Tackling Systematic Errors in Quantum Logic Gates with Composite Rotations

T ac kling Systematic Errors in Quan tum Logic Gates with Comp osite Rotations Holly K. Cummi ns, Ga vin Llew ellyn, and Jonathan A. Jones  Centr e for Quantum Computation, Clar endon L ab or atory, University of Oxfor d, Parks R o ad, O X1 3PU, Unite d Kingdom (Dated: Jan uary 29, 2003) W e describ e the use of comp osite rotations to com bat systematic errors in single qubit quan-tum logic gates and discuss three families of comp osite rotations whic h can b e used to correct o-resonance and pulse length errors. Although dev elop ed and describ ed within the con text of NMR quan tum computing these sequences should b e applicabl e t o a n y implemen tation of quan tum computation. P A CS n um b ers: 03.67.-a, 76.60.-k, 82.56.Jn I. INTR ODUCTION Quan tum computers [1] are information pro cessing de-vices that use quan tum mec hanical eects to implemen t algorithms whic h are not accessible to classical comput-ers, and th us to tac kle otherwise in tractable problems [2 ]. Quan tum computers are extremely vulnerable to the ef-fects of errors, and considerable eort has b een exp ended on alleviating the eects of random errors arising from decoherence pro cesses [3 , 4 , 5]. It is, ho w ev er, also im-p ortan t t o consider the eects of systematic errors, whic h arise from repro ducible imp erfections in the apparatus used to implem en t quan tum computations. The eects of systematic errors are clearly visible in n u-clear magnetic resonance (NMR) exp erimen ts [6 ] whic h ha v e b een used to implemen t small quan tum computers [7 ,8 ,9 ,1 0 , 1 1 ,1 2 ]. Implemen ting complex quan tum al-gorithms require a net w ork of man y quan tum logic gates, whic h for an NMR implemen tation translates in to ev en longer cascades of pulses. In these cases small systematic errors in the pulses (whic h can b e ignored in man y con-v en tional NMR exp erimen ts) accum ulate and ha v e sig-nican t eects. It mak es sense to consider systematic errors as some of them can b e tac kled relativ ely easily . I n the Blo c h pic-ture, where unitary op erations are visualized as rotations of the Blo c hv ector on the unit sphere, systematic errors are expressed as rotational imp erfections. The sensitiv-it y o f the nal state to these imp erfections can b e m uc h reduced b y replacing single rotations with comp osed ro-tations as discussed b elo w. II . SYSTEMA TIC ERR ORS IN NMR QUANTUM COMPUTERS An y implemen tatio n o f a quan tum computer requires quan tum bits (qubits) on whic h the quan tum informa-tion is stored, and quan tum logic gates whic h act on the qubits to pro cess the quan tum information. F ortunately  Electronic address: jonathan.jo ne s@qubi t.or g it is only necessary to implem en t a small set of quan tum logic gates, as more complex op erations can b e ac hiev ed b y joining these gates together to form logic circuits. A simple and con v enien t set comprises a range of single qubit gates together with one or more t w o qubit gates, whic h implem en t conditional ev olutions and th us logical op erations [13 ]. NMR quan tum computers are implemen ted [11 ] us-ing the t w o spin states of spin-1/2 atomic n uclei in a magnetic eld as the qubits. T ransitions b e t w een these states, and th us single qubit gates, are ac hiev ed b y the application of radio frequency (RF) pulses. Tw o qubit gates require some sort of spin{spin in teraction, whic h in NMR is pro vided b y the scalar spin{spin coupling ( J coupling) in teraction. While this do es not ha v e quite the form needed for standard t w o qubit gates, it can b e easily sculpted in to the desired form b y com bining free ev olu-tion under the bac kground Hamiltonian (whic h includes spin{spin coupling terms) with the application of single qubit gates [11 ]. As single qubit gates in v olv e the application of external elds they are vulnerable to systematic errors in these elds. In the ideal case, the application of a R F eld in resonance with the corresp onding transition with relativ phase  (in the rotating frame [6 ]) will driv e the Blo c h v ector through some angle ab out an axis orthogonal to the z -axis and at an angle  to the x -axis. The rotation angle,  , dep ends on the n utation rate induced b y the RF eld, usually written  1 , and the duration of the pulse,  . I n practice the RF eld is not ideal, and this leads to t w o imp ortan tt yp es of systematic errors, pulse length errors and o-resonance eects [6 ,1 4 ]. Pulse length errors o ccur when the duration of the RF pulse is set incorrectly , o r (equiv alen tly) when the RF eld strength deviates from its nominal v alue, so that the rotation angle ac hiev ed deviates from its theoretical v alue. Within NMR this eect is most commonly ob-serv ed as a result of spatial inhomogeneit y i n the applied RF eld, so that it is imp ossible for all the spins within a macroscopic sample to exp erience the same rotation an-gle. O-resonance eects arise when the RF eld is not quite in resonance with the relev an t transition, so that the rotation o ccurs around some tilted axis. Comp osite pulses [6 ,1 4 , 15] are widely used in NMR

