Effect of electron-nuclear spin interactions for electron-spin qubits localizedin InGaAs self-assembled quantum dotsSeungwon Lee, Paul von Allmen, Fabiano Oyafuso, Gerhard Klimeck, and K. Birgitta Whaley Citation: J. Appl. Phys. 97, 043706 (2005); doi: 10.1063/1.1850605 View online: http://dx.doi.org/10.1063/1.1850605 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v97/i4 Published by the American Institute of Physics. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Effect of electron-nuclear spin interactions for electron-spin qubitslocalized in InGaAs self-assembled quantum dots

Seungwon Lee,a! Paul von Allmen, Fabiano Oyafuso, and Gerhard KlimeckJet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109

K. Birgitta WhaleyDepartment of Chemistry, University of California, Berkeley, California 94720

sReceived 16 August 2004; accepted 22 November 2004; published online 25 January 2005d

The effect of electron-nuclear spin interactions on qubit operations is investigated for a qubitrepresented by the spin of an electron localized in an InGaAs self-assembled quantum dot. Thelocalized electron wave function is evaluated within the atomistic tight-binding model. The electronZeeman splitting induced by the electron-nuclear spin interaction is estimated in the presence of aninhomogeneous environment characterized by a random nuclear spin configuration, by the dot-sizedistribution, alloy disorder, and interface disorder. Due to these inhomogeneities, the electronZeeman splitting varies from one qubit to another by the order of 10−6, 10−6, 10−7, and 10−9 eV,respectively. Such fluctuations cause errors in exchange operations due to the inequality of theZeeman splitting between two qubits. However, the error can be made lower than the quantum errorthreshold if an exchange energy larger than 10−4 eV is used for the operation. This result shows thatthe electron-nuclear spin interaction does not hinder quantum-dot based quantum computerarchitectures from being scalable even in the presence of inhomogeneous environments. ©2005American Institute of Physics. fDOI: 10.1063/1.1850605g

I. INTRODUCTION

Quantum computerssQCd hold the promise of solvingproblems that would otherwise be beyond the practical rangeof conventional computers.1 Natural candidates for the fun-damental building block of quantum computerssqubitd arethe electronic and the nuclear spins, since they have a welldefined Hilbert space and a relatively long decoherence timecompared to the orbital degrees of freedom. Several QCimplementations have been proposed based on the use of thespin degree of freedom, such as using the nuclear spins in amolecule2 and in crystal lattices,3 the nuclear spins of donorsin Si4 and in endohedral fullerenes,5 and the spin of electronsconfined in quantum dots6,7 or donors.8 While small-scalequantum computing has been demonstrated with a few qubitsusing the nuclear spin9–11or trapped ions,12 large-scale quan-tum computing with many qubits used in parallel is yet to bedemonstrated.

Solid-state spin-based QC architectures are in principlescalable to many qubits. However, they are intrinsically in-homogeneous due to defects, impurities, alloys, interfaces,strain, etc. Although the inhomogeneous environment willcause inaccuracy in the qubit operations, fault-tolerant error-correction schemes have been shown to compensate for er-rors up to a certain threshold.13–16 For spin-based QC archi-tectures, the single-qubit operation typically uses the Zeemancoupling to an external magnetic fieldsgmBS·Bd, while thetwo-qubit operation relies on the exchange interaction be-tween two spinssJS1·S2d. In these operations, if the inhomo-geneous environment causes fluctuations in the Zeeman en-ergy EZ, it will lead to operational errors. For example, a

recent study17 has shown that a “swap” operation with aZeeman energy fluctuationDEZ yields an error ofsDEZ/Jd2.It is thus crucial to examine whether the proposed solid-statespin-based QC implementations are scalable within the quan-tum error limit, in presence of a realistic inhomogeneousenvironment. As a prototype of this examination, the presentpaper focuses on the scalability of architectures where thequbit is represented by an excess electron spin localized in anInGaAs self-assembled quantum dot. An array of InGaAsself-assembled quantum dots is an excellent candidate for ascalable QC architecture because recent advances in the fab-rication technology have substantially improved the controlof size and location of the nanostructures.18–24

