No-go theorem and optimization of dynamical decoupling against noise with soft cutoff

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by  Zhen-Yu Wang

No-gotheoremandoptimizationofdynamicaldecouplingagainstnoisewithsoftcutoff
Zhen-Yu Wang and Ren-Bao Liu∗
Department of Physics and Center for Quantum Coherence,
The Chinese University of Hong Kong, Shatin, N. T., Hong Kong, China
Westudytheperformanceofdynamicaldecouplinginsuppressingdecoherencecausedbysoft-cutoffGaus-
sian noise, using short-time expansion of the noise correlations and numerical optimization. For the noise
correlationswithoddpowertermsintheshort-timeexpansion,thereexistsnodynamicaldecouplingschemeto
eliminatethedecoherencetoarbitraryordersoftheshorttime,regardlessofthetimingorpulseshapingofthe
controlunderthepopulationconservingcondition. Weformulatetheequationsforoptimizingpulsesequences
fordynamicaldecouplingthatminimizesdecoherenceuptothehighestpossibleorderoftheshort-timeexpan-
sion. Inparticular,weshowthattheCarr-Purcell-Meiboom-Gillsequenceisoptimalinshort-timelimitforthe
noisecorrelationswithalinearorderterminthetimeexpansion.
2 PACSnumbers:03.67.Pp,03.65.Yz,03.67.Lx,82.56.Jn
1
0
2 I. INTRODUCTION periments, finding efficient DD schemes with fewer control
pulses is desirable. A remarkable advance is the Uhrig DD
t
c (UDD)[26–29]. UDDisoptimalintheshort-timelimitinthe
O Quantum information processing [1] relies on the coher- sensethatitsuppressesthepuredephasingofaqubitcoupled
enceofquantumsystems. Unavoidableinteractionsbetween toafinitebathtothe Mthorderusingonly Mqubitflips. The
4
aquantumsystemanditsenvironment(bath)introducenoise performance bounds for UDD against pure dephasing were

] on the system and lead to error evolutions (decoherence) of established [30]. Shaped pulses [31, 32] of finite amplitude
h the quantum system. Various methods have been proposed canbeincorporatedintoUDD[31]. Manyrecentexperimen-
p to combat the decoherence, including decoherence-free sub- tal studies [20, 23, 33–37] demonstrated the performance of
-
t spaces[2–4],error-correctioncodes[5,6],anddynamicalde- UDD.
n coupling(DD)[7–9]. Inparticular,theDDschemeusesrapid Itisimportanttofindefficientschemestosuppressgeneral
a
unitarycontrolpulsesactingonlyonthesystemstosuppress
u decoherence(includingpuredephasingandpopulationrelax-
theeffectsofthenoisefromtheenvironments.DDhasthead-
q ation). Yang and Liu extended UDD to the suppression of
[ vantages of suppressing decoherence without measurement, populationrelaxation[29].Thisinspiredefficientwaystosup-
feedback,orredundantencoding[8]. DDoriginatedfromthe pressthegeneraldecoherenceofsinglequbits,includingcon-
1 seminal spin echo experiment [10], in which the effect of a
v catenationofUDDsequences(CUDD)[38]andamuchmore
8 static random magnetic field (inhomogeneous broadening) is efficientonecalledquadraticDD(QDD)[25,39–42]discov-
canceled.AndmorecomplexDDpulsesequences,suchasthe
3 eredbyWestetal[39]. Basedontheproofin[29],Mukhtar
3 Carr-Purcell-Meiboom-Gill(CPMG)sequence[11,12],were et al generalized UDD to protect arbitrary multilevel sys-
1 designedtoprolongthespincoherencetime[13].
temswithfullpriorknowledgeoftheinitialstates[43]. One
.
0 The early DD schemes only eliminate low-order errors, can actually preserve the coherence of arbitrary multi-qubit
1 i.e., the errors of quantum evolutions up to some low order systems by protecting a mutually orthogonal operation set
2 in the Magnus expansion. By unitary symmetrization pro- (MOOS)[25]. BynestingUDDsequencesforprotectingthe
1
cedure [7, 8], DD cancels the first order (i.e., leading order) elementsintheMOOS,thenestedUDD(NUDD)[25,42,43]
:
v errors. To eliminate errors to the second order in short time, requires only a polynomially increasing number of pulses in
Xi mirror-symmetric arrangement of two DD sequences can be thedecouplingorder. TheseuniversalDDschemesalsowork
used [8]. The first explicit arbitrary Mth order DD scheme, foranalyticallytime-dependentbaths[44].
r
a which suppresses errors to O(TM+1) for short evolution time Theabove-mentionedvariationsofUDD,however,relyon
T,istheconcatenatedDD(CDD)[14,15]proposedbyKhod- the finiteness of the baths, i.e., the existence of hard high-
jastehandLidar.CDDsequencesagainstpuredephasingwere frequency cutoff in the noise spectra. Legitimate questions
investigatedforelectronspinqubitsinrealisticsolid-statesys- are:Forquantumsystemscoupledtoaninfinitequantumbath
tems with nuclear spins as baths [16–18]. Experiments [19– or affected by soft-cutoff noise, can any DD be designed to
23] have tested the performance of CDD. CDD works for eliminate the decoherence to arbitrary orders of precision in
generic quantum systems coupled to a finite bath [24, 25]. the short-time limit? And if yes, how can such DD be de-
However, since CDD uses recursively constructed pulse se- signed? Such questions have been previously addressed in
quences to suppress decoherence, the number of pulses in- somespecificnoisemodels. Comparingtheefficiencyofvar-
creases exponentially with the decoupling order. As pulse ious DD sequences in suppressing pure dephasing of a qubit
errors are inevitably introduced in each control pulse in ex- duetoclassicalnoise,Cywin´skietalobservedthatifthenoise
spectrumcutoffisnotreached,CPMGsequences[11,12]ac-
tually performs better than CDD and UDD sequences [45].