tominimizethesensitivityofthesystemto theseerrors by replacing simple rotations with comp osite rotations whichareless susceptible tosuch eects. However, con-ventionalcomp ositepulsesequences arerarely appropri-ateforquantumcomputationb ecausetheyusually incor-p orate assumptions ab outthe initialstate of the spins. Suchstartingstatesarenotknownforpulsesinthe mid-dleofcomplexquantumcomputations,anditistherefore necessary touse fully-comp ensating(typ e A)comp osite pulse sequences [15], which work for any initial state. Comp ositepulses of this kind, which do notoer quite thesamedegree of comp ensationas isfound withmore conventionalsequences, areof little use inconventional NMR,andhavereceivedrelativelylittlestudy. Theyare, however,ideallysuitedto quantumcomputation.

I I I. OFF-RESONANCEERRORS

The problemof tackling o-resonance errors was ini-tially studied by Tycko [16]; his results were then ex-tendedbyCumminsandJones[17,18].Herewedescrib e two families of comp osite pulses which can b e used to comp ensateforo-resonance errors, andshowhowthey canb e derivedusingquaternions.

Theoriginalmetho dusedtodevelopmanytyp eA com-p ositepulsesequences [16,17,18]wasbasedondividing thepropagatordescribingthe evolutionofthe quantum system into intended and error comp onents, and then seekingtominimisetheerrorterm. Whilethisapproach is eective, it is cumb ersome, and a much simpler ap-proach can b e adopted forsinglequbitgates, which are simplyrotationsontheBlo chsphereandsocanb emo d-elledby quaternions. Thequaternions corresp ondingto individual pulses can b e multiplied together to give a quaterniondescriptionofthecomp ositepulse,whichcan then b e compared with thequaternion of theideal sys-tem.

Aquaternionisoftenthoughtofasavectorwithfour co eÆcients, but when describing a rotation it is more usefultoregrouptheseco eÆcientsasascalaranda three-vector,

q=fs;v g (1)

where

s=cos ( =2) (2)

dep ends solelyontherotationangle, ,and

v=sin( =2)a (3)

dep ends onb oththerotationangle, ,andaunitvector alongtherotationaxis,a. Thus thequaternion describ-inganon-resonancepulsewithphaseangle is

q  

=fcos( =2);sin( =2)fcos ();sin();0gg: (4)

Ano-resonance pulse isconvenientlyparameterised by its o-resonance fraction f = Æ=

1

(where Æ is the o-resonance frequency, and 

1

the nutation rate), and is describ ed bythequaternion

q   =fcos( 0 =2); sin( 0 =2) p 1+f 2 fcos ();sin();fgg (5) where 0 = p 1+f 2

,andisnowthenominalrotation angle, that is the rotation achieved when f = 0. The quaterniondescribingasequence ofpulsesisobtainedby multiplyingthequaternions foreach pulse according to therule q 1 q 2 =fs 1
s 2 v 1
v 2 ;s 1 v 2 +s 2 v 1 +v 1 ^v 2 g: (6)

Finally, two quaternions can b e compared using the quaterniondelity[15] F(q 1 ;q 2 )=jq 1
q 2 j=js 1
s 2 +v 1
v 2 j (7)

(it is necessary to take the absolute value, as the two quaternions fs;v g and f s; v g corresp ond to equiva-lentrotations,dieringintheirrotationanglebyinteger multiplesof2 ).