The electron Zeeman energies of InGaAs self-assembledquantum dots can fluctuate due to inhomogeneous hyperfineenergies induced by the electron-nuclear spin interaction.This fluctuation of the hyperfine energies can be quite largesince all the nuclei in an InGaAs quantum dot and its envi-ronment GaAs buffer have nonzero magnetic moment andthe resulting numberN of nuclei interacting with the electronspin is in the range 104–106. The hyperfine energy generatedby such a large number of nuclear spins varies from dot todot, due to the various sources of the inhomogeneous envi-ronment. A recent study showed that when the nuclei areunpolarized, the fluctuation of the hyperfine energies due tothe random nuclear spin distribution is proportional to 1/ÎNand is as large as 10−7–10−6 eV for N=104–106.25 The fluc-tuation can be suppressed by polarizing the nuclear spinsusing dynamic polarization schemes.26–30However, the fluc-tuation of the hyperfine energies is persistent even in polar-ized nuclei because of other sources of inhomogeneity in theenvironment. For example, the dot-size distribution, alloydisorder, and interface disorder are inherent to self-adElectronic mail: seungwon.lee@jpl.nasa.gov

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assembled quantum dots. We find here that the electron-nuclear spin interaction in an inhomogeneous environmentcan lead to a fluctuation of the hyperfine energies as large asthat due to the random nuclear distribution in the case ofunpolarized nuclei.

In addition to the inhomogeneous hyperfine energies,fluctuation in the electron Zeeman energies can arise fromthe fluctuation of the electron Landé factorg and the fluc-tuation of the local magnetic field due to, e.g., magnetic im-purities. In particular, considerable fluctuations of the elec-trong factor are expected due to the inhomogeneous dot size,strain, and alloy disorder.31–33Moreover, the electrong factorcan be artificially fluctuatedsi.e., engineeredd by an externalelectric and magnetic field.34–36 The estimation of theg-factor fluctuation due to these sources is reserved for futurework.

The electron-nuclear spin interaction in III–V materialquantum dots has been intensively studied over the last fewyears.25,37–44The focus of these studies has been to under-stand the effect of the interaction on electron spin relaxation,ensemble spin dephasing, and spin decoherence. These in-vestigations have shown that the electron-nuclear spin inter-action is one of the dominant mechanisms for electron spinrelaxation and decoherence, prevailing over the mechanismrelated to spin-orbit coupling under certain circumstances.41

In clear distinction from these previous investigations, thefocus of the present work is to estimate the fluctuations ofthe electron hyperfine energies in the presence of an inhomo-geneous environment and to investigate its consequence forquantum gate operations. Furthermore, most of the previouswork has concentrated on gate-controlled GaAs quantumdots in GaAs–AlGaAs heterostructures,45,46where the lateralconfinement is much weaker than the vertical confinement.In contrast, a self-assembled quantum dot, the subject of thepresent work, is strongly confined in both lateral and verticaldirections. Due to the stronger lateral confinement, the self-assembled quantum dot has a larger energy spacing betweenconfined levels, which is typically about 10–100 meV incontrast to about 1 meV for gate-controlled GaAs quantumdots.

In this paper, we first estimate the fluctuations in thehyperfine energies resulting from the electron-nuclear spininteraction in the presence of inhomogeneities in theenvironments,—in particular dot-size distribution, alloy dis-order, and interface disorder. We calculate the electron den-sity using the atomisticsp3d5s* tight-binding model. Thetight-binding model is ideally suited for the description ofalloy and interface disorder with atomistic resolution, whichenables us to study the microscopic effect of the inhomoge-neous environment on the electron densities. In the secondstage, we evaluate the effect of the resulting fluctuations inthe electron hyperfine energies on qubit operations.

The paper is organized as follows. Section II describesthe treatment of the electron-nuclear spin interaction withinthe tight-binding model. Section III describes the fluctuationof the hyperfine energies of the electron spin from one quan-tum dot to another due to the inhomogeneous environment.

Section IV discusses the effect of the fluctuation of the hy-perfine energies on qubit operations. Finally, Sec. V summa-rizes the results of this work.