Also, Pasini and Uhrig derived the equations for minimizing
∗Electronicaddress:[e-mail] decoherenceforpower-lawspectra,andfoundthatthenumer-
2
ically optimized sequences resemble CPMG [46]. Chen and destroys the quantum coherence. We can suppress the deco-
LiuprovedthatfortelegraphlikenoisetheCPMGsequences herencebyDDcontrolonthequbit. Thereisonlyonenoise
are the most efficient scheme in protecting the qubit coher- sourceβ(t)inthemodelEq.(1),andtosuppressthedecoher-
ence in the short-time limit and the decoherence can be sup- ence we need to protect a MOOS which consists of a Pauli
pressed at most to the third order of short evolution time by operatorσ (moregenerallyσ cosφ+σ sinφwithφbeing
x x y
DD[47]. Theseresultssuggestthatfornoisewithsoftcutoff real) [25]. We will prove later that DD can suppress the de-
inthespectrum,therearecertainconstraintsontheefficacyof coherence (i.e., the protection of the MOOS {σ }) only to a
x
DD. However, no conclusion has been drawn on the perfor- certainorderofshortevolutiontimefornoisecorrelationsthat
manceofDDwitharbitrarytimingandpulseshapingforthe haveodd-powerexpansiontermsintime. Weexpectthatthe
generalcasesofsoft-cutoffnoise. proofalsoappliestootherquantumsystems(e.g.,multi-qubit
In this paper, we address the general question of the per- systems) when the noise correlations have odd-power terms,
formanceofDDagainstsoft-cutoffnoise,basedontheshort- sinceinthosesystemstherearemorenoisesourcesandmore
timeexpansionofthenoisecorrelations. Theessenceoftime- system operators (e.g., a MOOS consisting of L > 1 Pauli
expansionmethodistoavoiddivergenceintheintegrationof operators)shouldbeprotected.
soft-cutoffnoiseinthefrequencydomain. Wefindthatwhen Whenweapplyasequenceofinstantaneousunitaryopera-
the expansion of the noise correlation has odd-power terms tionsσ atthemomentsT ,T ,...,T ,thecontrolledevolu-
x 1 2 N
intime,thereexistsnoDDschemetoeliminatethedecoher- tionoperatorreads
encetoanarbitraryorderoftheshorttime, regardlessofthe
timing and shaping of the control pulses under the popula- U(T) = (σx)NU(TN+1,TN)σxU(TN,TN−1)···
tionconservingcondition. Theodd-powertermsinthetime- ×σ U(T ,T )σ U(T ,T ), (2)
x 2 1 x 1 0
expansionindicatethatthenoisehasnohardhigh-frequency
cutoff. Sinceforsoft-cutoffnoisethedecoherencecannotbe whereT0 =0,TN+1 =T,andthefreeevolutionoperator
eliminatedatacertainorderofshorttime,wederiveasetof
eexqpuaantisoinosntowmhiilneimeliizmeitnhaetilnegadtihneg-loorwdeerrtoerrdmerisn.thTehsehsoerte-tqiumae- U(Tj+1,Tj)=e−iσ2z(cid:82)TTjj+1[ωa+β(t)]dt. (3)
tionsarenumericallysolvedforoptimalsolutions. Inparticu-
NotethatwhenNisodd,wemayapplyanadditionalσ pulse
lar,fornoisecorrelationswithalinearordertermintime,we x
attheendofthesequencefortheidentityevolution. Using
provethattheCPMGsequencesareoptimal. Forothernoise
correlationswithodd-orderterms,theminimumpulseinterval
oftheoptimizedsequencesislargerthaninUDDsequences. σxU(Tj+1,Tj)σx =e−iσ2z(cid:82)TTjj+1[−ωa−β(t)]dt, (4)
This feature is important in realistic experiments when there
isaminimumpulseswitchingtime[48]. wewritetheevolutionoperatoras
This paper is organized as follows: In Sec. II, we analyze
theperformanceofDDviatheshort-timeexpansionofcorre- U(T)=e−iσ2z(cid:82)0TωaFπ(t)dte−iσ2z(cid:82)0Tβ(t)Fπ(t)dt, (5)
lationfunctions,andwegivetheconditionunderwhichdeco-
herencesuppressiontoarbitraryorderisimpossible.Therela-
where we have defined the modulation function for instanta-
tionbetweenthehigh-frequencycutoffandtheshort-timeex-
neousπ-pulsesequences[26,45]
pansionofcorrelationsisalsodiscussed.InSec.III,wederive
the equations for sequence optimization and obtain optimal 
DD for noise correlations with odd-power expansion terms. F (t)=(−1)j fort∈(Tj,Tj+1] . (6)
Finally,theconclusionsaredrawninSec.IV. π 0 fort>T, ort≤0
Under DD control, the qubit is flipped at different moments,
II. NO-GOTHEOREMONDYNAMICALDECOUPLING and the random field β(t) is modulated by the modulation
AGAINSTNOISEWITHSOFTCUTOFF function F (t). Formultilevelsystems, themodulationfunc-
π
tions resulting from instantaneous π-pulse sequences may
We consider the pure-dephasing Hamiltonian for a single havevaluesnotrestrictedto{±1}fort∈(0,T]. InRef.[49],it
spin(qubit) isshownthatforDDcomposedofspeciallyengineeredfinite-
duration pulses, the effective modulation functions can take
1 valuesfrom{+1,−1,0}alternatively. Wemayalsoencounter
H = σ [ω +β(t)], (1)
2 z a effectivemodulationfunctionswhicharetrianglewavefunc-
tions during the time of system evolution [50]. For a more
whereσ =|+(cid:105)(cid:104)+|−|−(cid:105)(cid:104)−|isthePaulioperatorofthequbit,ω
z a general analysis, we assume that the control conserves the
istheenergysplittingofthequbit,andβ(t)describesrandom
populations and the phase modulation function F (t) has a
noisewithaverageβ(t) = 0. Heretheoverbardenotesaver- π
generalformas
agingoverthenoiserealizations.Weassumethatthestatistics
ofthenoisefluctuationsareGaussian. 
