Optimal quantum gates for semiconductor qubits

0.24 MB·English

by  Ulrich Hohenester

Optimal quantum gates for semiconductor qubits
Ulrich Hohenester∗
Institut fu¨r Physik, Karl–Franzens–Universit¨at Graz, Universit¨atsplatz 5, 8010 Graz, Austria
(Dated: Octotber11, 2006)
7 Weemployoptimalcontroltheorytodesignoptimizedquantumgatesforsolid-statequbitssubject
0 to decoherence. At the example of a gate-controlled semiconductor quantum dot molecule we
0 demonstrate that decoherence due to phonon couplings can be strongly suppressed. Our results
2 suggest a much broader class of quantumcontrol strategies in solids.

n
PACSnumbers: 73.21.La,03.67.Lx,71.38.-k,02.60.Pn
a
J

5 Decoherence is the process within which a quantum coherence time,14,15 which provides a serious bottleneck
1
systembecomesentangledwithitsenvironmentandloses for solid state based quantum computation. In addition,

its quantum properties. It is responsible for the emer- future miniaturization of nanostructures will result in a
]
l gence of classicality1 and constitutes the main obsta- further increase of such decoherence owing to the larger
l
a cle in the implementation of quantum computers.2,3 For number of phonon modes to which carriersin small dots
h atoms,environmentcouplingscanbestronglysuppressed can couple.16
-
s by working at ultrahigh vacuum and ultralow tempera-
Let us consider the setup depicted in fig. 1a where a
e ture. For artificial atoms —the solid-state analogies to
m single electron is confined in a double dot structure. Al-
atoms—, things are more cumbersome because they are
though true quantum algorithms will involve the manip-
. intimatelyincorporatedinthesurroundingsolid-stateen-
t ulationoftwoormoreelectrons,thecasestudyofasingle
a vironmentandsuffer from variousdecoherencechannels.
m electron completely suffices to understand the problems
Anumberofquantumcontroltechniquesareknown,such
inherent to charge control within the solid state. The
- as quantum bang-bang control,4 decoherence-free sub-
d spaces,5 or spin-echo pulses,6 that allow to fight deco- tunnel coupling v(t) and the energy detuning ε(t) be-
n tween the left and right dot can be controlled through
herence. However, it is not the system–environment in-
o voltage pulses applied to the top gates L, T, and R, and
c teractionitselfthatleadstodecoherence,buttheimprint
[ of the quantum state into the environmental degrees of