II. ELECTRON-NUCLEAR SPIN INTERACTION

The electron-nuclear spin interaction originates from thecoupling of the nuclear magnetic moment to the magneticfield generated by the electron magnetic momentsor equiva-lently from the coupling of the electron magnetic moment tothe magnetic field generated by the nuclear magnetic mo-mentd. Both the spin and orbital angular momentum of theelectron contribute to its magnetic moment. Therefore, theelectron-nuclear spin interactions for an electron in ans-symmetry orbital and in a non-s-symmetry orbital are ex-pressed as follows:47

HHF =16p

3gImBmNdsr dfS · I g, l = 0 s1d

HHF =2gImBmN

r3 FsL − Sd · I + 3sS · r dsI · r d

r2 G, l Þ 0.

s2d

Here, gI, mB, and mN are the gyro-magnetic factor of thenuclear spin, the Bohr magneton, and the nuclear magneton,respectively.S and I are the spin operators for the electronand the nucleus, andL is the angular momentum operator forthe electron. When hydrogen-like atomic orbitals are consid-ered, the first-order energy shifts due to the electron-nuclearspin interaction for boths-symmetry and non-s-symmetryelectron orbitals are comparable.47

The conduction electron wave function in an InGaAsself-assembled dot is mostly composed ofs-symmetryatomic orbitals, as shown in Table I. For the lowest electronwave function in InAs quantum dots, the tight-binding cal-culation shows that the contribution ofs and s* orbitals isabout 94%, while the contributions ofp and d orbitals areabout 5% and 1%, respectively. This small admixture can beunderstood from the wave function symmetry in bulk. Whilethere is no admixture atG6c, Table I shows that the contribu-tions from p and d orbitals increase for wave vectors awayfrom the center of the Brillouin zone. The same trend isreported in psudopotential studies and another tight-bindingcalculation.48–50 The effective wave vector of the electronwave function in the quantum dot is roughlysp /Lx,p /Ly,p /Lzd whereLx,Ly, andLz are the dot dimen-sions. The wave vector is not atG although it is close toG.Therefore, the smallp and d orbital admixture in thequantum-dot wave function originates from the bulk wavefunction symmetry at a wave vector that is not atG.

The effective electron magnetic field generated by eachelectron orbital is roughly proportional to its weight in thewave function. This means that the contribution ofp and dorbitals to the effective magnetic field is about 6%. Since weare interested in the order of magnitude of the hyperfine en-ergy sand its fluctuationd, we therefore choose to ignore theelectron-nuclear spin interaction due to a non-s-symmetryorbital. As a result, the remaining electron-nuclear spin inter-action is described by the hyperfine Fermi contact term:

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HHF =16p

3mBmNo

j

g jsS · I jddsr − Rjd, s3d

where g j and I j are the gyro-magnetic factor and the spinoperator of thej th nucleus.r andR j are the position vectorsfor the electron and thej th nucleus. Since the energy of thehyperfine interactions,0.1 meVd is much smaller than theenergy spacing between the quantized electron levelssabout10–100 meVd, the hyperfine Hamiltonian for a given elec-tron level can be approximated with first-order perturbationtheory as:

HHF =16p

3mBmNo

j

g jucsR jdu2sS · I jd=oj

AjsS · I jd, s4d

wherecsR jd is the electron wave function at nuclear siteR j,and Aj is the effective hyperfine coupling constant betweenthe electron and thej th nuclear spin.

The coupling constantAj is proportional to the square ofthe electron wave function at the nuclear siteR j

Aj =16p

3mBmNg jucsRjdu2. s5d

Within the tight-binding model, the electron wave function isexpressed as a linear combination of atomic basis orbitalsfsr −R jd. The present tight-binding model includessp3d5s*

basis orbitals.51 Therefore, the total electron density atnuclear siteR j is given bys-symmetry orbital densities:

ucsR jdu2 = ua jfss0d + b jfs*s0du2, s6d

wherea j andb j are the tight-binding coefficients fors ands*

orbitals centered at siteR j, respectively. In terms of the ef-fective mass approximation, the tight-binding coefficientsloosely speaking correspond to the envelope functions whilethe tight-binding orbitals correspond to the Bloch wave func-tions.

The tight-binding coefficientsa j and b j depend on dotgeometry, material, strain profile, alloy disorder, etc. The ge-ometry of a quantum dot grown by molecular beam epitaxyvaries widely with the growth condition.52–54 Based on theexperimentally achievable geometries, we model a lens-shaped self-assembled InAs dot with diameter 15 nm andheight 6 nm, as shown in Fig. 1. Since the dots are embed-

ded in a GaAs matrix, InAs–GaAs self-assembled dots arestrongly strained due to the large lattice mismatch of 7%between InAs and GaAs. The equilibrium atomic positionsunder strain are calculated with an atomistic valence forcefield model.55 Following the strain calculation, the electronwave function in the strained nanostructure is obtained withthe empiricalsp3d5s* tight-binding model including the spin-orbit coupling.51 The strain effect on the electronic structureis captured by adjusting the atomic energies of the tight-binding model with a linear correction that is obtained withinthe Löwdin orthogonalization procedure.51,56We also modifythe nearest-neighbor coupling parameters for the strainedstructures according to the generalized version of Harrison’sd2 scaling law and the Slater–Koster direction-cosinerule.57,58

In the empirical tight-binding model,fss0d and fs*s0dare unknown because the model determines the Hamiltonianmatrix elements without introducing the real-space descrip-tion of the basis orbitals. For this work, the densities of thebasis orbitals at a nuclear site are determined empiricallyusing measurements of the Overhauser shift of the electronspin resonance.59 The details of determining the densities aregiven in the Appendix.