AfteradurationoffreeevolutiontimeT,thenoiseinduces F(t)=aboundedfunction fort∈(0,T] . (7)
(cid:82)T 0 fort>T ort≤0
between the two states |±(cid:105) a random phase shift β(t) that
0
3
AtthemomentT, theoff-diagonaldensitymatrixelement This kind of noise can be caused by Markovian (or instanta-
ofanensembleis neous) collisions in the bath [53]. We can see from Eq. (13)
thatwhenthenoisespectrumhashardhigh-frequencycutoff,
ρ↑↓(T)=ρ↑↓(0)e−i(cid:82)0TωaF(t)dte−i(cid:82)0Tβ(t)F(t)dt, (8) the noise correlation function β(t)β(0) is analytic as a func-
tionoftimeandtheexpansioncontainsonlyeven-orderterms,
where F(t) = Fπ(t) for ideal instantaneous σx pulses on β(t)β(0) =(cid:80)∞ C t2k.
the qubit. The coherence is characterized by the ensemble- cutoff k=0 2k
Expanding the noise correlation in time has the advantage
averagedphasefactor
of avoiding the divergence of integration over frequency for
noisewithasofthigh-frequencycutoffinthepowerspectrum.
W(T)≡e−i(cid:82)0Tβ(t)F(t)dt. (9) Thereforewecanderivecertaingeneralresultsontheefficacy
of general dynamical decoupling control against soft-cutoff
ForGaussiannoise,theensemble-averagedphasefactorW(T) noise. Also,theperformanceofDDintheshort-timelimitis
isdeterminedbythetwo-pointcorrelationfunctionβ(t1)β(t2) directlyderivedfromthetime-domainexpansion.
andW(T)becomes[45,51,52] UsingtheexpansionEq.(14),wewrite
W(T)=e−χ(T), (10) (cid:88)∞
χ(T)= C φ , (15)
k k
wherethephasecorrelation k=0
1(cid:90) T (cid:90) T wherethedecoherencefunctions
χ(T)= dt dt β(t )β(t )F(t )F(t ), (11)
2 0 1 0 2 1 2 1 2 φ ≡(cid:90) Tdt (cid:90) t1dt (t −t )kF(t )F(t ). (16)
k 1 2 1 2 1 2
can be written as the overlap between the noise power spec- 0 0
trumandafilterfunctiondeterminedbytheFouriertransform
ThereforeifthemodulationfunctionF(t)isdesignedtomake
ofthemodulationfunction[45]. TheaimofDDdesignisto
minimizeχ(T). {φ =0}M ≡{φ =0,··· ,φ =0}, (17)
k k=0 0 M
We assume the noise is stationary, i.e., of time translation
symmetry, β(t )β(t ) = β(t −t )β(0). The noise correlation the decoherence in the short-time limit is eliminated to the
functionhasth1esy2mmetry1β(t)β2(0) = β(0)β(t),sinceβ(t)and Mthorder, i.e., χ(T) = O(TM+3). Notethate−i(cid:82)0TωaF(t)dt = 1
inEq.(8)whenφ =0.Theeven-orderdecoherencefunctions
β(0)commute[Forquantumnoise,thismaynotbetrue.Butin 0
Eq.(11),t andt canbeexchangedwithoutchangingtheinte- φ2k describethedecoherenceduetothelow-frequencynoise,
1 2
gration. Sothenoisecorrelationcanalwaysbesymmetrized]. while the odd-order functions φ2k+1 arise from the soft high-
frequencycutoffinthenoisespectrum.
Thesesymmetriesindicatethatthenoisecorrelationisaneven
functionoftime,i.e.,
β(t)β(0)≡C (t)=C (|t|). (12) A. Analysisofthedecoherencefunctions
corr corr
The noise correlation can be transformed from the noise
Belowweshowthatwhenthenoisehassofthigh-frequency
powerspectrumS(ω)as cutoff,itisnotpossibletosuppressthedecoherencetoanar-
(cid:90) ∞ dω bitraryorder,nomatterhowtheDDmodulationfunctionF(t)
β(t)β(0)= S(ω)e−iωt. (13) isdesigned.
