Quantum Computation With Spins In Quantum Dots

Fikeraddis Damtie

June 10, 2013

1 Introduction

Physical implementation of quantum computating has been an active area of research since the

last couple of decades. In classical computing, the transmission and manipulation of classical

information is carried out by physical machines (computer hardwares, etc.). In theses machines

the manipulation and transmission of information can be described using the laws of classical

physics. Since Newtonian mechanics is a special limit of quantum mechanics, computers making

use of the laws of quantum mechanics have greater computational power than classical computers.

This need to create a powerful computing machine is the driving motor for research in the field of

quantum computing.

Until today, there are a few different schemes for implementing a quantum computer based on

the David Divincenzo chriterias. Among these are: Spectral hole burning, Trapped ion, e-Helium,

Gated qubits, Nuclear Magnetic Resonance, Optics, Quantum dots, Neutral atom, superconductors

and doped silicon. In this project only the quantum dot scheme will be discussed. In the year

1997 Daniel Loss and David P.DiVincenzo proposed a spin-qubit quantum computer also called

The Loss-DiVincenzo quantum computer. This proposal is now considered to be one of the

most promising candidates for quantum computation in the solid state. The main idea of the

proposal was to use the intrinsic spin-1/2 degrees of freedom of individual elecrons confined in

semiconductor quantum dots. The proposal was made in a way to satisfy the five requirements for

quantum computing by David diVincenzo which will be described in sec. 2 in the report. Namely

1. A scalable physical system with well characterized quibits

2. The ability to initialize the state of the qubits to a simple fiducial state such as |000...〉

3. Long relevant decoherence times, much longer than the gate operation time

4. A "universal" set of quantum gates

5. A qubit-specific measurement capability

A good candidate for such quantum computer is single and double lateral quantum dot systems.

This report is organized as follows. In section 2 general requirements for the physical implementa-

tion of quantum computation the so called diVincenzo criteria is discussed briefly. Section 3 will

introduce the basics of quantum dot physics. In this section the properties of single and double

quantum dot will be discussed. In section 4 a description of the Loss-diVincenzo proposal for quan-

tum computation based on single electron spin in semiconductor quantum dot will be given. In

section 5, analysis of the theoretical and experimental development in the last decade will be given.

The last section will be devoted to discussion and assesment of main qualitative and quantitative

1

obstacles for an ultimate realization of the spin based quantum computer.

For the report, I mainly based on the papers [9] and [5]

2 DiVincenzo requirements for the physical implementation

of quantum computation

In his paper "The Physical Implementation of Quantum Computation"[3], David P. DiVincenzo

and his co-workers described Five (plus two) requirements for the implementation of quantum

computations. Below I will try to briefly mention the requirements.

1. A scalable physical system with well characterized quibits : The requirement here is that a

physical system containing a collection of qubits is needed at the beginning. Here a qubit

being "well characterized" can mean different things. It can for example mean that its

physical parameters should be accurately known including the internal Hamiltonian of the

qubit, the presence of and couplings to other states of the qubit, the interaction with other

qubits and the couplings to external fields that might be used to manipulate the state of the

qubit.

2. The ability to initialize the state of the qubits to a simple fiducial state such as |000...〉:This is analogues to say that registers should be initialized to a known value before start of

computation, which is a straight forward computing requirement for classical computation.

Another reason for this is initialization requirement is from the point of quantum error

correction which requires a continous, fresh supply of qubits in a low entropy state. (link the

|0〉) state.

3. Long relevant decoherence times, much longer than the gate operation time: An overly sim-

plified definition for a coherence time can be described as the time for a generic qubit state

|ψ〉 = a|0〉 + b|1〉 to be transformed in to the mixture ρ = |a|2|0〉〉0| + |b|2|1〉〉1|. This time

helps characterize the dynamics of a qubit in contact with its enviroment. Decoherance is

an important concept in quantum mechanics. It is ientified as the principal mechanism for

the emergence of classical behaviour. For quantum computating, decherance can be very can

be very dangerous. If the qubit system has a fast decoherence time, the capability of the

quantum computer will not be very different from that of the classical ones.Hence in quan-

tum computing, it is desirable to have long enough decoherence time such that the uniquely

quantum features of quantum computating can have a chance to come in to play. The answer

to "How long is long enough?" is determined by quantum error correction which will not be

2

discussed in this report see for example [7].

4. A "universal" set of quantum gates :This fourth requirement can be considered as the most

important for quantum computing. One and two qubit gates are needed.

