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Solid state Q-‐bits

Realizing spin Q-‐bits

[Loss and Vincenzo, PRA 57, 120 (1998)]

Quantum computation with quantum dots

Daniel Loss1,2,* and David P. DiVincenzo1,3,†1Institute for Theoretical Physics, University of California, Santa Barbara, Santa Barbara, California 93106-4030

2Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland3IBM Research Division, T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598

~Received 9 January 1997; revised manuscript received 22 July 1997!

We propose an implementation of a universal set of one- and two-quantum-bit gates for quantum compu-tation using the spin states of coupled single-electron quantum dots. Desired operations are effected by thegating of the tunneling barrier between neighboring dots. Several measures of the gate quality are computedwithin a recently derived spin master equation incorporating decoherence caused by a prototypical magneticenvironment. Dot-array experiments that would provide an initial demonstration of the desired nonequilibriumspin dynamics are proposed. @S1050-2947~98!04501-6#

PACS number~s!: 03.67.Lx, 89.70.1c, 75.10.Jm, 89.80.1h

I. INTRODUCTION

The work of the past several years has greatly clarifiedboth the theoretical potential and the experimental challengesof quantum computation @1#. In a quantum computer thestate of each bit is permitted to be any quantum-mechanicalstate of a qubit ~quantum bit, or two-level quantum system!.Computation proceeds by a succession of ‘‘two-qubit quan-tum gates’’ @2#, coherent interactions involving specific pairsof qubits, by analogy to the realization of ordinary digitalcomputation as a succession of Boolean logic gates. It is nowunderstood that the time evolution of an arbitrary quantumstate is intrinsically more powerful computationally than theevolution of a digital logic state ~the quantum computationcan be viewed as a coherent superposition of digital compu-tations proceeding in parallel!.Shor has shown @3# how this parallelism may be exploited

to develop polynomial-time quantum algorithms for compu-tational problems, such as prime factoring, which have pre-viously been viewed as intractable. This has sparked inves-tigations into the feasibility of the actual physicalimplementation of quantum computation. Achieving the con-ditions for quantum computation is extremely demanding,requiring precision control of Hamiltonian operations onwell-defined two-level quantum systems and a very high de-gree of quantum coherence @4#. In ion-trap systems @5# andcavity quantum electrodynamic experiments @6#, quantumcomputation at the level of an individual two-qubit gate hasbeen demonstrated; however, it is unclear whether suchatomic-physics implementations could ever be scaled up todo truly large-scale quantum computation, and some havespeculated that solid-state physics, the scientific mainstay ofdigital computation, would ultimately provide a suitablearena for quantum computation as well. The initial realiza-tion of the model that we introduce here would correspond toonly a modest step towards the realization of quantum com-puting, but it would at the same time be a very ambitiousadvance in the study of controlled nonequilibrium spin dy-

namics of magnetic nanosystems and could point the waytowards more extensive studies to explore the large-scalequantum dynamics envisioned for a quantum computer.

II. QUANTUM-DOT IMPLEMENTATIONOF TWO-QUBIT GATES

In this paper we develop a detailed scenario for howquantum computation may be achieved in a coupledquantum-dot system @7#. In our model the qubit is realized asthe spin of the excess electron on a single-electron quantumdot; see Fig. 1. We introduce here a mechanism for two-qubit quantum-gate operation that operates by a purely elec-

*Electronic address: [email protected]†Electronic address: [email protected]

FIG. 1. ~a! Schematic top view of two coupled quantum dotslabeled 1 and 2, each containing one excess electron (e) with spin1/2. The tunnel barrier between the dots can be raised or lowered bysetting a gate voltage ‘‘high’’ ~solid equipotential contour! or‘‘low’’ ~dashed equipotential contour!. In the low state virtual tun-neling ~dotted line! produces a time-dependent Heisenberg ex-change J(t). Hopping to an auxiliary ferromagnetic dot ~FM! pro-vides one method of performing single-qubit operations. Tunneling(T) to the paramagnetic dot ~PM! can be used as a POV read outwith 75% reliability; spin-dependent tunneling ~through ‘‘spinvalve’’ SV! into dot 3 can lead to spin measurement via an elec-trometer E. ~b! Proposed experimental setup for initial test of swap-gate operation in an array of many noninteracting quantum-dotpairs. The left column of dots is initially unpolarized, while theright one is polarized; this state can be reversed by a swap operation@see Eq. ~31!#.

PHYSICAL REVIEW A JANUARY 1998VOLUME 57, NUMBER 1

571050-2947/98/57~1!/120~7!/$15.00 120 © 1998 The American Physical Society

trical gating of the tunneling barrier between neighboringquantum dots rather than by spectroscopic manipulation as inother models. Controlled gating of the tunneling barrier be-tween neighboring single-electron quantum dots in patternedtwo-dimensional electron-gas structures has already beenachieved experimentally using a split-gate technique @8#. Ifthe barrier potential is ‘‘high,’’ tunneling is forbidden be-tween dots and the qubit states are held stably without evo-lution in time (t). If the barrier is pulsed to a ‘‘low’’ voltage,the usual physics of the Hubbard model @9# says that thespins will be subject to a transient Heisenberg coupling,

Hs~ t !5J~ t !SW 1•SW 2 , ~1!