Followingourpreviouswork[17]weseektotackle o-resonanceerrors ina

x

pulse using asequence of three pulses appliedalong thex, xand xaxes; pulses with any other phase angle can then b e trivially derived by simplyadding the desired value to the phase angles of all the pulses in the sequence. Such sequences can b e describ ed completelyby the nominalrotation angles of thethreepulses,

1 ,

2 and

3

. Thecomp ositequaternion forthiscomp ositepulseiscomplicated,butthesituation canb egreatlysimpliedbyexpandingitasaMaclaurin seriesinf andneglectingalltermsab ovetherstp ower. Thisgives s=cos   1  2 + 3 2  (8) and v=  sin   1  2 + 3 2  ; sin  2 2 sin   1  3 2  ; f  2cos   2  3 2  sin  1 2 sin   1  2  3 2  ; (9)

whiletheidealquaternionhastheform

fcos( =2);fsin( =2);0;0gg: (10)

Itnowremainstochosethethreenominalrotationangles sothattheseequationsagree.

Firstwenotethatinordertoachievethecorrect rota-tionangle,s=cos ( =2),wemustcho oseouranglessuch that  + =+2a(whereaisanyinteger). We

also note that the y comp onentof v should equalzero, andthatthiscan b eachievedbycho osing

1 =

3 +2b (wherebis anyinteger). Thesetwochoices give

v=  sin  2 ; 0; f  sin  2 2sin   2  1  : (11)

Finally we cho ose  1

such that the z comp onent of v equalszero;thisgives

 1 =  2 arcsin  sin( =2) 2  : (12)

Combiningthisvaluewithourpreviousrelationsb etween theanglesgives

 1 =2n 1 +  2 arcsin  sin( =2) 2  (13)  2 =2n 2  2arcsin  sin( =2) 2  (14)  3 =2n 3 +  2 arcsin  sin( =2) 2  (15) where n 1 , n 2 and n 3

are integers, subject to the phys-ical restriction that the resulting pulse angles must b e p ositive.

These solutions have the same general form as those found previously [17]. Although they app ear to dier in detail the expressions are, in fact, identical: taking the values n 1 =1, n 2 =1 and n 3

=0 gives our previ-ousfamilyof solutions[17], referred to bythe acronym corpse (Comp ensationforO-Resonance with a Pulse SEquence). Thisfamilyisnowseen tob ejustone mem-b erofalargergroupoffamilies.Tocho oseb etweenthese itis necessary to lo okat higherorderterms, andthisis mostconvenientlyachievedusingthequaterniondelity, Equation 7. As abaseline we take thedelityof a sin-gle o-resonance 

pulse compared withits (ideal) on-resonanceform, F1+f 2  cos 1 4  (16) wheretermsinf 4

andhigherhaveb eenneglected. Note that the delity onlycontains even order termsinf as the comp ositepulse p erformsymmetricallyfor p ositive andnegativevaluesoff.

As exp ected all memb ersof ourgeneral group of so-lutions result inmuchb etter delities; inparticular the terminf

2

isalwayscompletelyremoved. Theb ehaviour of theterminf

4

ismuch morecomplicated,but itcan b e shown that this term dep ends only on the value of n=n 1 n 2 +n 3

,that isthetotalnumb erofadditional 2rotationsp erformedbythecomp ositepulsesequence, and has the smallest absolute value when the three in-tegers arechosen so that n=0. Asourprevious values (n 1 =1, n 2 =1 and n 3

= 0) are thesmallestnumb ers thattthiscriterion,itseemsthatthecorpsefamilyof pulsesequencesisindeedtheb estmemb erofthisgroup.