2π
−∞ The general filter function is defined as the Fourier trans-
Toanalyzethedecoherencesuppressioninshort-timelimit, formofthegeneralmodulationfunction,
weexpandthenoisecorrelationfunctioninpowerseries (cid:90) ∞
F˜(ω)≡ F(t)eiωtdt= F˜∗(−ω). (18)
(cid:88)∞ −∞
β(t)β(0)≡ C |t|k. (14)
k
Weexpanditinpowerseries
k=0
The existence of odd-order terms means that the correla- (cid:88)∞ (iω)m−1
F˜(ω)= Λ , (19a)
tionfunctionisnon-analytical,whichindicatesthatthenoise m! m
sourcecannotbeafinitequantumbath. Thenoisewithnon- m=1
(cid:90) T
analytical correlation functions must come from the fluctua- Λ ≡m F(t)tm−1dt. (19b)
tions of an infinite bath, since otherwise the unitary evolu- m
0
tion of a finite quantum system will always lead to analyti-
NotethatΛ = O(Tm). Forasequenceof N instantaneousπ
cal correlation functions. For example, the noise correlation m
β(t)β(0) = e−|t|/tc has odd-order terms in the time expansion, pulses,F(t)isgivenbyEq.(6)andΛmreads
and the noise has a Lorentz spectrum, which has a power- (cid:90) T (cid:88)N
lawdecreaseathighfrequencies. Wecallthispower-lawde- Λ(π) =m F (t)tm−1dt= (−1)j(Tm −Tm). (20)
creaseasofthigh-frequencycutoffinthenoisespectrum[46]. m 0 π j=0 j+1 j
4
AsF(t)=0fort(cid:60)(0,T],weextendtheboundsofintegra- whichisdecomposedas(withthechangesofsummationorder
tionforttoinfinityandtransformEq.(16)to andindices)
φk = ∂(∂iηk)k (cid:90)−∞∞ d2ωπ1 (cid:90)−∞∞ d2ωπ2 (cid:90)−∞∞dt1(cid:90)0t1dt2 φ2m = (2m)!(cid:90)−∞∞ d2ωπ 2(cid:88)rm=+112m(cid:88)n−=r1+22(iω−)(2−m1−)r+r3−nΛr!r Λn!n
whereweset×ηF˜→(ω01)aF˜f(teωr2d)eiff−ieωr1et1net−iaiωt2iot2ne.iη(Itn1−tet2g),rationsove(r2t12), +2(cid:88)m+12m(cid:88)−r+22(iω−)(2−m1−)rn+3−nΛr!r Λn!n
t andω give r=1 n=1
1 1
∂k (cid:90) ∞ dω F˜(ω) −(2m)!2(cid:88)m+1(−1)2m−rΛr Λ2m−r+2 . (29)
φ = [F˜(η)−F˜(−ω)]. (22) 2 r! (2m−r+2)!
k ∂(iη)k 2π i(ω+η) r=1
−∞
Theintegralsofthetermswithodd-powerofωvanish. And
Using
for even functions of ω, the sum n+r is an odd number, so
∂(∂iηr)rF˜(η)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (r+r!1)!Λr+1, forr≥0, (23) (th−e1c)rh+an(−ge1)onf=su0m.mThautisonthienidnetxegorfalthineElaqs.t(l2in9e)ivnanEiqsh.e(2s.9)W,withe
η=0
write
andchangingthesummationindex,weobtain
φk =k!(cid:90)−∞∞ d2ωπ (|−F˜i(ωω))k|+21 −(cid:88)kr=+11 (−iFω˜()ωk−)r+2Λr!r. (24) φ2m =(2m)!(−21)m (cid:32)(mΛ+m+11)!(cid:33)2−(cid:88)r=m1(−1)rΛr!r (2mΛ−2mr−r++2(23)0!).
FromEq.(30),wefindthatthefollowingtwosetsofequations
Note that the summation over r and the integration over fre-
areequivalent
quency cannot be exchanged when the integration does not
convergeforeachindividualterm. UsingEq.(19a),wehave {φ2m =0}mM=−01 ⇔{Λm =0}mM=1. (31)

φk = k!(cid:90)−∞∞ d2ωπ (|−F˜i(ωω))k|+21 −(cid:88)kr=+11k(cid:88)−n=r+12(iω(−)k1−)rk+−3r−nΛr!r Λn!n ofFthoereinqsutaatniotannseeotu{sΛπm-p=ulΛse(mπ)se=qu0e}mnM=c1esi,sthe optimal solution
− (cid:88)k+1 (cid:88)∞ (iω(−)k1−)rk+−3r−nΛr!r Λn!n. (25) TUjDD =Tsin2(cid:20)2Nπ+j 2(cid:21), (j=1,2,...,N), (32)
r=1n=k−r+3 which is the timing of UDD sequences [26]. Note that
WesimplifythelastlinebyusingEq.(19b)andtheequality the decoherence functions φ describe the noise with high-
2m
frequency cutoff in spectrum. Our result is consistent with
(cid:90) ∞ dω(cid:88)∞ (±iω)n−rtn = 1 tr−1 , forr≥1,t≥0, (26) Refs.[28,29]. TheconditionsEq.(31)aremoregeneralthan
2π n! 2(r−1)! the one that leads to UDD, and Eq. (31) may lead to more
−∞ n=r
generaldesignsofoptimalDD.
whichisprovedinAppendixA.Weobtain
 
φk =k!(cid:90) ∞ d2ωπ (|−F˜i(ωω))k|+21 −(cid:88)k+1k(cid:88)−r+2(iω(−)k1−)rk+−3r−nΛr!r Λn!n 2. Leadingodd-orderdecoherencefunctionφ2M−1
−∞ r=1 n=1
−k!(cid:88)k+1(−1)k−rΛr Λk−r+2 . (27) relFatoironthefunnocitsieonwβit(ht)sβo(f0t)hiisghn-oftreaqnuaelnyctiyc.cuIttosffe,xtphaennsoioisneccoonr--
2 r=1 r! (k−r+2)! tains odd-order terms C2m+1φ2m+1 (cid:44) 0. Suppose the DD
scheme has eliminated the leading order errors by making
{φ = 0}M−1, i.e., Eq. (31). We want to study the leading
2m m=0
1. Even-orderdecoherencefunctionsφ2m odd-order decoherence functions φ2M−1 due to the soft high-
frequencycutoffinthenoisespectrum.