where J(t)54t02(t)/u is the time-dependent exchange con-

stant @10# that is produced by the turning on and off of thetunneling matrix element t0(t). Here u is the charging en-ergy of a single dot and SW i is the spin-1/2 operator for dot i .Equation ~1! will provide a good description of the

quantum-dot system if several conditions are met. ~i! Higher-lying single-particle states of the dots can be ignored; thisrequires DE@kT , where DE is the level spacing and T is thetemperature. ~ii! The time scale ts for pulsing the gate po-tential low should be longer than \/DE in order to preventtransitions to higher orbital levels. ~iii! u.t0(t) for all t; thisis required for the Heisenberg exchange approximation to beaccurate. ~iv! The decoherence time G21 should be muchlonger than the switching time ts . Much of the remainder ofthe paper will be devoted to a detailed analysis of the effectof a decohering environment. We expect that the spin-1/2degrees of freedom in quantum dots should generically havelonger decoherence times than charge degrees of freedomsince they are insensitive to any environmental fluctuationsof the electric potential. However, while charge transport insuch coupled quantum dots has received much recent atten-tion @11,8#, we are not aware of investigations on their non-equilibrium spin dynamics as envisaged here. Thus we willcarefully consider the effect of magnetic coupling to the en-vironment.If G21 is long, then the ideal of quantum computing may

be achieved, wherein the effect of the pulsed Hamiltonian isto apply a particular unitary time evolution operator Us(t)5Texp$2i*0

t Hs(t8)dt8% to the initial state of the two spins:uC(t)&5UsuC(0)& . The pulsed Heisenberg coupling leadsto a special form for Us : For a specific duration ts of thespin-spin coupling such that *dtJ(t)5J0ts5p(mod2p)@12#, Us(J0ts5p)5Usw is the ‘‘swap’’ operator: If ui j& la-bels the basis states of two spins in the Sz basis with i , j50,1, then Uswui j&5u j i&. Because it conserves the total an-gular momentum of the system, Usw is not by itself sufficientto perform useful quantum computations, but if the interac-tion is pulsed on for just half the duration, the resultingsquare root of the swap operator is very useful as a funda-mental quantum gate: For instance, a quantum XOR gate isobtained by a simple sequence of operations

UXOR5ei~p/2 !S1ze2i~p/2 !S2

zUsw1/2eipS1

zUsw1/2 , ~2!

where eipS1z, etc., are single-qubit operations only, which can

be realized, e.g., by applying local magnetic fields ~see Sec.

III B! @13#. It has been established that XOR along withsingle-qubit operations may be assembled to do any quantumcomputation @2#. Note that the XOR of Eq. ~2! is given in thebasis where it has the form of a conditional phase-shift op-eration; the standard XOR is obtained by a simple basischange for qubit 2 @2#.

III. MASTER EQUATION

We will now consider in detail the nonideal action of theswap operation when the two spins are coupled to a magneticenvironment. A master equation model is obtained that ex-plicitly accounts for the action of the environment duringswitching, to our knowledge, the first treatment of this effect.We use a Caldeira-Leggett–type model in which a set ofharmonic oscillators are coupled linearly to the system spinsby Hint5l( i51,2SW i•bW i . Here bi

j5(agai j(aa ,i j1aa ,i j

† ) is afluctuating quantum field whose free motion is governed bythe harmonic-oscillator Hamiltonian HB5(va

i jaa ,i j† aa ,i j ,

where aa ,i j† (aa ,i j) are bosonic creation ~annihilation! opera-

tors ~with j5x ,y ,z) and vai j are the corresponding frequen-

cies with spectral distribution functionJi j(v)5p(a(ga

i j)2d(v2va) @14#. The system and environ-ment are initially uncorrelated with the latter in thermal equi-librium described by the canonical density matrix rB withtemperature T . We assume for simplicity that the environ-ment acts isotropically and is equal and independent on bothdots. We do not consider this to be a microscopically accu-rate model for these as-yet-unconstructed quantum-dot sys-tems, but rather as a generic phenomenological descriptionof the environment of a spin, which will permit us to explorethe complete time dependence of the gate action on thesingle coupling constant l and the controlled parameters ofHs(t) @15#.

A. Swap gate

The quantity of interest is the system density matrixr(t)5TrB r (t), which we obtain by tracing out the environ-ment degrees of freedom. The full density matrix r itselfobeys the von Neumann equation

r ̇ ~ t !52i@H , r #[2iLr , ~3!

where

L5Ls~ t !1Lint1LB ~4!

denotes the Liouvillian @16# corresponding to the full Hamil-tonian

H5Hs~ t !1Hint1HB . ~5!

Our goal is to find the linear map ~superoperator! V(t) thatconnects the input state of the gate r05r(t50) with theoutput state r(t) after time t.ts has elapsed, r(t)5V(t)r0 .V(t) must satisfy three physical conditions: ~i! trace preser-vation Trs Vr51, where Trs denotes the system trace; ~ii!Hermiticity preservation (Vr)†5Vr; and ~iii! complete posi-tivity, (V^1B) r >0. Using the Zwanzig master equation ap-proach @16#, we sketch the derivation for V in the Born and

57 121QUANTUM COMPUTATION WITH QUANTUM DOTS

Swap operaEon

XOR gate = any quantum computaEon

Coherent spin control

[Nowack et al, Science 318, 1430 (2007)]

periodically displacing the electron wave functionaround its equilibrium position (Fig. 1B).

