Quantum Computation with Spins in Quantum Dots

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...i

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 implementation 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 quantum 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...i: 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 |0i) 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 |ψi = a|0i + b|1i to be transformed in to the mixture ρ = |a|2|0ii0| + |b|2|1ii1|. 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 quantum 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 computation 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 1 only one single electron, each with an associated spin 2 operator SL(R) for the electron in the left 1 (right) dot. The | ↑i and | ↓i spin 2 states 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 lateral(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 | ↑i = |0i. 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 exchange 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 neighboring 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 interactions), 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 Bac , 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 corresponding 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 R dt R φ = J(t) . 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, n0i = |n0, ni, where n and n0 are two arbitrarily 0 oriented unit vectors and |n, n i indicates a simultaneous eigenstate of the two operators SL· n 0 and SR· n . U(π) implements the so-called swap operation. If the exchange is pulsed on for π 1 2 the shorter time τs/2, the resulting operation U( 2 ) = (U(π)) 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 Annu. Rev. Condens. Matter Phys. 2013.4:51-81. Downloaded from www.annualreviews.org by Lund University Libraries, Head Office on 05/02/13. For personal use only.

Table 1 Overview of the state of the art for quantum computing with spins in quantum dots (QDs), with references in parentheses or footnotesa

Self-assembled QDs Lateral QDs in 2DEGs QDs in nanowires

Electrons Holes Single spins S-T0 qubits Electrons Holes

Lifetimes T1 > 20 ms (68) T1: 0.5 ms (73, 74) T1 > 1 s (70) T1: 5 ms (204, 206) T1 1 ms(57) T1: 0.6 ms (232)

64 T2:3ms (120, 129) T2:1.1ms (158) T2: 0.44 ms (127) T2:276ms (131) T2: 0.16 ms (57) T2: n.a. T : b > : c : d : : : T2 0 1 ms T2 0 1 ms T2: 37 ns (127) T2 94 ns T2 8 ns (57) T2:na Si Si

Kloeffel > ∼ T1 1 s (210) T1 10 ms (211) ,Si: e T2 360 ns f h ∙ Operation tZ: 8.1 ps (129) tZ: 17 ps (158) tZ: n.a. tZ: 350 ps (126) tZ: n.a. n.a. Loss times tX: 4 ps (129, 157) tX: 4 ps (158) tX: 20 ns (188) tX: 0.39 ns (100) tX: 8.5 ns (57, 59) g tSW : 17 ps (161) tSW : 25 ps (159) tSW : 350 ps (126) tccpf :30ns tSW : n.a. 5 4 3 T2/top 1.8 3 10 4.4 3 10 22 9.2 3 10 n.a. n.a. Readout F ¼ 96% (166) Absorption (74, 159) V ¼ 65% (192) V ¼ 90%i F ¼ 70%–80%m Spin-dependent schemes and Resonance and emission (73, 158) VSi ¼ 88% (210) F ¼ 97%j Pauli spin blockade charge distribution fidelities F fluorescence in a QD spectroscopy Spin-selective Spin-dependent (sensor dot coupled (visibilities V) molecule (selection rules) tunneling charge distribution Other: via floating gate)o (rf-QPCi/SQDj) Dispersive readoutn Other: F ¼ 86% (193) Faraday (162) and Spin-selective V ¼ 81%k Kerr (156, 163, 165) tunneling (two spins) Spin-dependent rotation spectroscopy, tunneling rates resonance Other: fluorescence (164) Photon-assisted Other: tunneling (189) Dispersive readoutl

m Initialization Fin > 99% (154) Fin ¼ 99% (74) Spin-selective Pauli exclusion (126) Pauli spin blockade n.a. schemes and Optical pumping Optical pumping tunneling (192, 193, (single-hole

fidelities Fin 210), adiabatic regime not yet Fin > 99% (155) ramping to ground reached) Exciton ionization state of nuclear fieldp