5. A qubit-specific measurement capability :This is also called a readout. The result of a compu-

tation must be readout. This requires the ability to measure specific qubits.

For computation alone, the above five requirements are enough. But the advantages of

quantum information processing are not manifest solely for straight forward computation

only. There are different kinds of information- processing tasks that involve more that just

computation and for which quantum tools provide a unique advantage. One of such tasks

is quantum communication: the transmission of intact qubits from place to place. If one

consider additional information processing tasks than just computation only, two more re-

qurements are needed to be fulfilled.

6. The ability to interconvert stationary and flying qubit :

7. The ability faithfully to transmit flying qubits between specified locations :

[3]

Before discussing the Loss-diVincenzo proposal for quantum computer based on spin in quantum

dots, it might be a good idea to spend some time discussing the basics of quantum dot physics

briefly.

3 Semiconductor Quantum Dots

Quantum dot’s are artificial sub-micron structures in a solid, typically consisting of 103−109 atoms

and comparable number of electrons. [6] As they are confined in all three dimensions, the resulting

electronic states exhibit discreteness.

3

Figure 1: Top: Schematic representation of three, two, one, and zero-dimensional nanostructures

made using semiconductor hetrostructures. Bottom: The corresponding densities of electronic

states. [1]

In a three dimensional bulk semiconductor electrons are free in all three dimensions. In 2DEG

(Two dimensional electron gas) electrons are free to move in a plane in two dimensions. In one

Dimensional quantum wires, electrons are allowed only to move along the direction of length of

the wire. In zero dimensional quantum dots, electronic motion is restricted in all three dimensions

and the density of state is discrete.

By using the constant interaction model, it is possible to describe for example the current voltage

characterstics of a quantum dot system. In the constant interaction model, Coulomb interaction

between electrons in the dot and electrons in the surrounding enviroment is replaced by self-

capacitances.

3.1 Resonant Tunneling Through Quantum Dots

Due to the discreteness of the electronic states in quantum dots electron transport is restricted to

resonant conditions. For a quantum dot coupled to a source and drain contact, resonant tunneling

occurs when an electronic state which can either be occupied or empty in one of the contacts align

with any of the available states in the dot. Schematically this situation is shown as in the picture

below

4

Figure 2: Schematic diagram of resonant condition through a single quantum dot showing the

electrochemical potential level of a dot coupled to the source and drain reservoirs. [4]

As one can see from the above schematic, resonant tunneling occurs when there is available discrete

level of the dot in between the source and drain chemical potentials. As a result, electrons flow

through the dot sequentially and are detected as a change in source-drain current. [4]

Practically, in experiments, the chemical potentials of the source and drain contact can be varied

by varying the source drain bias as µs − µd = eVsd. Similarly, the electrochemical potential in

the dot can be adjusted by varying the gate voltage Vg. By plotting the gate voltage versus the

source drain bias on a two-dimensional map, which is called the charge-stability diagram, one can

get information about the sequential tunneling and total number of particles involved as there

are diamond-shaped regions with well-defined charge numbers in the dot. These diamond-shaped

regions are called Coulomb diamonds. [4] The typical charge stability diagram of a single quantum

dot coupled to source and drain contacts is shown in the figure below.

5

Figure 3: A typical charge stability diagram of a single quantum dot coupled to the source and

drain contact which shows Coulomb diamond due to a transport through the ground state of the

quantum dot and additional transport lines running parallel to the diamond edges at an energy

∆E. These lines can be attributed to transport due either to other single particle levels or inelastic

processes such as emission or absorption of phonon or phonon with energy ∆E. [4]

3.2 Resonant Tunneling Through Double Quantum Dots

Double quantum dots are sometimes called artificial molecules as they are made by coupling two

quantum dots either vertically or in parallel. Depending on how strongly the two dots are coupled,

they can form either ionic-like bonding (weak tunnel coupling) or covalent-like bonding (strong

tunnel coupling). [8] In the case of weak tunnel coupling, the electrons are localized on the

individual dots so that the binding occurs due to the coulomb interaction. On the other hand, in

the case of strong coupling, the two dots are quantum mechanically coupled by tunnel coupling.

As a result, electrons can tunnel between the two dots in a phase coherent way. During strong

coupling, the electrons cannot be regarded as particles that can reside in a particular dot, instead,

it must be thought of as a wave that is de localized over the two dots. The binding force between

the two dots in strong coupling case is a result of the fact that the bonding state of the strongly

coupled double dot has a lower energy than the energies of the original states of the individual

dots. This difference in energy forms the binding.