freedom: theenvironmentmeasuresthequantumsystem.
1
In this paper we show that optimal control theory7,8 al-
v
3 lowsto designcontrolstrategieswhere quantumsystems
2 can be controlled even in presence of such environment
3 couplingswithoutsufferingsignificantdecoherence. This
1 opens the possibility for a much broader class of quan-
0
tumcontrolthatmightrenderpossiblehigh-performance
7
quantum computation in solids.
0
/ Anattractivecandidateforasolid-statequbitisbased
t
a on semiconductor quantum dots, which allow controlled
m coupling of one or more electrons by means of voltage
pulses applied to electrostatic gates.9 The spin of elec-
-
d trons confined in such dots provides a viable quantum
n memory owingto its long life andcoherence times ofthe
o orderofmicrotomilliseconds.6,10 Inthe seminalworkof
c
Loss and DiVincenzo11 a mixed quantum computation FIG.1: (a)Schematicsketchofthedoubledotstructurecon-
:
v approachhadbeenenvisioned,wherethequantuminfor- sideredinourcalculations. Byapplyingvoltagepulsestothe
Xi mation is encoded in the spin degrees of freedom, thus topgatesL,T,andRthetunnelcouplingandenergydetun-
ing can be controlled. We consider dot radii of 60 nm, a dot
benefiting from the long spin coherence times, and the
r distance of 80 nm, and GaAs-based material parameters14.
a much strongercoupling to the chargedegreesof freedom
Thesurface plot atthebottom ofpanel(a) indicatesthelat-
is exploited for performing fast quantum gates. Recent
ticedistortation for theelectron in theleft dot. Upon apply-
experiments haveindeed demonstratedthe coherentma- ing atunnelcoupling v0 theelectron tunnelsfrom the left to
nipulation of charge states in coupled dots.6,12,13 Elec- therightdot. Panels(b)and(c),respectively,showsnapshots
tron charge, however, not only couples to the external wheretheelectronisdelocalizedoverthewholestructureand
controlgatesbutalsotothesolid-stateenvironment,e.g. localizedintherightdot. (d)Populationoftheleft(solidline)
to phonons, which introduces decoherence during gate and right (dashed line) dot state. The oscillation is damped
manipulations. Theoretical work has estimated for real- becauseofphonon-assisteddephasing. (e)Qualityfactorasa
istic quantum dot structures that typically ten to hun- functionofgatetimeforthreedifferenttemperaturesandfor
dred quantum gates can be performed within the charge constant pulses v0 (lines) and optimized pulses (symbols).
2
the hamiltonian describing the system is of the form 40
V) 30 A Vo
H0 =(cid:18)v(0t) vε((tt)) (cid:19) . (1) µe ( e 20 ↑ OCT
g
a
Suppose that the electron is initially in the left dot. Volt 10 ↓
Whenattimezeroaconstanttunnelcouplingv0isturned 0
0 20 40 60 80 100
on, the left-dot and right-dotstates become coupled and
1
the electron will start to tunnel back and forth between
B
the two dots, as experimentally demonstrated.12 How- n
o
edveegrr,eeesleocftfrroenesdoinm,sowlihdischa,lwfoarytsheinctoenraficntedweitlehcttrhoenlsattatticees pulati 0.5
o
underconsideration,resultsinaslightdeformationofthe P
lattice in the vicinity of the electron. Such interaction
0
is conveniently described within the independent Boson 0 20 40 60 80 100
model17,18 Time (ps)
V (t) / V ε(t) / V
T o o
sL 0
Hph =Xq gq(cid:16)b†q+b−q(cid:17) (cid:18) 0q sRq (cid:19) , (2) e (ns) 00..12 22.5 1
m 0.3 1.5 0
wthheerlaetqticise,thi.ee.wpahvoevneocntso,rgoqftthheecqouuapnltiinzgedcoenigsetnamntoodfesthoef Gate ti 0.4 C 01.5 D −1
0.5
bulk material (piezoelectric and deformation potential),
0 0.5 1 0 0.5 1
b†q the bosoniccreationoperatorfor phonons,andsiq the Time / (Gate time) Time / (Gate time)
usualformfactorfortheelectron-phononcouplinginthe
left or right dot.14,18 Through the different form factors FIG. 2: Results of optimal control calculations for a lattice
siq, the electroncouples differently to the phonons in the temperature of 50 mK. (a) Optimized voltage pulse v(t) and
left and right dot, respectively. The surface plot at the (b)timeevolutionofleft-dot(solidline)andright-dot(dashed
bottom of fig. 1a shows the lattice displacement17 line) population, for a gate time of 100 ps. (c) Density plot
of the optimized pulses v(t), in units of the constant pulse
u(r)= (2ρω )−12 eiqrhb i+c.c. (3) height v0, for different gate times. (d) Same as (c), but for
q q ε(t).
Xq
for the electron initially localized in the left dot as com-
puted within a density-matrix framework,19,20,21,22,23 quantumgate,withthegatefidelity2 beingdirectlygiven
with ρ the mass density of the semiconductor, ω = cq byQ. Figure1ereportsthedependenceofQongatetime
q
thephononenergy,andcthesoundvelocity. Conversely, and lattice temperature.23 Over a range of experimen-
when the electron is localized in the right dot, fig. 1c, tally accessible temperatures Q has a minimum for gate