With the resultinga j, b j, css0d, andcs*s0d, Aj is calcu-lated according to Eqs.s5d and s6d. The spatial distributionsof Aj along the directions of the dot diameter and dot heightare plotted in Fig. 2. The maximum value ofAjs7 neVd isfound at the As nucleus located at the center of the quantumdot. TheAj value associated with an As nucleus is about 1.7times larger than that associated with the In and Ga nuclei.

TABLE I. Projection of the lowest electron wave function onto the atomic orbitals of different symmetry forlens-shaped InGaAs quantum dotssQDd with diameter 15 nm and height 6 nm. The wave function is calculatedwith an sp3d5s* tight-binding model. Three different alloy materials are considered for quantum dots. Forcomparison, the wave functions of the lowest conduction band atG and X in bulk InAs and GaAs are alsoprojected onto the atomic orbitals.

s symmetry p symmetry d symmetry

Material s s* p d

InAs QD 0.738 0.195 0.051 0.015In0.8Ga0.2As QD 0.762 0.184 0.042 0.012In0.6Ga0.4As QD 0.783 0.174 0.023 0.010InAs Bulk at G6c 0.827 0.173 0.000 0.000GaAs Bulk atG6c 0.844 0.156 0.000 0.000InAs Bulk at X6c 0.204 0.001 0.337 0.457GaAs Bulk atX6c 0.055 0.019 0.434 0.492

FIG. 1. Geometry of the modeled self-assembled quantum dot. The quantumdot is lens shaped with a base diameter of 15 nm and a height of 6 nm. Thelines x and z are the transverse and vertical axes along which the spatialdistributions ofAj are plotted in Fig. 2.

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This large difference is due to the larger electron density onanions than on cations. The global distribution ofAj reflectsthe localization of the electron density inside the dot. Al-though the electron confinement inside the dot is quite effec-tive along the lateral direction, along the vertical axis theelectron density extends farther outside the dot. This causesthe electron spin to interact with a large number of nucleioutside the dot, in addition to the interaction with the nucleiinside the dot. The number of nuclei for whichAj is largerthan 0.01*maxsAjd is about 60 000, whereas the number ofnuclei inside the dot is about 30 000.

III. FLUCTUATION OF THE HYPERFINE SPLITTING

A typical quantum-computer architecture involves an ex-ternal magnetic filedBz to define the qubit space. The spinup and spin down states of the electron in the external mag-netic field Bz are split bygmBB. The two levels are furthersplit by the electron-nuclear spin interaction. According toEq. s4d, the hyperfine splitting is given by

EHF = E↑ − E↓ = oj

AjI jz, s7d

where I jz is the expectation value of operatorI j

z over thenuclear spin states. The hyperfine splitting is determined bythe spatial distributions of bothAj andI j

z. Hence, ifAj andI jz

vary from dot to dot, the hyperfine splitting fluctuates aswell. The variations ofAj andI j

z arise from inhomogeneity inthe environment, such as the nuclear spin orientation, thedot-size distribution, the alloy and interface disorder.

To estimate the fluctuation of the hyperfine splitting, wetake the following approach. First, the hyperfine splittingfluctuation is estimated for one inhomogeneity source at atime while other inhomogeneity sources are assumed to beeliminated. Second, the inhomogeneity source is treated as arandom source since the correlation among quantum dots foreach of the inhomogeneity sources is negligible. Third, theeffect of the inhomogeneity source on the electron wavefunction si.e., Ajd is examined. If the effect is negligible, the

electron wave function is assumed to be unchanged. Other-wise, the electron wave function is recalculated for severalconfigurations of the dot environment.