 1 2 3 30 Æ 367.6 345.1 7.6 45 Æ 371.5 337.9 11.5 90 Æ 384.3 318.6 24.3 180 Æ 420.0 300.0 60.0

TABLEI:Pulserotationanglesforacorpsecomp ositepulse withatargetrotationof

x

;corpsepulsephasesare+x, x, +x.

-1

-0.5

0

0.5

1

0

0.5

1

FIG.1:Fidelityofsimple(dashed line)andcorpsecomp os-itepulses(solidline)asafunctionoftheo-resonancefraction fforpulseswithatargetrotationangleof180

Æ .

The only other familyof interest is that with n 1 = 0, n 2 = 1 and n 3

= 0, previously referred to as short-corpse[18];while this p erformsless well thancorpse it is somewhat shorter. Numerical values of individual pulserotationangles forcorpse sequences with a vari-etyoftarget anglesaregiveninTableI.

The p erformanceof the corpse sequence for a 180 Æ

pulseisdemonstratedinFigure1;thecorpsepulsep er-formsb etter thanasimplepulse as longasjfj0:663. Forsmallervaluesoftheeectiverangeoff isreduced, butnotdramaticallyso: fora30

Æ

pulsethecorpsepulse outp erformsasimplepulse aslongas jfj0:297.

IV. PULSELENGTHERRORS

A similarapproachcan b e used to developcomp osite pulsesto tacklepulse lengtherrors. As b efore we b egin withasequence of three pulses, but thesubsequent de-velopmentis quite dierent. In particular we allowthe three pulses to have arbitrary phase angles, as well as arbitraryrotationangles,thusgivingussixvariable pa-rameters,althoughthisnumb erisso onreducedtothree. Thequaternion corresp ondingto each pulsetakesthe simpleform q   =fcos ( 0 =2);sin( 0 =2)fcos();sin();0gg: (17) where 0

= (1+g ) istheactual rotationangleachieved by apulse with nominal rotationangle  , and g is the fractionalerror inthepulse p ower. Thequaternion de-scribingthecomp ositepulseisverycomplicated,butcan b esimpliedbyrestrictingattentiontothetime symmet-riccase,where = and = . Thisautomatically

ensures that thecomp ositequaternionhasno z comp o-nent,asanytimesymmetricsequence ofrotationsab out axesinthexyplane isitselfarotationab outanaxisin thexyplane.

Even after this simplication, the comp osite quater-nion remains extremely complicated. To make further progresswenotethat acomp ositepulseofthiskindhas b eenpreviouslydescrib ed forthecaseofa180

Æ rotation: thesequence 180 60 180 300 180 60 (18) willp erforma180 x

rotationwithcomp ensationforpulse lengtherrors (see [19],but notethe corrected phase an-gles). It seems likely that other memb ers of this fam-ily will have either 

1

=  or  2

=  ; b oth p ossibili-tieswereinitiallyexplored,butthesecondchoiceseemed more pro ductive and formsthe basis of oursubsequent work.

Asb efore,thecomp ositequaternioncanb e expanded as aMaclaurin seriesing , and itis mostusefulto con-centrate on the rst order error term. This can b e set equaltozerobycho osing

 2 = 1 arccos (  =2 1 ) (19)

and, for consistency with equation 18, we will use the minus sign in future. Sequences ob eying this equation willb einsensitivetopulselengtherrors;therotationand phase angle can then b e adjusted by cho osing suitable values for

1 and

1

. Asb eforewe willderive values for a

x

pulse;pulseswithotherphaseanglescanb eobtained byosettingallthephase anglesbythedesiredamount. Solvingthese equations is complex,but thesolutions arefairlystraightforward:

 1 = 3 =arcsinc  2cos( =2)   (20)  2 = (21)  1 = 3 =arccos  cos 1 2 1 sin( =2)  (22)  2 = 1 arccos (  =2 1 ) (23)

where sinc(x)isdened as sin(x)=x. Werefer to thisas aShortComp osite ROtationFor UndoingLength Over andUnder Sho otorscrofulous sequence.

Numerical values of individual pulse rotation and phase angles for a variety of target angles are given in Table I I. The p erformance of scrofulous and plain 180

Æ

pulsesarecomparedinFigure2.