Wefirstconsidertheeven-orderdecoherencefunctionsφ , The odd-order decoherence function φ of Eq. (27)
2m 2M−1
whichiscausedbythenoisewithhardhigh-frequencycutoff. reads
Fgroarlevvaennisnhuems.bEeqr.2(m27,)(−g|Fi˜iω(vω)e2)m|s2+1 isanoddfunctionanditsinte- φ = (2M−1)!(cid:88)2M(−1)rΛr Λ2M+1−r
2M−1 2 r! (2M+1−r)!
  r=1
φ2m =(2m)!(cid:90)−∞∞ d2ωπ 2(cid:88)rm=+112m(cid:88)n−=r1+2(iω−)2(m−−1r)+r3−nΛr!r Λn!n +(2M2−π 1)!(cid:90)−∞∞dω(cid:34)(|−F˜i(ωω))2|M2

−(2m2)!2(cid:88)m+1(−1)2m−rΛr!r (2mΛ−2mr−r++22)!, (28) +(cid:88)2M 2M(cid:88)−r+1(iω)(2−M1−)rr+2−nΛr!r Λn!n. (33)
r=1 r=1 n=1
5
TheconditionEq.(31)gives C = 0 andC = −1. We assume that the noise correlation
1 3
decreasessmoothlyatlongcorrelationtimes,thatis,
(cid:90) ∞ dω(cid:34) |F˜(ω)|2
φ2M−1 =(2M−1)! −∞ 2π (−iω)2M (34) lim dk C (t)=0, fork=0,1,..., (39)
+ (cid:88)2M 2M(cid:88)+1−r (iω)(2−M1−)rr+2−nΛr!r Λn!n. and t→∞dtk corr
r=M+1 n=M+1 (cid:90) ∞ dL
Noticeinthesummation2M+1−r< M+1. Weobtain IL(ω)≡ eiωtdtLCcorr(t)dt, (40)
0
φ =(−1)M(2M−1)!(cid:90) ∞ dω|F˜(ω)|2. (35) vanishes at large ω for L = 0,1,... [54]. The noise correla-
2M−1 −∞ 2π ω2M tionsthatdecayinthecorrelationtimesmoothlywithoutfast
oscillationsatisfyEq.(40).Forexample,thenoisecorrelation
InEq.(35),theintegrand|F˜(ω)|2/ω2M ≥0anditcannotvan-
e−|t|hasI (ω)=i(−1)L/(ω+i)→0whenω→∞.
ish for all ω from −∞ to ∞. Thus we have the following L
We consider the high frequency behavior of the Fourier
theorem.
transformofC (t),
corr
Theorem1. Forthenoisecorrelationβ(t)β(0) = (cid:80)∞k=0Ck|t|k (cid:90) ∞
with{C2m (cid:44)0}mM=−01andC2M−1 (cid:44)0,thereisnoDDschemethat S2K(ω)= Ccorr(t)eiωtdt. (41)
caninduceafilterfunction F(t)toeliminatetheerrorsupto −∞
O(T2M+1)forshortevolutiontimeT. Thatis,thereisnoDD IntegrationbypartsL≥(2K+1)timesgives
schemesatisfying{φ =0}M−1andφ =0.
2m m=0 2M−1  
tioTnhFis(tt)h,eaonrdemithsohlodwssfothraatrbointrearcyannnoont-zseurpopmreossduthlaetidoencfouhnecr-- S2K(ω)=2(cid:60)(cid:88)r=L1 ((r−−iω1))r!Cr−1+ (I−Li(ωω))L, (42)
encetoarbitraryorderwhenthenoisecorrelationisnotana-
wherewehaveusedEqs.(37)and(39).
lytic. Forexample,forthenoisewiththecorrelationfunction
e−γ|t|,onecannotsimultaneouslyeliminatethetwoleadingde- UsingEqs.(38)and(40),weobtainforlargeω,
coherencetermsC0φ0 andC1φ1,andtheerrorinducedbythe (2K−1)! (cid:32) 1 (cid:33)
noiseisatleastoftheorderO(T3). S2K(ω)≈2 (iω)2K C2K−1+O ω2K+1 , (43)
TheresultinRef.[47]thatDDcansuppressdecoherenceat
mosttothethirdorderofshortevolutiontimefortelegraphlike whichisapower-lawdecreaseathighfrequencies.
noiseisgeneralfornoisewitharbitrarystatistics. Inderiving When the noise correlation expansion contains only even-
Theorem1,wehavemadetheassumptionthatthestatisticsof order terms, from Eq. (42) we have the noise spec-
thenoiseareGaussianandthedecoherenceisdeterminedby trum S (ω) = 2(cid:60)I (ω)/(−iω)L for an arbitrarily large
even L
thetwo-pointnoisecorrelation.Fornon-Gaussiannoise,there L. From the assumption lim I (ω) = 0, we have
ω→∞ L
existsthedecoherenceeffectfromhigher-ordernoisecorrela- lim S (ω)ωL = 0 for an arbitrarily large L and there-
ω→∞ even
tions. forethenoisespectrumhasahardhigh-frequencycutoff. One
example is the correlation function e−t2, which has the noise
spectrumofexponentialform∼ exp(−ω2),andobviouslythe
B. Noisecorrelationexpansionandhigh-frequencycutoff UDD sequence can suppress the noise4effect order by order.
Thelargeωbehaviorofothercorrelationfunctionsoftheform
Here we discuss further on the relation between the noise exp(−(cid:80)p α t2j)canbecalculatedbythesaddlepointinte-
correlationexpansionandhigh-frequencycutoff. Letuscon- j=1 2j
gration method, which gives a result of an exponential de-
siderthecorrelationfunctionsoftheexpansionform crease at high frequencies (i.e., hard cutoff). For example,
(cid:88)∞ when ω is very large, (cid:82)∞ e−t4eiωtdt (cid:39) 1(cid:61)(cid:113) 2π eg(ω), where
β(t)β(0)=C (t)= C |t|k, (36) −∞ 2 a(ω)
corr k=0 k g(ω)=3(ω4)43e−i2π/3anda(ω)=12(ω4)2/3ei2π/3.
