits’ states due to the different symmetry properties of their spatial wave functions. When the qubit is let to an Y-shaped junction via SAW-propagation, the singlet will localize to the Y-branch with the lower potential while the triplet stays split between the branches. The time-evolution of the qubit’s states is simulated in the system using the continuum exact diagonalization model and the requirements for the scheme’s successful implementation, namely the adibaticity of the charge transition, are discussed. The charge state transitions in singlet-triplet qubit systems are also the topic of Publication II. We simulated the charge transition to the doubly occupied state of the low detuned qubit using the continuum exact 43 Discussion diagonalization model. If the detuning pulse is too fast, a Landau-Zener transition may occur so that a part of the wave function does not tunnel to the doubly occupied state. This can results in errors in the measurement of the qubit. The charge state leakage was also studied in the case of two capacitatively coupled qubits, where it can result in errors in the two-qubit gate operation. The capacitative coupling is the main topic of Publication III and Publication IV. In the former, we propose a three-qubit generalization to the capacitative CPHASE gate in which the qubits are placed in a triangular formation. The entanglement properties of the gate are discussed, specifically, its ability to generate GHZ-type maximal three-qubit entanglement. The three-qubit systems are simulated using the extended Hubbard model. The analysis of three-qubit capacitative coupling is continued in Publication IV, extending it to other geometries of the qubit locations. The main point of this publication is the analysis of the effect of the geometry in the coupling of the qubits this way. It turns out that there are certain optimal geometries allowing strong coupling while other, so-called dead angle geometries, result in the coupling diminishing completely. A strong coupling arising from the geometry would allow the use of smaller charge distribution differences and hence also smaller susceptibility to charge noise. Publication V concentrates on mitigating the charge noise problem in the case of the one-qubit exchange gate. In asymmetric double dot systems, the exchange energy in the doubly occupied region of the larger dot can be squeezed to low enough levels for convenient exchange gate operations. The qubit is much more resilient to charge-based dephasing in the doubly occupied region than in the transition region where the exchange gates are conventionally operated. This is also justified in the paper by configuration interaction calculations. However, the exchange splitting still stays large enough in the smaller dot of the system so that the projective measurement can be done there. The charge noise is also the topic of Publication VI. The proposition for moving to many-electron singlet-triplet qubits in order to alleviate the noise problem is discussed. 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