V. THEBB1 FAMILY

Anotherapproachtocomp ositepulse designhasb een describ ed by Wimp eris [20]. While sequences such as corpse andscrofulous seekasingle comp ositepulse whichp erformsthedesired rotationwith reduced

sensi- 1 1 2 2 30 Æ 93.0 78.6 180.0 273.3 45 Æ 96.7 73.4 180.0 274.9 90 Æ 115.2 62.0 180.0 280.6 180 Æ 180.0 60.0 180.0 300.0

TABLEI I:Pulserotationandphaseanglesforascrofulous comp ositepulsewithatargetrotationof

x ;notethat 3 = 1 and3=1.

-1

-0.5

0

0.5

1

0

0.5

1

FIG.2:Fidelityofsimple(dashedline)andscrofulous com-p ositepulses(solid line) asafunctionofthefractional pulse lengtherrorg forpulseswithatargetrotationangleof180

Æ .

naive pulse,whichp erformsthedesired rotation,with a sequenceoferror-correcting pulses,whichpartially com-p ensateforimp erfections. Thisapproachapp earsto sim-plifythedesignofcomp ositepulsesequences,andforthe caseofpulselengtherrors pro duces excellentresults.

Theformoftheerrorcorrectingpulsesequenceisquite tightlyconstrained,asitmusthavenooveralleectinthe absence oferrors, butit mustalsoretainsuÆcient exi-bilitythatitcan actagainsterrors whenthey doo ccur. Hereweconcentrateononeparticularformsuggestedby Wimp eris[20] 180  1 360  2 180  1 (24)

wherethevaluesof  1

and 2

remainto b edetermined. When placed in front of a 

x

pulse Wimp eris refers to theentiresequenceasBB1[20],butaswewillgeneralise his approach we refer to the error correcting sequence (Equation24)asW1.

Asb eforeweevaluatethequaternionforthecomp osite rotation(W1followedby a

x

pulse) inthepresence of pulselength errors, and thenexpandthis quaternion as aMaclaurinseriesing ,the fractionalerror inthepulse p ower. The y andz comp onentsinthe rstordererror termareeasilyremovedbysetting

2 =3

1

;the remain-ingcomp onentscanthenb e eliminatedbycho osing

 1 =arccos   4  : (25)

Thep ositivesolutionis thenidenticaltothat previously describ ed[20]. Examininghigherordererrortermsshows that this pulse sequence is even b etter than it rst ap-p ears,asthesechoicesalsocompletelyremovethesecond

 1 2 30 Æ 92.4 277.2 45 Æ 93.6 280.8 90 Æ 97.2 291.5 180 Æ 104.5 313.4

TABLEI I I:PulsephaseanglesforaW1correctionsequence witha target rotation of 

x

; pulse rotation angles are  1 = 180 Æ and2=360 Æ .

-1

-0.5

0

0.5

1

0

0.5

1

FIG.3: Fidelity ofsimple (dashed line) andBB1 comp osite pulses(solid line)asafunctionofthefractionalpulse length errorg forpulseswithatargetrotationangleof180

Æ .

to b e aprop erty of theW1 sequence andits close rela-tions.

It iseasyto imaginearangeofvariationsoftheBB1 sequence. MostsimplytheW1errorcorrection sequence can b e placed after the 

x

pulse, instead of b efore it. Unsurprisinglythishasnoeect: thesolutionisthesame asb efore,andthep erformanceofthisreversedsequence is identical to that of BB1. More surprisingly the W1 sequence can b e place in themiddleof the 

x

pulse, so thattheoverallsequence

( =2) x

W1( =2) x

(26)

is timesymmetric. Indeed theW1 pulse can b e placed at any p oint withinthe 

x

pulse, with almostidentical eects. The formof the comp osite quaternion dep ends slightlyonwheretheW1pulseisplaced,butthedelity ofthepulsesequence isunchanged: allerrortermsb elow sixthorderarecancelled,withthesizeofthesixthorder termdep endingonthevalueof .