The experiment consists of four stages (Fig.1C). The device is initialized in a spin-blockaderegime where two excess electrons, one in each

dot, are held fixed with parallel spins (spintriplet), either pointing along or opposed to theexternal magnetic field [the system is neverblocked in the triplet state with antiparallelspins, because of the effect of the nuclear fields

in the two dots combined with the small interdottunnel coupling; see (16) for details]. Next, thetwo spins are isolated by a gate voltage pulse,such that electron tunneling between the dots orto the reservoirs is forbidden. Then, one of thespins is rotated by an ac voltage burst applied tothe gate, over an angle that depends on thelength of the burst (17) (most likely the spin inthe right dot, where the electric field is expectedto be strongest). Finally, the readout stage allowsthe left electron to tunnel to the right dot if andonly if the spins are antiparallel. Subsequent tun-neling of one electron to the right reservoir givesa contribution to the current. This cycle is re-peated continuously, and the current flow throughthe device is thus proportional to the probabilityof having antiparallel spins after excitation.

To demonstrate that electrical excitation canindeed induce single-electron spin flips, we ap-ply a microwave burst of constant length to theright side gate and monitor the average currentflow through the quantum dots as a function ofexternal magnetic field Bext (Fig. 2A). A finitecurrent flow is observed around the single-electron spin resonance condition, i.e., when|Bext| = hfac/gmB, with h Planck’s constant, facthe excitation frequency, and mB the Bohrmagneton. From the position of the resonantpeaks measured over a wide magnetic fieldrange (Fig. 2B), we determine a g factor of |g| =0.39 ± 0.01, which is in agreement with otherreported values for electrons in GaAs quantumdots (18).

In addition to the external magnetic field, theelectron spin feels an effective nuclear field BNarising from the hyperfine interaction withnuclear spins in the host material and fluctuatingin time (19, 20). This nuclear field modifies theelectron spin resonance condition and is gener-ally different in the left and right dot (by DBN).The peaks shown in Fig. 2A are averaged overmany magnetic field sweeps and have a widthof about 10 to 25 mT. This is much larger thanthe expected linewidth, which is only 1 to 2 mTas given by the statistical fluctuations of BN

(21, 22). Looking at individual field sweepsmeasured at constant excitation frequency, wesee that the peaks are indeed a few mT wide(Fig. 2C), but that the peak positions change intime over a range of ~20 mT. Judging from thedependence of the position and shape of theaveraged peaks on sweep direction, the origin ofthis large variation in the nuclear field is mostlikely dynamic nuclear polarization (4, 23–26).

To demonstrate coherent control of the spin,we varied the length of the microwave burstsand monitored the current level. In Fig. 3A weplot the maximum current per magnetic fieldsweep as a function of the microwave burstduration, averaged over several sweeps (this is amore sensitive method than averaging the tracesfirst and then taking the maximum) (17). Themaximum current exhibits clear oscillations as afunction of burst length. Fitting with a cosinefunction reveals a linear scaling of the oscilla-

Fig. 1. (A) Scanning elec-tron micrograph of a de-vice with the same gatestructure as the one usedin this experiment. MetallicTiAu gates are depositedon top of a GaAs hetero-structure that hosts a two-dimensional electron gas90 nm below the surface.Not shown is a coplanarstripline on top of themetallic gates, separatedby a dielectric [not usedin this experiment; see also(4)]. In addition to a dcvoltage, we can apply fastpulses and microwaves tothe right side gate (as indi-cated) through a homemadebias-tee. The orientation ofthe in-plane external magnetic field is as shown. (B) The electric field generated upon excitation of thegate displaces the center of the electron wave function along the electric field direction and changesthe potential depth. Here, D is the orbital energy splitting, ldot = !/

!!!!!!!!!

m!Dp

the size of the dot, m* theeffective electron mass, ! the reduced Planck constant, and E(t) the electric field. (C) Schematic of thespin manipulation and detection scheme, controlled by a combination of a voltage pulse and burst, V(t),applied to the right side gate. The diagrams show the double dot, with the thick black lines indicatingthe energy cost for adding an extra electron to the left or right dot, starting from (0,1), where (n,m)denotes the charge state with n and m electrons in the left and right dot. The energy cost for reaching(1,1) is (nearly) independent of the spin configuration. However, for (0,2), the energy cost for forming asinglet state [indicated by S(0,2)] is much lower than that for forming a triplet state (not shown). Thisdifference is exploited for initialization and detection, as explained further in the main text.

A

0.3 µm

Initialize Isolate Read-outManipulateC

V (

mV

)

time

–6

02 µs0

Position

B

+

Idot

dot2

! eE (t)

Bext

E

S(0,2)

DC+E

nerg

y

Fig. 2. (A) The currentaveraged over 40 mag-netic field sweeps isgiven for eight differ-ent excitation frequen-cies, with a microwaveburst length of 150 ns.The traces are offsetfor clarity. The micro-wave amplitude Vmwwas in the range 0.9to 2.2 mV, dependingon the frequency (esti-mated from the outputpower of the micro-wave source and tak-ing into account theattenuation of the co-axial lines and theswitching circuit usedto create microwavebursts). (B) Position ofthe resonant responseover wider frequencyand field ranges. Error bars are smaller than the size of the circles. (C) Individual magnetic field sweepsat fac = 15.2 GHz measured by sweeping from high to low magnetic field with a rate of 50 mT/min. Thetraces are offset by 0.1 pA each for clarity. The red trace is an average over 40 sweeps, including the onesshown and scaled up by a factor of 5.