6

3.3 Charge Stability Diagrams of Double Quantum Dots

Understanding the charge stability diagrams helps to visualize the equilibrium charge states in a

double dot system. [8] It is sometimes called a honeycomb diagram as it has a similar structure

with a honeycomb. A typical stability diagram for a double quantum dot with small, intermediate

and large interdot Coulomb coupling is shown in the schematic below.

Figure 4: Schematic stability diagram of a double dot system for (a) Small (b) intermediate, and

(c) large interdot coupling. The equilibrium charge in each dot in each domain is denoted by

(N1, N2). (d) represents the two kinds of triple points in the honey comb which illustrates the

electron transfer processes (•) and hole transfer processes (◦) [8]

Having the basics of single and quantum dot, it is now time to discuss the main point of the

project, the Loss-Divincenzo proposal.

7

Figure 5: A double quantum dot. Top-gates are set to an electrostatic voltage configuration that

confines electrons in the two-dimensional electron gas (2DEG) below to the circular regions shown.

Applying a negative voltage to the back- gate, the dots can be depleted until they each contain

only one single electron, each with an associated spin 12operator SL(R) for the electron in the left

(right) dot. The | ↑〉 and | ↓〉 spin 12states of each electron provide a qubit (two-level quantum

system).[9]

4 Loss-diVincenzo proposal for quantum computation based

on single electron spins in semiconductor quantum dots.

In their original proposal (Loss and DiVincenzo 1998) the qubit is realized as the spin of excess

electron on a single-electron quantum dot as shown schematically in fig.5

In fig. 5 the Voltages applied to the top gates create confining potential for electrons in a two-

dimensional electron gas (2DEG), created below the surface. In order to deplete the 2DEG locally,

a negative voltage can be applied to a back-gate. This allow the number of electrons in each dot

to be reduced down to one (the single electron regime). This step can be considered as the first in

the diVincenzo criteria Creating a well characterized qubit, the single electron.

To ensure a single two-level system is available to be used as a qubit, it is practical to consider

single isolated electron spins (with intrinsic spin 1/2) confined to single orbital levels. By operating

a quantum dot in a Coulomb blockade regime (where the energy for the addition of an electron to

the quantum dot is larger than the energy that can be supplied by electrons in the source or drain

leads) it is possible to demonstrate control over the charging of a quantum dot electron-by-electron

in a single gated quantum dot. In this case, the charge on the quantum dot is conserved, and no

8

electrons can tunnel onto or off of the dot.

With the current technology in material fabrication and gating techniques, it is possible to reach

single electron regime for example in in single vertical (Tarucha et al. 1996) and gated lat-

eral(Ciroga et al.) dots, as well as double dots (Elzerman et al. 2003, Hayashi et al. 2003,

Petta et al. 2004).

The second step in the diVincenzo criteria will be initialization. One way to do this is to initialize

all qubits in the quantum computer to the Zeeman ground state | ↑〉 = |0〉. This could be

achieved by allowing all spins to reach thermal equilibrium at temperature T in the presence of a

strong magnetic field B, such that |gµBB| > kBT , with g-factor g < 0, Bohr magneton µB, and

Boltzmann’s constant KB (Loss and DiVincenzo 1998).

Once we have the qubits initialized to some state, we want them to remain in that state until a

computation can be executed. The spins-1/2 of single electrons are intrinsic two-level systems,

which cannot "leak" into higher excited states in the absence of environmental coupling.

In addition because of the fact that electron spins can only couple to charge degrees of freedom

indirectly through the spin-orbit (or hyperfine) interactions, they are relatively immune to fluctua-

tions in the surrounding electronic environment. This way we address the diVincenzo requirement

"Long relevant decoherence times, much longer than the gate operation time" for electron spins in

quantum dots.

By varying the Zeeman splitting on each dot individually (Loss and DiVincenzo 1998) the Single-

qubit operations in the Loss-DiVincenzo quantum computer could be carried out.

This can be done in a number of different ways. Through the g-factor modulation (Salis et al.

2001), the inclusion of magnetic layers (Myers et al. 2005) (see Figure 6), modification of the local

Overhauser field due to hyperfine couplings (Burkard et al. 1999), or with nearby ferromagnetic

dots (Loss and DiVincenzo 1998) are some of the ways.

Within the Loss-DiVincenzo proposal, two-qubit operations can be performed by pulsing the ex-

change coupling between two neighboring qubit spins "on" to a non-zero value (J(t) = J0 6= 0, t ∈{−τs/2...τs/2) for a switching time τs, then switching it "off" (J(t) = 0, t /∈ {−τs/2...τs/2).