We first consider the hyperfine splitting fluctuation dueto the nuclear spin orientation. When the nuclear spins areunpolarized, the nuclear spins in the magnetic field at ther-

mal equilibrium are distributed among the eigenstates ofI jz

with a probability proportional to the Boltzmann factorexpsg jmNI j

zB/kBTd. Since the thermal energykBT is typicallymuch larger than the nuclear Zeeman energyg jmNI j

zB, it issafe to assume that the nuclear spins are randomly distrib-

uted among the eigenstates ofI jz with a uniform probability

distribution. It is also reasonable to assume that the randomnuclear spin orientation does not change the electron wavefunction and thus the coupling constantAj, because theelectron-nuclear spin interaction is much smaller than theelectron-nuclear charge interaction. Taking into account theabove reasoning, the fluctuation of the hyperfine splitting dueto the random nuclear spin orientations is given by

DEHF = ÎkEHF2 lens− kEHFlens

2

=Îoj

fAj2ksI j

zd2lens− Aj2kI j

zlens2 g

=Îoj

Aj2I jsI j + 1d/3, s8d

wherek¯lens is an average over the ensemble of dots. Notethat the fluctuation is only due to the random nuclear spinorientation while other inhomogeneity sources are sup-pressed; the dot geometry and atomic configuration in thealloy and near the interface are identical for all the dots. Dueto the random distribution of the nuclear spin within theensemble,kI j

zlens is zero andksI jzd2lens is I jsI j +1d /3 for all j .

Furthermore, the nuclear spins are assumed to be uncorre-lated, i.e.,kI j

zIkzlens=kI j

zlenskIkzlens for j Þk.

The fluctuationDEHF due to the random distribution ofnuclear spins can be suppressed by polarizing the nuclearspins. The polarization can be achieved via the electron-nuclear spin interaction using spin-polarized current or opti-cal pumping.27–30,60,61However, even when the nuclear spinsare fully polarized,EHF can still be broadened by othersources of inhomogeneity in the dot array. For example, anensemble of self-assembled quantum dots has typically abouta 10% size distribution, which is inherent to the nonequilib-rium growth process of the molecular beam epitaxy.62 As thedot size changes, the effective number of nuclei interactingwith the confined electron and the spatial distribution ofAj

change. This leads to different hyperfine splittings for dotswith different sizes. The fluctuation inEHF due to the sizedistribution will be estimated here by comparing calculatedvalues ofEHF for three different dot geometries with basediameter and height values ofs14 nm, 5.5 nmd, s15 nm,6 nmd, ands16 nm, 6.5 nmd, respectively. From the smallestto the largest dot, the number of nuclei inside the dot in-creases from 22 304 to 35 161.

We further consider two additional sources for thebroadening ofEHF in self-assembled dots: Alloy and inter-face disorder. The alloy disorder stems from the fact that a

FIG. 2. Spatial distributions of the hyperfine coupling coefficientsAjd for alens-shaped InAs quantum dot embedded in a GaAs buffersad along thexaxis andsbd along thez axis of Fig. 1. The coupling coefficientAj is givenby Eq. s5d and is proportional to the electron density. The dashed linesindicate the interface between the InAs dot and the GaAs buffer.

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large number of atomic configurations will yield the samecompositional ratio in an InGaAs quantum dot. The inter-faces between an unalloyed InAs dot and the GaAs buffer isaffected by In–Ga intermixing over a length scale of1.25 nm, which is the origin of interface disorder.63 Whenthermal annealing is applied after the growth of an InAs dotin order to tune its electronic structure, the In–Ga intermix-ing length becomes even larger.64 The alloy and interfacedisorder lead to the broadening ofEHF in two ways. First, Inand Ga have different nuclear spin quantum numberssI In

=9/2, IGa=3/2d. Second, they have different ionic potentialsthat will lead to a change in the electron densitiessor Ajd.

When the nuclear spins are polarized and the dot size isuniform, the fluctuationDEHF due to the alloy and interfacedisorder is given by

DEHF =Îoj

D2sAjI jzd, s9d

whereD2sAjI jzd is the variance ofAjI j

z. The varianceD2sAjI jzd

is studied by examining three different atomic configurationsfor an In1−xGaxAs dot. The three atomic configurations areconstructed by randomly choosing the cation atoms as In orGa with probability 1−x and x, respectively. The overlapbetween the wave functions for two arbitrary configurationsis about 0.997, indicating that the fluctuation of the energydensitysor Ajd is very small. Furthermore, the average of thedensity fluctuation per site is only about 10−4% of the aver-age density. Therefore, we may justifiably choose to ignorethe fluctuation ofAj and approximateDEHF as

DEHF < Îoj

Aj2D2I j

z. s10d

For the case of the alloy disorder in In1−xGaxAs dots,D2I jz is

calculated to bexs1−xdsI In− IGad2 for all In and Ga atoms andis zero for all As atoms. For the case of the interface disorderin InAs dots,D2I j

z is 0.25sI In− IGad2 for the In and Ga atomsin the interface region and is zero for all other atoms. Eachcation atom site in the interface is taken to have probability0.5 to be occupied by either an In or a Ga atom.