2.76 2.77 2.78 2.79

0

0.2

0.4

0.6

0.816.00 GHz

15.79 GHz

15.57 GHz

15.36 GHz

15.14 GHz

14.93 GHz

14.71 GHz

14.50 GHz

2.6 2.8 30

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2 2.5 30

5

10

15B

A C

Bext (T)

f(G

Hz)

Bext (T)

Bext (T)

acD

ot c

urre

nt (

pA)

Dot

cur

rent

(pA

)

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periodically displacing the electron wave functionaround its equilibrium position (Fig. 1B).

The experiment consists of four stages (Fig.1C). The device is initialized in a spin-blockaderegime where two excess electrons, one in each

dot, are held fixed with parallel spins (spintriplet), either pointing along or opposed to theexternal magnetic field [the system is neverblocked in the triplet state with antiparallelspins, because of the effect of the nuclear fields

in the two dots combined with the small interdottunnel coupling; see (16) for details]. Next, thetwo spins are isolated by a gate voltage pulse,such that electron tunneling between the dots orto the reservoirs is forbidden. Then, one of thespins is rotated by an ac voltage burst applied tothe gate, over an angle that depends on thelength of the burst (17) (most likely the spin inthe right dot, where the electric field is expectedto be strongest). Finally, the readout stage allowsthe left electron to tunnel to the right dot if andonly if the spins are antiparallel. Subsequent tun-neling of one electron to the right reservoir givesa contribution to the current. This cycle is re-peated continuously, and the current flow throughthe device is thus proportional to the probabilityof having antiparallel spins after excitation.

To demonstrate that electrical excitation canindeed induce single-electron spin flips, we ap-ply a microwave burst of constant length to theright side gate and monitor the average currentflow through the quantum dots as a function ofexternal magnetic field Bext (Fig. 2A). A finitecurrent flow is observed around the single-electron spin resonance condition, i.e., when|Bext| = hfac/gmB, with h Planck’s constant, facthe excitation frequency, and mB the Bohrmagneton. From the position of the resonantpeaks measured over a wide magnetic fieldrange (Fig. 2B), we determine a g factor of |g| =0.39 ± 0.01, which is in agreement with otherreported values for electrons in GaAs quantumdots (18).

In addition to the external magnetic field, theelectron spin feels an effective nuclear field BNarising from the hyperfine interaction withnuclear spins in the host material and fluctuatingin time (19, 20). This nuclear field modifies theelectron spin resonance condition and is gener-ally different in the left and right dot (by DBN).The peaks shown in Fig. 2A are averaged overmany magnetic field sweeps and have a widthof about 10 to 25 mT. This is much larger thanthe expected linewidth, which is only 1 to 2 mTas given by the statistical fluctuations of BN

(21, 22). Looking at individual field sweepsmeasured at constant excitation frequency, wesee that the peaks are indeed a few mT wide(Fig. 2C), but that the peak positions change intime over a range of ~20 mT. Judging from thedependence of the position and shape of theaveraged peaks on sweep direction, the origin ofthis large variation in the nuclear field is mostlikely dynamic nuclear polarization (4, 23–26).

To demonstrate coherent control of the spin,we varied the length of the microwave burstsand monitored the current level. In Fig. 3A weplot the maximum current per magnetic fieldsweep as a function of the microwave burstduration, averaged over several sweeps (this is amore sensitive method than averaging the tracesfirst and then taking the maximum) (17). Themaximum current exhibits clear oscillations as afunction of burst length. Fitting with a cosinefunction reveals a linear scaling of the oscilla-

Fig. 1. (A) Scanning elec-tron micrograph of a de-vice with the same gatestructure as the one usedin this experiment. MetallicTiAu gates are depositedon top of a GaAs hetero-structure that hosts a two-dimensional electron gas90 nm below the surface.Not shown is a coplanarstripline on top of themetallic gates, separatedby a dielectric [not usedin this experiment; see also(4)]. In addition to a dcvoltage, we can apply fastpulses and microwaves tothe right side gate (as indi-cated) through a homemadebias-tee. The orientation ofthe in-plane external magnetic field is as shown. (B) The electric field generated upon excitation of thegate displaces the center of the electron wave function along the electric field direction and changesthe potential depth. Here, D is the orbital energy splitting, ldot = !/

!!!!!!!!!

m!Dp

the size of the dot, m* theeffective electron mass, ! the reduced Planck constant, and E(t) the electric field. (C) Schematic of thespin manipulation and detection scheme, controlled by a combination of a voltage pulse and burst, V(t),applied to the right side gate. The diagrams show the double dot, with the thick black lines indicatingthe energy cost for adding an extra electron to the left or right dot, starting from (0,1), where (n,m)denotes the charge state with n and m electrons in the left and right dot. The energy cost for reaching(1,1) is (nearly) independent of the spin configuration. However, for (0,2), the energy cost for forming asinglet state [indicated by S(0,2)] is much lower than that for forming a triplet state (not shown). Thisdifference is exploited for initialization and detection, as explained further in the main text.

A

0.3 µm

Initialize Isolate Read-outManipulateC

V (

mV

)

time

–6

02 µs0

Position

B

+

Idot

dot2

! eE (t)

Bext

E

S(0,2)

DC+

Ene

rgy

Fig. 2. (A) The currentaveraged over 40 mag-netic field sweeps isgiven for eight differ-ent excitation frequen-cies, with a microwaveburst length of 150 ns.The traces are offsetfor clarity. The micro-wave amplitude Vmwwas in the range 0.9to 2.2 mV, dependingon the frequency (esti-mated from the outputpower of the micro-wave source and tak-ing into account theattenuation of the co-axial lines and theswitching circuit usedto create microwavebursts). (B) Position ofthe resonant responseover wider frequencyand field ranges. Error bars are smaller than the size of the circles. (C) Individual magnetic field sweepsat fac = 15.2 GHz measured by sweeping from high to low magnetic field with a rate of 50 mT/min. Thetraces are offset by 0.1 pA each for clarity. The red trace is an average over 40 sweeps, including the onesshown and scaled up by a factor of 5.