The way to achieve this switching can be by briefly lowering a center-gate barrier between neigh-

boring electrons, resulting in an appreciable overlap of the electron wavefunctions (Loss and DiVin-

cenzo 1998), or alternatively, by pulsing the relative back-gate voltage of neighboring dots (Petta

et al. 2005a).

Under such an operation (and in the absence of Zeeman or weaker spin-orbit or dipolar interac-

tions), the effective two-spin Hamiltonian takes the form of an isotropic Heisenberg exchange term,

9

Figure 6: A series of exchange-coupled electron spins. Single-qubit operations could be performed

in such a structure using electron spin resonance (ESR), which would require an rf transverse

magnetic field B||ac , and a site-selective Zeeman splitting g(x)µBB⊥ , which might be achieved

through g−factor modu lation or magnetic layers. Two-qubit operations would be performed by

bringing two electrons into contact, introducing a nonzero wavefunction overlap and corresponding

exchange coupling for some time (two electrons on the right). In the idle state, the electrons can

be separated, eliminating the overlap and cor- responding exchange coupling with exponential

accuracy (two electrons on the left).[9]

given by (Loss and DiVincenzo 1998, Burkard et al. 1999)

Hex(t) = J(t)SL·SR (1)

where SL(R) is the spin 1/2 operator for the electron in the left (right) dot, as shown in Figure 5.

The exchange Hamiltonian Hex(t) generates the unitary evolution U(φ) = exp[−iφSL·SR], where

φ =∫J(t)dt~ . If the exchange is switched such thatφ =

∫J(t)dt/~ = J0τs/~ = π, U(φ) exchanges

the states of the two neighboring spins, i.e.: U(π)|n, n′〉 = |n′, n〉, where n and n′ are two arbitrarily

oriented unit vectors and |n, n′〉 indicates a simultaneous eigenstate of the two operators SL·nand SR·n′. U(π) implements the so-called swap operation. If the exchange is pulsed on for

the shorter time τs/2, the resulting operation U(π2) = (U(π))

12 is known as the "square-root-of-

swap" (√swap). The √swap operation in combination with arbitrary single-qubit operations is

suffcient for universal quantum computation (Barenco et al. 1995a, Loss and DiVincenzo 1998).

The √swap operation has been successfully implemented in experiments involving two electrons

confined to two neighboring quantum dots. (Petta et al. 2005a, Laird et al. 2005). Errors during

the √swap operation have been investigated due to nonadiabatic transitions to higher orbital

states (Schliemann et al. 2001, Requist et al. 2005), spin-orbit-interaction (Bonesteel et al. 2001,

Burkard and Loss 2002, Stepanenko et al. 2003), and hyperfine coupling to surrounding nuclear

spins (Petta et al. 2005a, Coish and Loss 2005, Klauser et al. 2005, Taylor et al. 2006). The

10

isotropic form of the exchange interaction given in Equation 1 is not always valid.

In realistic systems, a finite spin-orbit interaction leads to anisotropic terms which may cause

additional errors, but could also be used to perform universal quantum computing with two-spin

encoded qubits, in the absence of single-spin rotations (Bonesteel et al. 2001, Lidar and Wu 2002,

Stepanenko and Bonesteel 2004, Chutia et al. 2006).

In the Loss-DiVincenzo proposal, readout could be performed using spin- to-charge conversion.

This could be accomplished with a "spin filter" (spin- selective tunneling) to leads or a neighboring

dot, coupled with single-electron charge detection .

5 Analysis of the theoretical and experimental development

during the last decade

In this section, analysis and discussion of the progress in relation to the diVincenzo criteria for a

successful hardware implementation of a quantum computer will be addressed. Many researchers

worldwide in the field have been working hard toward the successful implementation of quantum

computer since its first proposal in 1997. A short summary in tabular form about progress in

QD systems will be given from a recent review paper. ("Prospects for Spin-Based Quantum

Computing in Quantum Dots" by Christoph Kloffel and Daniel Loss 2013) followed by an outlook

and coclusion.