After calculating the fluctuation ofEHF due to the inho-mogeneities in the environment as described above, we haveobtained the following results. When the nuclear spins areunpolarized, a random nuclear spin configuration yieldsDEHF on the order of 10−6 eV. When the nuclear spins arepolarized, a 10% distribution of dot sizes also yieldsDEHF ofthe order of 10−6 eV. When the nuclear spins are polarizedand the dot size is uniform, the alloy disorder results in fluc-tuations on the order of 10−7 eV, while the interface disordergives rise to fluctuations of the order of 10−9 eV. These re-sults indicate that an unpolarized nuclear spin configurationand quantum dot size fluctuation are the dominant sources ofthe inhomogeneous hyperfine splitting. Since the electron lo-calized in each quantum dot has a different hyperfine split-ting, this leads to an ensemble dephasing characterized by adephasing timeT2

* =" /DEHF. The fluctuation ofEHF can bealso seen as the fluctuation of the effective nuclear magneticfield BN=EHF/gemB. The values ofDE, T2

* , andDBN resultingfrom the inhomogeneous environments studied here are sum-marized in Table II.

IV. EFFECT OF ELECTRON-NUCLEAR SPININTERACTION ON QUBIT OPERATIONS

First, we examine the effect of the electron-nuclear spininteraction on a two-qubit operation. As shown in Sec. III,the electron-nuclear spin interaction in the presence of theinhomogeneous environment leads to the fluctuation of thehyperfine splittingEHF from qubit to qubit. Consequently, thefluctuation ofEHF causes the Zeeman energy to be differentin the two qubits. When a two-qubit operation such as a“swap” operation uses the exchange interactionJS1·S2, theZeeman-energy differenceDEZ between two qubits causes anerror probability ,sDEZ/Jd2.17 It has been shown that inprinciple this swap error can be fixed by additional, subse-quent exchange gate operations ifDEZ is known and stable.65

However, the new pulse sequence adds complexities to swapoperation, and the fidelity of the complex swap operation issensitive to the pulse rise–fall times.65 Therefore, we concen-trate on the original swap operation and its swap error due tothe Zeeman-energy difference.

A quantum error up to a certain threshold can be fixed bya fault-tolerant error correction code. The error threshold var-ies widely depending on the code and the assumption aboutthe system and the error type.66 For our analysis, we choosea conservative estimation of the error threshold 10−4.66,67

This threshold is obtained under the assumption that the con-stant process of error correction takes place after every logi-cal gate. We note that the error threshold here is for the“probability” of getting the output orthogonal to the correctoutput, rather than for the “amplitude” of the output orthogo-nal to the correct one.

For the error correction codes to be able to repair errorsin the swap operation, we requiresDEZ/Jd2,10−4. TheZeeman-energy fluctuationDEZ due to the four differentsources of inhomogeneity considered in this paper rangesfrom 10−9 to 10−6 eV. Hence, the exchange energyJ shouldbe larger than 10−4 eV. At the same time, to prevent theelectron from being excited to higher-lying orbitals,J shouldbe smaller than the electron energy spacingsDEed betweenthe ground and the excited orbital. The excitation probabilitydue to the exchange interaction is roughly on the order of

TABLE II. Hyperfine-splitting fluctuationsDEHFd, ensemble-spin dephasingtime sT2

* =" /DEHFd, and effective nuclear magnetic field fluctuationsDBN

=DEHF/gemBd caused by various inhomogeneities for an InGaAs quantumdot embedded in a GaAs buffer. For unpolarized nuclei, each nuclear spindirection is chosen randomly. For dot-size fluctuations, the base diameter isset to 15±1 nm and the height to 6±0.5 nm. For alloy disorder, In0.5Ga0.5Asdots are examined and each cation atom is randomly chosen to be an In ora Ga atom. For interface disorder, each cation within a 1.25 nm thick inter-face between the dot and the buffer is randomly chosen to be an In or a Gaatom, reflecting the experimental observation of In–Ga mixing near theinterfacesRef. 63d.