2.76 2.77 2.78 2.79

0

0.2

0.4

0.6

0.816.00 GHz

15.79 GHz

15.57 GHz

15.36 GHz

15.14 GHz

14.93 GHz

14.71 GHz

14.50 GHz

2.6 2.8 30

0.05

0.1

0.15

0.2

0.25

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5

10

15B

A C

Bext (T)

f(G

Hz)

Bext (T)

Bext (T)

acD

ot c

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pA)

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(pA

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tion frequency with the driving amplitude (Fig.3B), a characteristic feature of Rabi oscillationsand proof of coherent control of the electronspin via electric fields.

The highest Rabi frequency we achieved is~4.7 MHz (measured at fac = 15.2 GHz), cor-responding to a 90° rotation in ~55 ns, which isonly a factor of 2 slower than those realized withmagnetic driving (4). Stronger electrical drivingwas not possible because of photon-assisted tun-neling. This is a process whereby the electricfield provides energy for one of the followingtransitions: tunneling of an electron to a reser-voir or to the triplet with both electrons in theright dot. This lifts spin blockade, irrespective ofwhether the spin resonance condition is met.

Small Rabi frequencies could be observed aswell. The bottom trace of Fig. 3A shows a Rabioscillation with a period exceeding 1.5 ms(measured at fac = 2.6 GHz), corresponding toan effective driving field of only about 0.2 mT,one-tenth the amplitude of the statistical fluctua-tions of the nuclear field. The oscillations arenevertheless visible because the dynamics of the

nuclear bath are slow compared to the Rabiperiod, resulting in a slow power-law decay ofthe oscillation amplitude on driving field (27).

We next turn to the mechanism responsiblefor resonant transitions between spin states.First, we exclude a magnetic origin becausethe oscillating magnetic field generated uponexcitation of the gate is more than two orders ofmagnitude too small to produce the observed Rabioscillations with periods up to ~220 ns, whichrequires a driving field of about 2 mT (17).Second, we have seen that there are in principlea number of ways in which an ac electric fieldcan cause single-spin transitions. What isrequired is that the oscillating electric field giverise to an effective magnetic field, Beff(t), actingon the spin, oscillating in the plane perpen-dicular to Bext, at frequency fac = gmB|Bext|/h.The g-tensor anisotropy is very small inGaAs, so g-tensor modulation can be ruledout as the driving mechanism. Furthermore, inour experiment there is no external magneticfield gradient applied, which could otherwiselead to spin resonance (5). We are aware of

only two remaining possible coupling mech-anisms: spin-orbit interaction and the spatialvariation of the nuclear field.

In principle, moving the wave function in anuclear field gradient can drive spin transitions(5, 28), as was recently observed (26). However,the measurement of each Rabi oscillation lastedmore than 1 hour, much longer than the timeduring which the nuclear field gradient isconstant (~100 ms to a few s). Because thisfield gradient and, therefore, the correspondingeffective driving field, slowly fluctuates in timearound zero, the oscillations would be stronglydamped, regardless of the driving amplitude(26). Possibly, a (nearly) static gradient in thenuclear spin polarization could develop as aresult of electron-nuclear feedback. However,such polarization would be parallel to Bext andthus cannot be responsible for the observedcoherent oscillations.

In contrast, spin orbit–mediated driving caninduce coherent transitions (12), which can beunderstood as follows. The spin-orbit interactionin a GaAs heterostructure is given by HSO =a(pxsy ! pysx) + b(!pxsx + pysy), where a and bare the Rashba and Dresselhaus spin-orbit co-efficient, respectively, and px,y and sx,y are themomentum and spin operators in the x and ydirections (along the [100] and [010] crystal direc-tions, respectively). As suggested in (13), the spin-orbit interaction can be conveniently accounted forup to the first order in a, b by applying a (gauge)transformation, resulting in a position-dependentcorrection to the external magnetic field. This ef-fective magnetic field, acting on the spin, is pro-portional and orthogonal to the field applied

Beff !x,y" # n"Bext; nx #2m#

!!!ay ! bx";

ny #2m#

!!ax$ by"; nz # 0 !1"

An electric field E(t) will periodically andadiabatically displace the electron wave func-tion (Fig. 1B) by x(t) = (eldot

2/D)E(t), so theelectron spin will feel an oscillating effectivefield Beff(t) $ Bext through the dependence ofBeff on the position. The direction of n can beconstructed from the direction of the electricfield as shown in Fig. 4C and together withthe direction of Bext determines how effec-tively the electric field couples to the spin.The Rashba contribution always gives n$E,while for the Dresselhaus contribution thisdepends on the orientation of the electric fieldwith respect to the crystal axis. Given the gategeometry, we expect the dominant electric fieldto be along the double dot axis (Fig. 1A), whichhere is either the [110] or [110] crystallographicdirection. For these orientations, the Dresselhauscontribution is also orthogonal to the electric field(Fig. 4C). This is why both contributions willgive Beff % 0 and lead to coherent oscillations inthe present experimental geometry, where E || Bext.In (26), a very similar gate geometry was used,but the orientation of Bext was different, and it

Fig. 3. (A) Rabi oscilla-tions at 15.2 GHz (blue,average over five sweeps)and 2.6 GHz (black, av-erage over six sweeps).The two oscillations at15.2 GHz are measuredat different amplitudes ofthe microwaves Vmw,leading to different Rabifrequencies. (B) Lineardependence of the Rabifrequency on applied mi-crowave amplitude mea-sured at fac = 14 GHz.