Wenowconsiderthenoisespectrum
wheretheexpansioncoefficients
1
C ≡ 1 dkCcorr(t)(cid:12)(cid:12)(cid:12)(cid:12) (37) SP(ω)≈ ωP, forω≥Ω, (44)
k k! dtk (cid:12)(cid:12)
t→0+ approximated by a power law (P > 0) decrease beyond the
frequencyΩ. Thenoisecorrelationfunctionis
arefiniterealnumberswith
C2k−1 =0, fork< K, (38a) β(t)β(0)=CΩ,cutoff(t)+CΩ,P(t), (45)
C2K−1 (cid:44)0. (38b) wherethecorrelationduetohigh-frequencynoise
The leading odd-order term in the short-time expansion of (cid:90) ∞ dω 1
Ccorr(t) is C2K−1|t|2K−1. An example is Ccorr(t) = e−|t|3 with CΩ,P(t)=2(cid:60) Ω 2π ωPe−iωt, (46)
6
for K ∈ {1,2,...}, which can be transformed to a more gen-
eral form of noise c by using some scaling of the
a+(ω/ωd)2K
amplitude and ω. For example, the measured ambient noise
for ions in a Penning trap has an approximate 1/ω4 spec-
trum at high frequencies and a flat dependence at low fre-
quencies[33],whichapproximatelycorrespondstothenoise
spectrumEq.(52)withK =2. Thecorrespondingcorrelation
functionofEq.(52)isobtainedbytheinversetransform
(cid:90) ∞ dω e−iωt
FIG.1: Thepathsfortheintegralof1/zP. β(t)β(0) = . (53)
2K 2π 1+ω2K
−∞
Usingtheresiduetheorem,wehave
andthenoisecorrelationatlowfrequencies
CΩ,cutoff(t)=2(cid:60)(cid:90) Ω d2ωπSP(ω)e−iωt, (47) β(t)β(0)2K = 2iK (cid:88)K−1exp[eixπp[(−2inei+2πK1(2)n(+21K)|t|−] 1)]. (54)
0 n=0 2K
causesadditionaldecoherence. TheeffectfromCΩ,cutoff(t)can Expandingtheright-handsideinpowersoft,weget
beeliminatedefficientlybyDDsequences, suchasUDDse-
quences,asthenoisespectrumofCΩ,cutoff(t)hasahardcutoff i (cid:88)∞ (−i|t|)m (cid:88)K−1
Ω. β(t)β(0)2K = 2K m! ei2πK(2n+1)(m−2K+1), (55)
AsCΩ,P(t)isanevenfunction,wejustcalculatetheintegral m=0 n=0
forthecaseoft>0. Wehave(fort>0)
andsummingthetermsgives
(cid:34)(cid:90) (cid:90) (cid:90) (cid:35)
(cid:60) 1
CΩ,P(t)= π cΩ+ ci+ c∞ zPeiztdz, β(t)β(0)2K = 2−Ki (cid:88)∞ (−mi|t!|)mei(m−e22iKK(+m1+)π1[)(π−/K1)−m1+1]. (56)
m=0
wherethepathscΩ,ci,andc∞areshowninFig.(1).
Since the maximum of 1/zP → 0 as |z| → ∞ in the upper InEq.(56),theleadingodd-ordertermofthetimeexpansion
(cid:82)
half-plane,thecontribution c∞ z1Peiztdz=0. Thecontribution is 2((2−K1−)K1)!|t|2K−1,aspredictedbyEq.(51). ByTheorem1,we
(cid:60)(cid:90) 1 (cid:60)(cid:90) ∞ havethefollowingcorollary.
eiztdz= i1−Py−Pe−ytdy (48)
π ci zP π Ω Cdeocroohlelarernyc1e.upDytonaamnicearlrodrecOo(uTp2lKin+g1)ciannthoenltyimseu-pepxrpeasnssitohne
vanishesforevenP. And ofχ(T)forthenoisespectrumwithhigh-frequencyasymptote
(cid:60)(cid:90) 1 eiztdz= (cid:60)(cid:90) π2 1 eiΩteiθ(iΩ)eiθdθ ∼ω−2K.
π zP π ΩPeiPθ For a more general noise spectrum with high-frequency
cΩ 0
= I(1)+I(2), (49) asymptote ω−α, one can not suppress the error higher than
cΩ cΩ the order of O(Tmα+1) with mα being the smallest even num-
where ber larger than α. This is because comparing with ω−mα, the
asymptoticnoisespectrum 1 withα<m decreasesslower
I(1) = (cid:60) (cid:88)∞ 1 Ω1−P(iΩt)r (cid:16)ir+1−P−1(cid:17), (50a) andtheoverlapbetweenS(1ω+)ωαand|F˜(ω)|2iαslarger.
cΩ π r!(r+1−p) The 1/f noise is a special case (the case of P = 1) to
Ic(2Ω) = 21(cid:60)r=(0Pi,Pr(cid:44)−tPP−−111)!(cid:88)r∞=0δr,P−1. (50b) fbπ1ue(cid:82)nΩcd∞ttiiscoconusszχs/ePzd=d1.z(TdT)ivh[eeErgqne.os(is1ae1t)tc]o→irnrde0lua;cteihodonwbefyuvnethrc,teitohhneigdChe-Ωcf,roPe=hq1eu(rtee)nncc=ye
noiseisfinite. WhenthereisnoDDcontrol,i.e.,forthecase
ForevenP,I(1)isanexpansionofevenpowersoft,and offreeinductiondecay,
cΩ
Ic(2Ω) = 2((−P1−)P1/2)!|t|P−1, forevenP, (51) χP=1(T)= 2π1Ω2 [1−cos(ΩT)+ΩTsin(ΩT)]
T2 (cid:90) ∞ cosz
+ dz, (57)
istheleadingoddorderexpansiontermofthenoisecorrela- 2π ΩT z
tionfunction.