the lattice becomes distorted around the right dot. In timesaround50ps,andissignificantlyenhancedatboth
a sense, this finding is reminiscent of molecular physics shorter and longer times. The appearance of this mini-
where electronic excitations are accompanied by varia- mum has been discussed in length in ref. 14 and can be
tions of the molecular structure, though in our case the qualitatively understood as follows. At short gate times
coupling is much weaker and to a continuum of phonon T the electron moves on a timescale much shorter than
modes rather than to a few vibronic states. theresponsetime ofthephononcloud,andconsequently
Aswewillshownext,thiscouplingtoaphononcontin- the electrontransportis notaffected bythe muchslower
uum hasa drastic influence onthe coherentchargeoscil- latticedynamics(dynamic decoupling): hereQincreases
lations. Uponturningonthe tunnelcouplingv between withdecreasingT. Ontheotherhand,atlonggatetimes
0
the two dots, the electron starts to oscillate and the lat- the phononcloudfollowsalmostadiabatically,andQ in-
tice distortion follows, as shown in figs. 1a–c. However, creaseswithincreasingT. Theminimumoccursatatime
sincethephononcloudcannotfollowinstantaneouslydue τ ∼ω−1 wherethephononwavevectorq ∼2π/dmatches
q
to the finite phonon frequencies ω , part of the quan- the interdot distance d, as will be discussed in more de-
q
tum coherenceis transferredfromthe electronsystemto tail below. This behavior of Q suggests that coherent
the phonons,resulting ina coherence lossofthe electron electron transport should be much faster or slower than
motion as evidenced by the damping of the oscillations τ, which imposes serious constraints on quantum gates.
showninpanel(d). Toquantifysuchloss,weintroducein In reality the situation is even more adverse. For fast
accordancetoref.14thequalityfactorQthatdetermines gating other quantum dot states might become excited,
how many charge oscillations can be resolved within the wheras for slow gating additional environment couplings
decoherencetime. Weshallalsofinditconvenienttorefer mightgainimportance. Alsoafurtherminiaturizationof
totheelectrontransportfromthelefttotherightdotasa the double-dot structure will lead to a further decrease
3
FIG. 3: Time evolution of left-dot (solid lines) and right-dot (dasehed lines) population, and of purity (dotted lines), for (a)
δ-like excitation at time zero, (b) constant voltage pulse v0, and (c) optimized control fields v(t) and ε(t). The lower panels
show real-space maps of the electron-phonon entanglement (for definition see text) at three selected times.
of Q. perform best for gate times where the constant v field
0
performs worst. Thus, optimal control theory allows to
Inref.20weshowedfortheopticalcontrolofquantum
design control strategies that can drastically outperform
dot exciatations that such phonon-assisted decoherence
more simple schemes.
can be strongly suppressed through laser-pulse shaping.
To understand the drastic improvement of optimized
Accordingly,wemightexpectthatforthetunnel-coupled
pulses, in fig. 3a we first analyze the more simplified sit-
double dot an optimization of the control fields v(t) and
uation of a δ-like voltage pulse where at time zero the
ε(t)couldimprovethequantum-gateperformance. Inthe
electron is brought instantaneously from the left dot to
following we thus employ the framework of optimal con-
troltheory7,8tosearchforcontrolfieldsv(t)andε(t)that a superposition state between the two dots, and the dot
coupling is turned off at later times. Consequently, the
maximize Q, i.e., we are seeking for voltage pulses that
left- and right-dot populations shown in fig. 3a remain
minimizedecoherencelossesduringgating. Optimalcon-
constant. We additionally plot the purity trρ2 of the
troltheory(OCT)accomplishesthe searchforoptimized
electron system, which, starting from the initial value of
controlfieldsbyconvertingtheconstrainedminimization
one, gradually decreases, thus indicating the transition
to anunconstrainedone,by meansofLagrangemultipli-
from an inital pure state to a final mixture. Similar to
ers, and formulating a numericalalgorithm which, start-
the entropy, the purity is a measure of the degree of en-
ing from an initial guess for the control fields, succeed-
tanglementbetweenthe electronicandphononic system,
ingly improves them. Details of our numerical approach
i.e.,howmuchinformationthephononsposessaboutthe
can be found in refs. 20,21,24. Figure 2 shows results of
quantum properties of the electron state. In analogy to
our OCT calculations. The optimized v(t) differs from
eq. (3) we define
the constant v one in that the strength is strongly re-
0
duced at the beginning and at the end of the quantum u (r)= (2ρω )−21 eiqrhhb σ ii+c.c. (4)
