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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. Using the CI method, it is found out that
four-electron QDs are considerably better protected from the charge noise
than two-electron ones. Another main topic of the paper is the validity of
a shell filling model similar to the one in atomic physics in the context of
QDs. In the four-electron QDs, the shell filling picture justifies the singlet
44
Discussion
and triplet states having very similar charge densities and thus also weak
coupling to the charge environment.
45
Discussion
46
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