Asanexample,weconsidertheflowingnoisespectrum, whichhastheshort-timeexpansion
1 1 (cid:16) (cid:17)
S2(cid:48)K(ω)≡ 1+ω2K, (52) χP=1(T)= 4π(−2ln(ΩT)+3−2γe)T2+O Ω2T4 , (58)
7
with the Euler constant γ ≈ 0.577. Notice the interesting
e 18
resultthatχP=1(T)doesnothavethesimplepowerlawscaling
atshorttimeT. Aftereliminatingtheeffectoflow-frequency 16
noisebyhigh-orderDDsequenceswhichsatisfy{Λ =0}M 14
m m=1
(Eq. (31)) w(cid:16)ith 2(cid:17)M > P−1, χP(T) has a power la(cid:16)w s(cid:17)caling 12
χP(T) = O TP+1 [46],andwehaveχP=1(T) = O T2 when 10
Λ =0.
1 8
6
ODD
III. SEQUENCEOPTIMIZATIONINSHORT-TIMELIMIT 4 UDD
CPMG
2
TheaimofoptimalDDistominimizeχ(T). Asindicated
0 0.2 0.4 0.6 0.8 1
in Eq. (35), a smaller F˜(ω) at low frequencies will give a
smallerφ . HerewefocusonDDwithidealinstantaneous
2M−1 FIG.2: (Coloronline). ComparisonoftheODD,UDDandCPMG
π pulses. We use more pulses to construct a more efficient sequences for different pulse number N. Squares (blue), triangles
modulation function F(t) = Fπ(t) to minimize φ2M−1, with (green),andcircles(red)correspondtoUDD,CPMG,andODD.The
the conditions {φ2m = 0}mM=−01 [Eq. (31)]. Using the method ODDsequencesareoptimizedtominimizeφ3 undertheconstraints
of Lagrange multipliers, we need to solve a set of nonlinear φ =φ =0.
0 2
equationsas
∇{y,T}GM =0, (59a) anN-pulseCPMGsequencereads
(cid:88)M
GM ≡ yjΛ(jπ)+φ2M−1, (59b) TCPMG = 2j−1T, for j=1,...,N. (62)
j=1 j 2N
∂ ∂ ∂ ∂
∇ ≡( ,..., , ,..., ). (59c) TheCPMGsequencesobviouslysatisfytheconstraintΛ(π) =
{y,T} ∂T ∂T ∂y ∂y 1
1 N 1 M 0 [see Eq. (20)], which is the so-called echo condition that
The introduced variables {y } are the Lagrange multipliers. eliminates the effect of static inhomogeneous broadening.
j
NotethatthesequenceoptimizationinRef.[46]alsousedthe Eqs.(61)and(20)give
constraints {Λ(mπ) = 0}, but the constraints were used there to ∂φ (cid:12)(cid:12) T2 (cid:104) (cid:105)
guarantee the convergence of the calculation of χ(T). Here, 1(cid:12)(cid:12) =(−1)k+1 1+(−1)N , (63a)
the constraints eliminate the lowest orders of errors ({φ2m = ∂Tk(cid:12)CPMG (cid:12) 4N2
d0}emMc=o−0h1)er[esneceeEfqro.m(31in)]hoinmoshgoenrte-otiumseblriomaidte.niInngpiasrteicliumlairn,attehde y1∂∂ΛT(1π)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =2(−1)k+1y1. (63b)
whenφ =0. k (cid:12)CPMG
0
WecalculateEq.(16)byseparatingthedomainofintegra- Thus the CPMG sequences also satisfy Eq. (59) with y =
tion according to the value of F (t )F (t ). For k ≥ 0, we (cid:104) (cid:105) 1
π 1 π 2 − T2 1+(−1)N ,sotheyareatleastthelocallyoptimalpulse
obtain 8N2
sequences. It has been proved that CPMG sequences are the

φk = (k+1−)(1k+2)4(cid:88)N (cid:88)j−1(Tj−Ti)k+2(−1)i+j menocsetaegffiaicniestnttepleuglsreapshe-qliukeencneosisienipnrtohteecsthinogrt-thtiemqeulbimiticto[h4e7r]-.
j=2 i=1 Withnumericalevidence,weconjecturethatitisalsotruethat
(cid:88)N theCPMGsequencesareoptimalintheshort-timelimitwhen
+(TN+1−T0)k+2(−1)N+1+2 (Tj−T0)k+2(−1)j the time expansion of the noise correlation function has the
j=1 twoleadingtermsC0andC1|t|.
+2(cid:88)N (TN+1−Ti)k+2(−1)N+1+i. (60) {ΛF(mπo)r=ot0h}emMr=1c,aosense coafnmsieneimthiaztinthgeφs2hMo−r1t-twimitehotphteimcioznedditDioDn
i=1 (ODD) coincides with UDD for pulse number N ≤ M. And
as N increases, the ODD sequences gradually approach the
Thenwehave
CPMG sequences. For example, ODD for the noise correla-

∂φ∂2TM−1 = (−M1)k 2 (cid:88)N (Tj−Tk)2M(−1)j−(Tk−T0)2M tion
k j=k+1 β(t)β(0)=C0+C2t2+C3|t|3+O(t4), (64)

+(TN+1−Tk)2M(−1)N+1−2(cid:88)k−1(Tk−Tj)2M(−1)j. (61) isshowninFig.2incomparisonwithUDDandCPMG.The
j=1 ODD sequences resemble the CPMG sequences when N is
large.