0 200 400 600 800

50

100

150

0 500 1000 1500 2000 250020

40

60

0 1.50

1

2

3

Burst time (ns)

Dot

curr

ent(

fA)

f(M

Hz)

A B

V (mV)mw

V ~0.8 mVmw

V ~1.4 mVmw

V ~3 mVmw

Rab

i

Fig. 4. (A) Rabi frequency re-scaled with the applied electricfield for different excitation frequen-cies. The error bars are given by

fRabi/E·!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!dE=E"2$ !dfRabi=fRabi"2q

where dfRabi and dE are the error inthe Rabi frequency and electric fieldamplitude, respectively. The graylines are the 95%confidence boundsfor a linear fit through the data(weighting the data points by theinverse error squared). (B) Estimatedelectric field amplitudes at which theRabi oscillations of (A) were mea-sured at the respective excitationfrequencies (17). (C) Constructionof the direction of n resulting fromthe Rashba and Dresselhaus spin-orbit interaction for an electric fieldalong [110] following Eq. 1. Thecoordinate system is set to thecrystallographic axis [100] and [010]. fac (GHz)

f/E

(MH

z·cm

/V)

A

E

BextEx

Ey

Rashba

Dresselhaus

n

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Working points

dimensional electron gas 90 nm below the sur-face of a GaAs/(Al,Ga)As heterostructure (Fig.1A, inset). We tune the device to the few-electronregime (Fig. 1A) and adjust the tunnel couplingbetween the dots and to the leads via the gatevoltages. In all measurements, the inter-dot tun-nel coupling ranges from 2 to 8 meV, determinedby microwave spectroscopy (21). Quantum pointcontacts (QPCs) on both sides of the structureallow us to monitor the charge occupation of thedouble dot. We used room-temperature IV con-verters to record the current from the left andright QPCs (IL;RQPC), and we monitored it in realtime (22) so that individual electrons can be seento leave and enter the dots. To set the electro-chemical potentials in the left and right dot in-dependently, we used combinations of voltageson gates left plunger (LP) and right plunger (RP),which compensate for capacitive cross-coupling.An in-plane magnetic field Bext = 6.5 T is applied

to split the spin-up (!) and spin-down (") energyof both electrons by the Zeeman energy (EZ !130 meV) defining a qubit in each of the dots. Theelectron temperature is typically 250 mK.

The read-out protocol consists of two steps(Fig. 1, B and C). Starting with one electron ineach dot, the spin in the right dot is read out byapplying a gate voltage pulse, which lines up thechemical potentials for spin-up and spin-downjust below and above the Fermi level in the res-ervoir (position 1 in Fig. 1, A and B). A spin-upelectron will then remain in the dot as it doesnot have enough energy to reach the unoccupiedstates in the reservoir, but a spin-down electronwill tunnel out; soon afterwards, the dot will berefilled with a spin-up electron (14). Throughoutthis process, the QPC current is monitored. Froma flat QPC response, we infer the electron was !;if the response shows a step, we conclude theelectron was " (Fig. 1C) [see supporting online

material (SOM) text for details on the thresholdanalysis]. Subsequently, the spin in the left dot isread out in a similar manner (position 2 in Fig. 1,A and B) (23).

Each single-shot measurement trace containstwo segments, reflecting the spin states of theright and left dot, respectively (Fig. 1C). Becausewe can read out both spins starting from thesame state preparation, themeasurement protocolachieves independent single-shot read-out of twosolid-state spins. By construction, the read-outprotocol resets both qubits to !.

As a first test of the measurement protocol,we inject a random spin in each of the dots byemptying the dot and then rapidly pulsing thelevels down (Fig. 1D). We then wait in the (1,1)charge region for a variable time and read out

Fig. 1. (A) Charge stability diagram, with ILQPCshown in color scale as a function of voltages ap-plied to gates LP and RP (a background plane hasbeen subtracted). The occupation in the left andright dots is indicated by numbers in brackets.(Inset) Scanning electron micrograph of a devicesimilar to the one used in our experiment. Gates LPand RP are connected to high-frequency linesvia bias-tees. The direction of Bext is indicated.(B) Electrochemical potential diagrams showing thedouble-dot configuration in the two read-out stages[positions (1) and (2) in (A)]. Tunnel events thatoccur for a !! state are indicated. (C) Single-shotread-out traces, displaying the difference of thecurrent through the two QPCs, which are oppositelybiased (22). The first and second parts correspondto read-out of the right and left dots, respectively.Four typical responses are shown (offset for clarity),one for each of the possible two-spin states. (D)Diagrams illustrating initialization into a randomspin state [positions (3) and (4) in (A)]. (E) Measuredprobabilities to find the spin states"","!,!", and!! as a function of wait time before the read-out.Circles denote two-spin probabilities; crosses andgray lines indicate the product of single-spinprobabilities (e.g., PL" ! PR! for "!), where the linesare based on exponential fits to the single-spinprobabilities, with fitted spin relaxation times, T1, inthe left and right dot of 4.9 T 1.7 ms and 3.8 T 0.7ms, respectively (see fig. S3).