11

Tab

le1

Overview

ofthestateof

theartforqu

antum

compu

ting

withspinsin

quan

tum

dots(Q

Ds),w

ithreferences

inpa

renthesesor

footno

tesa

Self-assem

bled

QDs

Lateral

QDsin

2DEGs

QDsin

nano

wires

Electrons

Holes

Sing

lespins

S-T0qu

bits

Electrons

Holes

Lifetim

esT1>

20ms(68)

T2:3

ms(120

,129

)T

� 2T

0:1msb

T1:0

.5ms(73,

74)

T2:1

.1ms(158

)T

� 2>

0:1msc

T1>

1s(70)

T2:0

.44ms(127

)T

� 2:37ns

(127

)T

Si 1>

1s(210

)

T1:5

ms(204

,206

)T2:2

76ms(131

)T

� 2:94ns

d

TSi 1∼

10ms(211

)T

�,Si

2:360

nse

T1

1ms(57)

T2:0

.16ms(57)

T� 2:8ns

(57)

T1:0

.6ms(232

)T2:n

.a.

T� 2:n:a:

Operation

times

t Z:8

.1ps

(129

)t X

:4ps

(129

,15

7)t S

W:1

7ps

(161

)

t Z:1

7ps

(158

)t X

:4ps

(158

)t S

W:2

5ps

(159

)

t Z:n

.a.f

t X:2

0ns

(188

)t S

W:3

50ps

(126

)

t Z:3

50ps

(126

)t X

:0.39ns

(100

)t c

cpf:30

nsg

t Z:n

.a.h

t X:8

.5ns

(57,

59)

t SW:n

.a.

n.a.

T2/t

op

1.83

105

4.43

104

229.23

103

n.a.

n.a.

Reado

utschemes

and

fidelitiesF

(visibilities

V)

F¼

96%

(166

)Reson

ance

fluo

rescence

inaQD

molecule

Other:

Farada

y(162

)an

dKerr(156

,16

3,16

5)rotation

spectroscopy

,resona

nce

fluo

rescence

(164

)

Absorption(74,

159)

andem

ission

(73,

158)

spectroscopy

(selection

rules)

V¼

65%

(192

)VSi¼

88%

(210

)Sp

in-selective

tunn

eling

F¼

86%

(193

)Sp

in-selective

tunn

eling(twospins)

Other:

Photon

-assisted

tunn

eling(189

)

V¼

90%

i

F¼

97%

j

Spin-dependent

charge

distribu

tion

(rf-QPC

i /SQD

j )

V¼

81%

k

Spin-dependent

tunn

elingrates

Other:

Dispersiveread

outl

F¼

70%–80

%m

Paulispinblocka

de

Other:

Dispersiveread

outn

Spin-dependent

charge

distribu

tion

(sensordo

tcoup

led

viafloa

ting

gate)o

Initialization

schemes

and

fidelitiesF i

n

F in>

99%

(154

)Optical

pumping

F in¼

99%

(74)

Optical

pumping

F in>

99%

(155

)Exciton

ionization

Spin-selective

tunn

eling(192

,193

,21

0),a

diab

atic

ramping

togrou

ndstateof

nuclearfieldp

Pauliexclusion

(126

)Pa

ulispinblocka

dem

n.a.

(single-ho

leregimeno

tyet

reached)

Scalab

ility

Scalingseem

schalleng

ing

Seem

sscalab

le(185

)[e.g.,viafloa

ting

gates(237

)]Seem

sscalab

le[e.g.,viafloa

ting

gates(237

)]

a For

each

ofthesystem

sdiscussedin

thetext,the

tablesummarizes

thelong

estlifetim

es,the

shortestop

erationtimes,the

high

estreado

utfidelities(visibilities),an

dthehigh

estinitializationfidelities

repo

rted

sofarin

experiments.Informationon

establishedschemes

forread

outa

ndinitializationisprov

ided,along

witharating

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llsing

le-qub

itop

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,tZrefers

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j1æÞ=

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64 Kloeffel ∙ Loss

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.Abb

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).dFrom

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113.

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,20

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When the first proposal for electron spins in QDs for quantum computation were proposed, the

experimental capabilities for efficient implementation was not encouraging. Gate-controlled QDs

within 2DEGs were limited to around 30 or more confined electrons each, and techniques for

single-qubit manipulation and readout were not available (Sacharajda AS 2011). Decoherence

from interactions with the environment was also considered as an almost insurmountable obstacle

in the early days of the proposal. Luckily, within the past decade, this situation has changed

dramatically, owing to continuous experimental and theoretical progress.

It has now been shown that QDs are routinely controlled down to the last electron (hole), owing

to a clever gate design based on plunger gates (Ciorga et al. 2000, Sacharajda AS 2011, Hanson

et al. 2007) and various schemes have been applied for both qubit initialization and readout.