Inhomogeneity DEHFseVd T2*ssd DBNsGd

Unpolarized nuclei 10−6 10−10 100Dot-size fluctuation 10−6 10−10 100

Alloy disorder 10−7 10−9 10Interface disorder 10−9 10−7 0.1

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sJ/DEed2. Therefore, a ratioJ/DEe,10−2 would ensure thatthe electron stays in the qubit space with leakage probabilitybelow 10−4.

The dual conditionsDEz!J!DEed can be met with ver-tically stacked self-assembled dots.68 A recent calculationwith harmonic double-well confinement potentials suggeststhat J can be varied from 10−2 to 10−4 eV as the inter-dotdistance increases from 5 to 20 nm.69 Self-assembled dotswith vertical inter-dot distance as small as 2 nm can be easilyfabricated.68 With a given physical inter-dot distance, an ef-fective inter-dot distance can be electronically tuned withgate voltages, which turn on and off the exchange interac-tion. The electron energy spacingDEe of a self-assembleddot is about 10−2–10−1 eV, depending on the geometry andsize of the dot.70 Therefore, the value ofJ that satisfies thedual condition lies between 10−4 and 10−3 eV, which isachievable with vertically stacked self-assembled dots. Inconclusion, withJ between 10−4 and 10−3 eV, the error dueto hyperfine-coupling induced inhomogeneities in Zeemanenergies is smaller than the threshold for error correction,and the qubit leakage to higher orbitals can be effectivelyprevented.

Second, we consider the effect of the electron-nuclearspin interaction on a single-qubit operation. The single-qubitoperation using the Zeeman coupling to an electron spinresonancesESRd field sBac cosvactd involves the tuning ofthe ESR field frequency to the electron-spin precession fre-quency or vice versa. This tuning can be achieved by tuningthe ESR field frequency11 or by tuning the electrong-factorvia an applied voltage.6,8,34,36 The tuning process becomescomplicated in the presence of the effective nuclear magneticfield BN generated by the electron-nuclear spin interaction,becauseBN affects the electron-spin precession frequency.The effective nuclear magnetic fieldBN fluctuates in spacefrom one qubit to another and evolves in time. The spatialfluctuation ofBN can be compensated by calibrating the gatevoltage for each qubit separately. However, the temporalevolution ofBN is difficult to compensate since a gate cali-bration cannot be done immediately before each operation.The temporal evolution is determined by many competinginteractions such as the Zeeman coupling of the nuclear spinto the external magnetic field, the nuclear spin interactionwith the electron spin, the nuclear spin dipolar interaction,and the nuclear spin–lattice interaction. Detailed studies ofthe temporal evolution ofBN that include these interactionsare needed to determine how long a single-qubit gate cali-bration is valid.

Here, we estimate the upper limit of the temporal changeof BN for a single-qubit gate calibration to be valid. We as-sume that a static magnetic fieldB0 of the order of 1 T isapplied, and that an ESR field of the order of 10−3 T is usedfor the spin rotation.4,6,8 The precession frequency of theelectron spin is given byve=gemBÎsB0+BN

i d2+sBN'd2, where

BNi andBN

' are theBN component parallel and perpendicularto B0, respectively. A single-qubit gate will be calibrated bytuning the frequency of the ESR fieldvac to ve. After sometime,71 BN

i andBN' will change byDBN

i andDBN' due to the

nuclear spin dynamics, andve will change accordingly. Thiswill lead to detuning of the ESR field. The frequency differ-

ence vac−ve is approximatelygemBsDBNi +sBN

' /B0dDBN'd,

using the fact thatB0@BN for unpolarized nuclear spins,where BN is on the order of 0.01 T. The error due to thedetuning in the single-qubit operation is proportional tosvac−ved2/ sgemBBacd2. To ensure that the error is smallerthan the threshold values10−4d given by error correctioncodes,66,67 DBN

i and DBN' should be smaller than 10−5 and

10−3 T, respectively. IfDBNi andDBN

' during the time inter-val between calibration and the termination of a computationare bigger than these upper limits, the single-qubit gate cali-bration becomes invalid, and thus QC architectures usingonly exchange interactions should be employed.72–74The ex-change interaction alone can provide universal quantumgates with a minimal cost of increasing the number of re-quired physical qubits by three times and of increasing thenumber of gate operations by five to seven times.73

V. CONCLUSION

The effect of electron-nuclear spin interactions on qubitoperations is investigated here for a qubit represented by theelectron spin localized in an InGaAs self-assembled quantumdot. The localized electron wave function is evaluated withinthe atomistic tight-binding model. The hyperfine splitting in-duced by the electron-nuclear spin interaction is estimated inthe presence of an inhomogeneous environment character-ized by random nuclear configurations, dot-size fluctuations,and alloy and interface disorder. Due to the inhomogeneities,the hyperfine splitting fluctuates from dot to dot on the orderof 10−6, 10−6, 10−7, and 10−9 eV, respectively.