ForthespecialcaseofM =1,G =y Λ(π)+φ ,wefindthat WeshowinFig.3(a)theperformanceofthreeDDschemes
1 1 1 1
theCPMGsequencesaresolutionstoEq.(59). Thetimingof againstthenoisedescribedbyEq.(64). Acomparisonisalso
8
model,weexpectthattheresultoftheexistenceofthelargest
decoupling order in short-time limit can be extended to the
generaldecoherencemodel(includingbothdephasingandre-
laxation)ofquantumsystems. ItisdesirabletostudytheDD
in suppressing the general decoherence of quantum systems
inthesoft-cutoffnoiseinthefuture.
Acknowledgments
WethankYi-FanLuoandBoboWeifordiscussions. This
workwassupportedbytheHongKongGRFCUHK402209,
the CUHK Focused Investments Scheme, and the National
FIG.3:(Coloronline).Thedecoherencefunctionχ(T)asafunction
ofpulsenumberN (a)withouthardcutoffand(b)withahardhigh- NaturalScienceFoundationofChinaProjectNo. 11028510.
frequency cutoff ω = 40. Here the noise spectrum S(ω) = κ ,
c 1+ω4
with κ = 105 and T = 0.5. Squares (blue), triangles (green), and
circles(red)correspondtoUDD,CPMG,andODD. TheODDse- AppendixA:ProofofEquation(26)
quencesarethesameasthoseshowninFig.2.
ToproveEq.(26),wejustneedtoprove
showninFig.3(b)byconsideringahardhigh-frequencycut- (cid:90) R (cid:88)∞ (±ix)n−r π
offω =40.InFig.3(a),wecanseethatODDsequencesgive lim dx = , forr≥1, (A1)
bettercperformance than UDD and CPMG sequences. These R→∞ −R n=r n! (r−1)!
ODDsequencesareoptimalforawiderangeofnoisewhich
wheretheboundsintheintegralguaranteethatthemodulation
has the noise correlation given by Eq. (64). When we intro-
duceahigh-frequencycutoffinthenoisespectrum,asthecase functionF(t)isarealfunction. Using
inFig.3(b),theODDisthebestinitiallywhenthenumberof
pulses N (cid:46) ωcT/2 ≈ 10, and the UDD sequences become (cid:90) Rdx(cid:88)∞ (±ix)n−r =(cid:88)∞ Rn (in−1+c.c.), (A2a)
better and suppress the decoherence order by order when N n! (n+r−1)!n
islargeandthehardcutoffisreached. InFig.3(b), forlarge −R1 n=(cid:90)r R sinx n(cid:88)=∞1 Rn 1
N UDD is better than ODD, since the ODD sequences are dx= (in−1+c.c.), (A2b)
optimized for soft-cutoff noise rather than hard-cutoff noise. (r−1)! −R x n=1 n!n(r−1)!
InFig.3(a)thedecreasingofχ(T)isalineardecreaseinthe
double-logarithmplot,butinFig.3(b)itismuchfaster. This andlim (cid:82)R sinxdx=π,wejustneedtoprove
confirmsthatDDisnotsoefficientagainstsoft-cutoffnoise. R→∞ −R x
(cid:88)∞ (cid:34) Rn Rn (cid:35)
lim − (in−1+c.c.)=0. (A3)
R→∞ (n+r−1)!n n!n(r−1)!
IV. SUMMARYANDCONCLUSIONS n=1
For r = 1, it obviously holds. For r ≥ 2, we can show the
Wehavestudiedthedynamicaldecouplingcontrolofdeco- difference
herence caused by Gaussian noise with soft cutoff. We have
tpperaronmvse,iodDnTDohfecnoaorneismoenc1loywrrsheulipacpthirosenhssohwadssectthohehate(,2rweKnhc−een1t)tohtheOos(dhTdo2rKet-x+t1pi)ma.nesTieohxne- ∆≡(cid:88)n∞=1 R(nn(+in−r1−+1c).!cn.)(r−1)!−(cid:89)rj−=11(n+ j)=O(cid:32)R1(cid:33),
existenceofodd-ordertermsintheshort-timeexpansioncor- (A4)
respondstoasofthigh-frequencycutoffinthenoisespectrum solimR→∞∆=0. Byexpandingthetermsinthesquarebrack-
(i.e., a power-law asymptote at high frequencies). For these etsofEq.(A4),wehave
noise spectra, we have derived the equations for pulse se-
 
qdueceonhceeroenpctiemfiuznactitoionn,wanhdicehlimmiinniamteizseesvtehne-olerdaedrindgecoodhde-roerndceer ∆=(cid:88)∞ R(nn(i+n−1r+−c1.)c!.)(cid:88)r−2aknk, (A5)
functions of lower orders. The ODD sequences obtained by n=1 k=0
thismethodcoincidewiththeUDDsequenceswhenthepulse
wherea isanumberindependentofn. Wearrangetheterms
k
numberNissmall,andtheyresembleCPMGsequenceswhen
inthesquarebracketsandget
N islarge. Forthespecialcasethattheshort-timecorrelation
function expansion has a linear term in time, the ODD se-  
quencesareexactlytheCPMGsequences. ∆=(cid:88)∞ R(nn(i+n−1r+−c1.)c!.)(cid:88)r−2bk(cid:89)k [(n+r− j)]+b0, (A6)
Although we derived Theorem 1 from a pure dephasing n=1 k=1 j=1
9
with b independent of n. After some simplification it be- Hence∆=O(1),andEq.(26)isproved.
j R
comesforr≥2
(cid:88)r−1 ibkr+−1kR−1k eiR−(cid:88)k−1 Rn!nin+c.c.=O(cid:32)R1(cid:33). (A7)
k=1 n=0
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