For high-fidelity quantum computation, reducing the occupation number of QDs to the minimum

is desirable. Also larger fillings with a well-defined spin-1/2 ground state are useful (Meier et al.).

Efficient single- and two-qubit gates have also been demonstrated, which allow for universal quan-

tum computing when combined.

One challenge which still exisits is achieving long decoherence time. The achieved gating times

are much shorter than measured lifetimes, and it seems that one will soon be able to overcome

decoherence to the required extent.

While the field is very advanced for the workhorse systems such as lateral GaAs QDs or self

assembled (In)GaAs QDs, rapid progress is also being made in the quest for alternative systems

with further optimized performance.

First, this includes switching to different host materials. For instance, Ge and Si can be grown

nuclear-spin-free, and required gradients in the Zeeman field may be induced via micromagnets.

Second, both electron- and hole-spin qubits are under investigation, exploiting the different prop-

erties of conduction and valence bands, respectively. Finally, promising results are obtained from

new system geometries, particularly nanowire QDs.

Future tasks can probably be divided into three categories.

1. Studying new quantum computing protocols (such as the surface code), which put very low

requirements on the physical qubits.

2. Further optimization of the individual components listed in Table [1]. For instance, longer

lifetimes are certainly desired, as are high-quality qubit gates with even shorter operation

times. However, as decoherence no longer seems to present the limiting issue, particular

focus should also be put on implementing schemes for highly reliable, fast, and scalable

qubit readout (initialization) in each of the systems.

14

3. Because the results in Table 1 are usually based on different experimental conditions, the

third category consists of merging all required elements into one scalable device, without

the need for excellent performance. Such a complete spin-qubit processor should combine

individual single-qubit rotations about arbitrary axes, a controlled (entangling) two-qubit

operation, initialization into a precisely known state, and single-shot readout of each qubit.

While [2] presents an important step toward this unit, prototypes of a complete spin-qubit

processor could present the basis for continuous optimization.

Based on the impressive progress achieved within the past decade, one can cautiously be optimistic

that a large-scale quantum computer can indeed be realized.

References

[1] Dieter Bimberg. Semiconductor Nanostructures. Springer, 2008. ISBN 978-3-540-77898-1.

[2] R. Brunner, Y.-S. Shin, T. Obata, M. Pioro-Ladrière, T. Kubo, K. Yoshida, T. Taniyama,

Y. Tokura, and S. Tarucha. Two-qubit gate of combined single-spin rotation and inter-

dot spin exchange in a double quantum dot. Phys. Rev. Lett., 107:146801, Sep 2011. doi:

10.1103/PhysRevLett.107.146801. URL http://link.aps.org/doi/10.1103/PhysRevLett.

107.146801.

[3] David P. DiVincenzo. The Physical Implementation of Quantum Computation. arXiv, 2008.

[4] C C Escott, F A Zwanenburg, and A Morello. Resonant tunnelling features in quantum dots.

Nanotechnology, 21(27):274018, 2010. URL http://stacks.iop.org/0957-4484/21/i=27/a=

274018.

[5] Christoph Kloeffel and Daniel Loss. Prospects for spin-based quantum computing in quan-

tum dots. Annual Review of Condensed Matter Physics, 4(1):51–81, 2013. doi: 10.1146/

annurev-conmatphys-030212-184248. URL http://www.annualreviews.org/doi/abs/10.

1146/annurev-conmatphys-030212-184248.

[6] L. P. Kouwenhoven, T. H. Oosterkamp, M. W. S. Danoesastro, M. Eto, D. G. Austing,

T. Honda, and S. Tarucha. Excitation spectra of circular, few-electron quantum dots. Sci-

ence, 278(5344):1788–1792, 1997. doi: 10.1126/science.278.5344.1788. URL http://www.

sciencemag.org/content/278/5344/1788.abstract.

[7] Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A,

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52:R2493–R2496, Oct 1995. doi: 10.1103/PhysRevA.52.R2493. URL http://link.aps.org/

doi/10.1103/PhysRevA.52.R2493.

[8] W. G. van der Wiel, S. De Franceschi, J. M. Elzerman, T. Fujisawa, S. Tarucha, and L. P.

Kouwenhoven. Electron transport through double quantum dots. Rev. Mod. Phys., 75(1):1–22,

Dec 2002. doi: 10.1103/RevModPhys.75.1.

[9] W.A.Coish and Daniel Loss. Quantum Computing with spins in solids. arXiv, 2008.

16