The inhomogeneous hyperfine splitting causes an errorin a two-qubit operation due to the inequality of the Zeemansplitting of the two qubits. However, these errors can bemade smaller than the quantum error threshold, as long asthe exchange energy for the two-qubit operation is largerthan 10−4 eV. Recent work indicates that an exchange energyof 10−4 eV or larger is easily achievable with verticallystacked quantum dots.69,75,76At the same time, the large en-ergy spacing between the ground and the excited orbitals10−2–10−1 eVd of the quantum dots ensures that the electronqubit stays in the ground orbital while the two-qubit opera-tion is conducted.70 This result shows that the electron-nuclear spin interaction does not hinder quantum-dot basedquantum computer architectures from being scalable even inthe presence of inhomogeneous environments causing inho-mogeneous hyperfine splittings.

We also estimated the upper limit of the temporal changeof the nuclear magnetic field for a single-qubit gate calibra-tion to be valid when an electron resonance field is used fora single-qubit operation. The changes of the nuclear mag-netic field parallel and perpendicular to the external staticmagnetic field should be smaller than 10−5 and 10−3 T, re-spectively. When this condition is not met, QC architecturesusing only exchange interactions should be employed.72–74

As shown above, the two-qubit operation using the exchangeinteraction is reliable even in the presence of the inhomoge-neous environment of self-assembled quantum dots.

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ACKNOWLEDGMENTS

This work was performed at Jet Propulsion Laboratory,California Institute of Technology under a contract with theNational Aeronautics and Space Administration. The simula-tions were implemented in and performed with the Nano-electronic Modeling toolsNEMO-3Dd77 that is publiclyavailable through the open channel foundation. This workwas supported by grants from NSA/ARDA, ONR, and JPLinternal Research and Development.

APPENDIX: DENSITY OF AN ATOMIC BASIS ORBITALAT A NUCLEAR SITE

In principle, a measurement of the Overhauser shift ofthe electron spin resonance provides the density of the con-duction electron at the nuclear site.78 Although Overhausershifts for bulk InAs and GaAs have not been measured, thedensities of an atomic orbital at a nuclear site for InAs andGaAs can be deduced from the Overhauser shifts measuredfor bulk InSb.59 The measured densities for InSb are 9.431025 cm−3 at In nuclear site and 1.631026 cm−3 at Sbnuclear site. From the measured densities, Pagetet al.79 de-duced the densities for InAs and GaAs, using the similarityof the ionicity of InAs and GaAs compared to that of InSb.80

ucs0dusIn in InAsd2 . 9.43 1025 cm−3, sA1d

ucs0dusAs in InAsd2 . 9.83 1025 cm−3, sA2d

ucs0dusGa in GaAsd2 . 5.83 1025 cm−3, sA3d

ucs0dusAs in GaAsd2 . 9.83 1025 cm−3. sA4d

The densityucs0du2 for each atom in bulk is related to thetight-binding orbitalsfs*s0d andfss0d as follows:

ucs0du2 = uafss0d + bfs*s0du2, sA5d

where a and b are the tight-binding coefficients for theconduction-band edge wave function in bulk. These coeffi-cients are listed in Table III. Determining the unknown val-uesfss0d andfs*s0d requires one more equation in additionto Eq. sA5d. We assume that the ratiofs*s0d /fss0d is equalto the ratio of the corresponding atomic orbitals. The atomicorbital can be described by a hydrogen-like atomic orbitalwith an effective nuclear charge.81,82The ratiosfs*s0d /fss0dfor In, Ga, and As atoms are 0.53, 0.44, and 0.30, respec-tively. Finally, by inserting the deduced densityucs0du2, theconduction-band edge coefficientsa ,b, and the orbital ratiofs*s0d /fss0d into Eq. sA5d, the densities of thes and s*

orbitals are obtained. The resulting densities are listed inTable III.

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