gate, and enhanced in the middle (indicated by arrows). 1 q q 1
Xq
At the same time, the energyoffsetε(t) [seepanel(d)] is
varied during the gate from positive to negative detun- as an entanglement measure, with hhb σ ii = hb σ i −
q 1 q 1
ing. Panels (c) and (d) report that these control strate- hb ihσ ithecorrelationbetweenphononmodeb andthe
q 1 q
gies prevail over a wide range of gate times. As appar- (real part) quantum coherence σ .23 When at time zero
1
ent from fig. 2b, the transfer process from the left to the electron superposition state is apruptly prepared, it
the right dot is not drastically altered by the optimized requires a time τ ∼ ω−1 for the phonons to aquire in-
q
fields v(t) and ε(t)in comparisonto that ofthe constant formation about the modified electron state. In particu-
field v (gray lines). On the other hand, the quality fac- lar phonon modes with wavevector 2π/d, where d is the
0
tor Q of the OCT gates [symbols in fig. 1(e)] becomes interdot distance, carry information about the superpo-
boosted by severalordersof magnitude when using opti- sition properties of the electron state, and thus set the
mized pulses. Even more striking is that the OCT fields timescale for the entanglement buildup. The snapshots
4
ofu (r)intheleftcolumnoffig.3reportsuchbuildupin requires smooth voltage variations on the timescale of
1
thevicintyofthedots. However,duetothephononiner- tens of picoseconds, rather than sub-picosecond pulses
tia the lattice distortion overshoots,instead of smoothly needed for quantum bang-bang control,4 where the sys-
approachingthe new equilibriumposition, anda phonon tem has to become dynamically decoupled from the en-
wavepacket is emitted from the dots (see arrow)18,25 vironment. Such short pulses, which are at the fron-
which imprints the quantum information about the su- tier of presentday technology, have been also proposed
perposition state into the environment and thus reduces for other quantum control applications,27 and might in-
its quantumproperties: the systemsufferesdecoherence. troduce additional decoherence channels due to voltage
Similar behaviour of overshooting and wavepacket emis- fluctuations28,29 orsampleheating. Whetherthiswillaf-
sion is observed for the quantum gate with constant v . fect the control performance will have to be determined
0
In contrast, the optimized quantum gate shown in the experimentally.
right column of fig. 3 strongly suppresses the emission In summary, we have employedoptimal quantum con-
of a phonon wavepacketby reducing the tunnel coupling troltheorytodesignquantumgatesforsolid-statequbits
betweenthedotsintheinitialstageoftheelectrontrans- interactingwiththeirenvironment. Foragate-controlled
fer, see arrowinfig.2a,and thus allowsthe lattice to re- semiconductor double quantum dot subject to phonon
act smoothly to the time varying electron configuration. couplings, we have shown that optimized gates can
Similarconclusionsholdforthefinalstageofthetransfer. strongly suppress phonon-assisted decoherence and can
As regarding the peak in the middle of the controlpulse boost the fidelity by several orders of magnitude. We
shown in fig. 2a, we find that its shape strongly depends attribute our finding to the fact that in the process of
onthedetailedpropertiesofthephononcouplingg ,and decoherence it takes some time for the system to be-
q
is strong in case of piezoelectric coupling and absent in come entangled with its environment. If during this
case of deformation-potential coupling. entanglement buildup the system is acted upon by an
Our optimal quantum control strategy differs appre- appropriately designed control, it becomes possible to
ciably from other control strategies. The inherent cou- channel back quantum coherence from the environment
pling of electrons to phonons excludes quantum state to the system. We therefore believe that our findings
manipulations in decoherence free subspaces5 or other are relevant for a much broader class of solid state sys-
quantum-optical controltechniques, such as, e.g., stimu- temswherequantuminformationisencodedinlong-lived
lated Raman adiabatic passage,26 where quantum state quasi-groundstates with small energy separations, such
transfer is achieved through states fully decoupled from as electon or nuclear spins, resulting in slow scattering
theenvironment. Furthermore,thetimedynamicsofthe processesthatcanbe manipulatedbymeansofquantum
phonon degrees of freedom disables spin-echo techniques control.
torestorepurequantumstatesbymeansofeffectivetime Work supported in part by the Austrian Science Fund
reversal through π pulses. Finally, optimal control only FWF under projet P18136–N13.
∗ Electronic address: [email protected] (1998).
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