Scalability Scaling seems challenging Seems scalable (185) Seems scalable [e.g., via floating gates (237)] [e.g., via floating gates (237)]

aFor each of the systems discussed in the text, the table summarizes the longest lifetimes, the shortest operation times, the highest readout fidelities (visibilities), and the highest initialization fidelities reported so far in experiments. Information on established schemes for readout and initialization is provided, along with a rating on scalability. All single-qubitpffiffiffi operation timespffiffiffi correspond to rotations of p (about the z and x axis, respectively) on the Bloch sphere. For a qubit with eigenstates j0æ and j1æ, tZ refers to operations of type ðj0æ þj1æÞ= 2 → ðj0æ j1æÞ= 2, whereas tX refers to rotations of type j0æ → j1æ. Two-qubit gates are characterized by the SWAP time tSW , describing operations of type j01æ → j10æ. The ratio T2/top , where top is the longest of the three operation times, gives an estimate for the number of qubit gates the system can be passed through before coherence is lost. Referring to standard error correction schemes, this value should exceed ∼104 for fault- tolerant quantum computation to be implementable. Using the surface code, values above ∼102 may already be sufficient. Experiments on self-assembled QDs have predominantly been carried out in (Footnotes continued) Annu. Rev. Condens. Matter Phys. 2013.4:51-81. Downloaded from www.annualreviews.org by Lund University Libraries, Head Office on 05/02/13. For personal use only.

Table 1 (Footnotes continued) (In)GaAs. Unless stated otherwise, GaAs has been the host material for gate-defined QDs in two-dimensional electron gases (2DEGs). The results listed for nanowire QDs have been achieved in InAs (electrons) and Ge/Si core/shell (holes) nanowires. We note that the experimental conditions, such as externally applied magnetic fields, clearly differ for some of the listed values and schemes. Finally, we wish to emphasize that further improvements might already have been achieved that we were not aware of when writing this review. Abbreviation: n.a., not yet available. bMeasured for single electrons in a narrowed nuclear spin bath (105) and for two-electron states in QD molecules with reduced sensitivity to electrical noise (170), both via coherent population ∼ : trapping. Without preparation, T2 0 5 10 ns (see Section 4.1). c ¼ From Reference 99. Measured through coherent population trapping. Other experiments revealed T2 2 21 ns attributed to electrical noise (158, 159). d ∼ From Reference 113. Achieved by narrowing the nuclear spin bath. Without narrowing, T2 10 ns (113, 126, 206). eFrom Reference 212. Hyperfine-induced. fWhile Reference 127 gets close, we are currently not aware of a Ramsey-type experiment where coherent rotations about the Bloch sphere z axis have explicitly been demonstrated as a function of

time. Thus no value is listed. However, tZ should be short, on the order of 0.1 ns assuming a magnetic field of 1 T and g ¼0.44 as in bulk GaAs. www.annualreviews.org g From Reference 195. SWAP gates for S-T0 qubits have not yet been implemented. We therefore list the duration tccpf of a charge-state conditional phase flip. hIn Reference 57, rotations about an arbitrary axis in the x-y plane of the Bloch sphere are reported instead, controlled via the phase of the applied microwave pulse. iFrom Reference 125. rf-QPC: radio-frequency quantum point contact (201, 202). jFrom Reference 199. rf-SQD: radio-frequency sensor quantum dot (194). kFrom Reference 204. The paper demonstrates readout of the singlet and triplet states in a single quantum dot. lFrom Reference 203. A radio-frequency resonant circuit is coupled to a double quantum dot. mFrom Reference 57. The readout and initialization schemes in this experiment only determine whether two spins in neighboring qubits are equally or oppositely oriented. nFrom Reference 59. A superconducting transmission line resonator is coupled to a double quantum dot.

oFrom Reference 232. The scheme, operated in the multihole regime, distinguishes the spin triplet states from the spin singlet. pnBsdQatmCmuigi unu Dots Quantum in Computing Quantum Spin-Based pFrom Reference 126. Information about the nuclear field is required for the electronic ground state to be known. 65 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 quantum 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 properties 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.

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