Fig. 2. (A) Charge stability diagram including the(2,0) charge region. Symbols indicate positions rel-evant for the measurements shown in (B), as wellas in Figs. 3 and 4. (B) Two-spin probabilities as afunction of wait time in the (2,0) charge region atthe position of the blue square in (A). Circles andsolid gray lines denote two-spin probabilities; crossesand dashed gray lines indicate products of single-spin probabilities. The solid gray lines are expo-nentials with saturation values determined by thetwo-spin probabilities calculated from independent-ly determined read-out fidelities (see SOM text) andthe ideal values P""= P!!=0,P"!= P!"=1/2. Thetime constant is determined from an exponential fitto the P"" data, and the initial values account forthe spin-up and spin-down injection probabilities.

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both spins. Repeating these steps, we collect sta-tistics and determine the two-spin probabilitiesP!!, P!", P"!, and P"". As expected because ofrelaxation at low temperature, the ground-stateprobability P!! increases exponentially, at the

expense of the three other probabilities (Fig. 1E).Given the injection of random spins in each of thedots [the actual probability for injecting spin-down is typically only 25 to 30%, dependingon where the electrons are injected (24)], the

probabilities should not display any correla-tions. Indeed, the products of the single-dotprobabilities—for example, PL

" ! P"! " P""—overlap with the corresponding two-spin proba-bilities (circles versus crosses in Fig. 1E), asexpected for uncorrelated spins.

Correlations between the states of the twospins are induced when after injection of randomspins in (1,1), we pulse into the (2,0) charge re-gion for a variable time (Fig. 2A), during whichrelaxation to the (2,0) spin-singlet ground statewill take place. When we subsequently separatethe two electrons by pulsing into the (1,1) chargeregion and measure both spins, we see that cor-relations build up in the measurement outcomesconsistent with singlet preparation [random localnuclear fields may dephase the singlet, but theantiparallel correlations survive (25)]: Whenevermeasurement of the left dot gives a spin-up out-come, measurement of the right dot most likelygives spin-down, and vice versa (Fig. 2B).

The (anti-)correlations are further elucidatedby comparing the four two-spin probabilities (cir-cles in Fig. 2B) with the product of the respectivesingle-spin probabilities (crosses). For each spinby itself, the spin-down probability is ideally 1/2.However, the joint probability for "" is not 1/2 !1/2 = 1/4, but instead 0. This gap between circlesand crosses develops in Fig. 2B as the singletprobability increases. The deviation from the idealvalues is quantitatively understood on the basisof the estimated read-out fidelities (gray lines, seebelow and SOM text). This demonstrates that thejoint single-shot read-out allows us to directlyprobe correlations between two spins.

We next examine whether the read-out ofboth spins is truly independent, in the sense thatthe measurement outcome of one spin is not in-fluenced by the measurement and/or the state ofthe other spin. First, with proper alignment of therespective chemical potentials (Fig. 1B) and suf-ficiently small QPC bias (typically 400 meV) (26),one electron stays in its dot while the other isbeing measured. A more subtle possible cross-talk effect is that the second dot will not be linedup in the proper read-out configuration if the firstdot is empty (due to the cross-capacitance). Wetherefore discard those traces (here, <5%) wherethe right dot was emptied but not refilled duringthe first read-out stage (see SOM text and fig. S2).

The most relevant remaining possible originof cross-talk arises from the tunnel coupling be-tween the dots, which results in exchange cou-pling of the two spins. To address this issue, weinitialize the left spin deterministically in ! bywaiting sufficiently long in the (1,0) charge re-gion for the spin to relax. Subsequently, we injectan electron with random spin into the right dot, atgate settings for which (1,0) is lower in energythan (0,1) so the left electron stays in its dot whilethe right electron tunnels in. We observe no de-cay for the left dot (Fig. 3B), whereas the rightdot shows the usual exponential decay. From theamplitude of this exponential decay compared tothe standard deviation of the red data points in

Fig. 3. (A) Schematic of the energy levels close tothe (1,1)-(2,0) boundary, along the green arrow inFig. 2A. Due to the Pauli exclusion principle, theground state in the (2,0) charge region is a spinsinglet. The (1,1) and (2,0) charge states with thesame spin hybridize due to the inter-dot tunnelcoupling. S(1,1) and S(2,0) denote the spin sin-glets in the (1,1) and (2,0) charge configuration.T+, T0, and T– are the three (1,1) triplets withmagnetic quantum number = +1, 0, and –1. T+and T– are split off due to Bext. (B) Single-spinprobabilities to find! as a function of wait time atthe position of the green star (see also Fig. 2A)when initializing the left spin deterministically in" and the right spin in a random spin state. (C)After initialization as in (B), the double dot ispulsed for 25 ns (circles) or 10 ms (diamonds) to aposition close to the (1,1)-(2,0) charge transition.Single-spin probabilities to find ! as a function ofthis position are shown. Gray lines are a guide forthe eye.

Fig. 4. (A to D) Two-qubit exchange gate on a full set of input states. The four panels correspond to fourdifferent mixtures of initial states, as indicated, taken with otherwise identical settings. Again, spin-downinjection probabilities are below 50%. Gray lines are fits to damped oscillations, including a correction forpulse imperfections. We first fit P"! in (A) and P!" in (B) and allow only the amplitude and offset of theoscillations to change for the other probabilities in the respective panel. In (C) and (D), we use the fitparameters of (A) and allow only amplitude and offset to change. The oscillations in (A) and (B) run out ofphase with each other for longer wait times. We attribute this to subtle distortions of the pulses arriving at thesample due to the bias tees (22). (E and F) Visualized theoretical and experimental truth tables for a p rotationand a 2p rotation of the exchange oscillation (details and actual numbers are given in the SOM and fig. S5).

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Probability oscilla5ons:

both spins. Repeating these steps, we collect sta-tistics and determine the two-spin probabilitiesP!!, P!", P"!, and P"". As expected because ofrelaxation at low temperature, the ground-stateprobability P!! increases exponentially, at the

expense of the three other probabilities (Fig. 1E).Given the injection of random spins in each of thedots [the actual probability for injecting spin-down is typically only 25 to 30%, dependingon where the electrons are injected (24)], the

probabilities should not display any correla-tions. Indeed, the products of the single-dotprobabilities—for example, PL

" ! P"! " P""—overlap with the corresponding two-spin proba-bilities (circles versus crosses in Fig. 1E), asexpected for uncorrelated spins.

Correlations between the states of the twospins are induced when after injection of randomspins in (1,1), we pulse into the (2,0) charge re-gion for a variable time (Fig. 2A), during whichrelaxation to the (2,0) spin-singlet ground statewill take place. When we subsequently separatethe two electrons by pulsing into the (1,1) chargeregion and measure both spins, we see that cor-relations build up in the measurement outcomesconsistent with singlet preparation [random localnuclear fields may dephase the singlet, but theantiparallel correlations survive (25)]: Whenevermeasurement of the left dot gives a spin-up out-come, measurement of the right dot most likelygives spin-down, and vice versa (Fig. 2B).

The (anti-)correlations are further elucidatedby comparing the four two-spin probabilities (cir-cles in Fig. 2B) with the product of the respectivesingle-spin probabilities (crosses). For each spinby itself, the spin-down probability is ideally 1/2.However, the joint probability for "" is not 1/2 !1/2 = 1/4, but instead 0. This gap between circlesand crosses develops in Fig. 2B as the singletprobability increases. The deviation from the idealvalues is quantitatively understood on the basisof the estimated read-out fidelities (gray lines, seebelow and SOM text). This demonstrates that thejoint single-shot read-out allows us to directlyprobe correlations between two spins.

We next examine whether the read-out ofboth spins is truly independent, in the sense thatthe measurement outcome of one spin is not in-fluenced by the measurement and/or the state ofthe other spin. First, with proper alignment of therespective chemical potentials (Fig. 1B) and suf-ficiently small QPC bias (typically 400 meV) (26),one electron stays in its dot while the other isbeing measured. A more subtle possible cross-talk effect is that the second dot will not be linedup in the proper read-out configuration if the firstdot is empty (due to the cross-capacitance). Wetherefore discard those traces (here, <5%) wherethe right dot was emptied but not refilled duringthe first read-out stage (see SOM text and fig. S2).

The most relevant remaining possible originof cross-talk arises from the tunnel coupling be-tween the dots, which results in exchange cou-pling of the two spins. To address this issue, weinitialize the left spin deterministically in ! bywaiting sufficiently long in the (1,0) charge re-gion for the spin to relax. Subsequently, we injectan electron with random spin into the right dot, atgate settings for which (1,0) is lower in energythan (0,1) so the left electron stays in its dot whilethe right electron tunnels in. We observe no de-cay for the left dot (Fig. 3B), whereas the rightdot shows the usual exponential decay. From theamplitude of this exponential decay compared tothe standard deviation of the red data points in

Fig. 3. (A) Schematic of the energy levels close tothe (1,1)-(2,0) boundary, along the green arrow inFig. 2A. Due to the Pauli exclusion principle, theground state in the (2,0) charge region is a spinsinglet. The (1,1) and (2,0) charge states with thesame spin hybridize due to the inter-dot tunnelcoupling. S(1,1) and S(2,0) denote the spin sin-glets in the (1,1) and (2,0) charge configuration.T+, T0, and T– are the three (1,1) triplets withmagnetic quantum number = +1, 0, and –1. T+and T– are split off due to Bext. (B) Single-spinprobabilities to find! as a function of wait time atthe position of the green star (see also Fig. 2A)when initializing the left spin deterministically in" and the right spin in a random spin state. (C)After initialization as in (B), the double dot ispulsed for 25 ns (circles) or 10 ms (diamonds) to aposition close to the (1,1)-(2,0) charge transition.Single-spin probabilities to find ! as a function ofthis position are shown. Gray lines are a guide forthe eye.

Fig. 4. (A to D) Two-qubit exchange gate on a full set of input states. The four panels correspond to fourdifferent mixtures of initial states, as indicated, taken with otherwise identical settings. Again, spin-downinjection probabilities are below 50%. Gray lines are fits to damped oscillations, including a correction forpulse imperfections. We first fit P"! in (A) and P!" in (B) and allow only the amplitude and offset of theoscillations to change for the other probabilities in the respective panel. In (C) and (D), we use the fitparameters of (A) and allow only amplitude and offset to change. The oscillations in (A) and (B) run out ofphase with each other for longer wait times. We attribute this to subtle distortions of the pulses arriving at thesample due to the bias tees (22). (E and F) Visualized theoretical and experimental truth tables for a p rotationand a 2p rotation of the exchange oscillation (details and actual numbers are given in the SOM and fig. S5).

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