Numericalvaluesofpulsephase anglesforavarietyof targetanglesaregiveninTableI I I. Thep erformanceof theBB1andplain180

Æ

pulsesarecomparedinFigure3. Foralltargetanglesb elow180

Æ

theBB1comp ositepulse outp erformsasimplepulsewhenjg j<1.

Another simplevariationis to use twoor more error correcting sequences; as b efore these can b e placed at various dierent p oints around or within the 

x pulse. For simplicity we assume that all the error correcting sequencesareidenticaltooneanother,andhavethesame general formas W1. In this case it can b e shown that

anglesgivenby 2 =3 1 and  1 =arccos   4n  : (27)

wherenisthetotalnumb erofsequencesused. Asb efore thedelityisindep endentofwheretheWnsequencesare placed,butitdo esdep endonthevalueofn. Thesecond and fourth order error terms are cancelled inall cases, andthesizeofthesixthordererrortermnowdep endson b othand n. Thesmallestsixthordertermisachieved when n = 2, but the term is not completelyremoved. Thegainover n=1is fairlysmall,and inpractice the simplercomp ositepulsesbased on theW1sequence are likelytob e themosteective.

Having varied the p osition and numb er of the error correctingpulsesequencesthenextlogicalstepistovary theirform. Inprincipleanysequencethathasnooverall eectintheabsence oferrors couldb e used. Inpractice we nd that many p ossible sequences allow the second ordererrorterminthedelityexpressiontob eremoved, but the simultaneous cancellation of second and fourth ordererrorsseemstob easp ecialfeatureoftheWn fam-ilyofsequences.

Given the success of this approach to tackling pulse lengtherrors, it seems obvious to apply the metho d to tackle o-resonance eects. As yet, however, this ap-proachhashadnosuccess.

VI. SIMULTANEOUSERRORS

So far we have only considered thecase of either o-resonanceeectsor pulselengtherrorsb eingpresent. In realityb oth problemsmaywello ccursimultaneously. It isthereforeimp ortanttoconsiderhowsuchsimultaneous errorsmightb etackled. Ideally we wouldlike todesign pulsesequenceswhichcancomp ensateforb othproblems atthesametime;this, however,isacomplicated andas yet unresolvedproblem,and herewesimplyanalysethe sensitivity of each of our pulse sequences to the other kindoferror.

Wepro ceedasb efore,calculatingcomp ositepulseand simplepulsequaternionsinthepresenceoferrors,and de-terminingthequaternion delity. Thisdelitycan then b eexpanded asaMaclaurin seriesintheerror, andthe lower order terms examined. Note that this pro cedure stillassumes that onlyone typ e of error is presentat a time;inorder to detailwiththe case where b oth errors arepresent simultaneously it would b ep ossible to usea Maclaurinexpansion inb oth errors, but this isunlikely toleadtomuchinsight. Insteadwe willsimplyplotthe delityas functionofb otherrorsforsomechosen target angle.

We b egin by considering theresp onse of thecorpse pulsesequence topulse lengtherrors. In theabsence of o-resonanceeectstheb ehaviourofcorpseistrivialto calculate,as the three pulses areapplied alongthe +x,

thatofasimplepulse. Theb ehaviourofa180 Æ

pulsein thepresence ofsimultaneouserrors isshowninFigure4.

The b ehaviour of the scrofulous pulse sequence is diÆcult to calculate for general target rotation angles, duetothedep endence of

1

onthearcsincfunction,and so we concentrate on the case of 180

Æ

pulses. For this case the dep endence of the delity on o-resonance ef-fectsisgivenbyF1 2f

2

,whileasimplepulsehasa delityF1 f

2

=2(seeEquation16). Ingeneral scro-fulous is considerably more sensitive to o-resonance eectsthanplainpulses.

FinallyweconsidertheBB1familyofpulsesequences, taking the time-symmetrised version of BB1, Equa-tion 26, as our standard. In this case we can solve the problemforanytargetrotationangle,anduptosecond ordertheresultisidenticaltothatofaplainpulse, Equa-tion 16. Thus, unlike scrofulous, the BB1 sequence achieves its impressive tolerance to pulse length errors at littleornocost insensitivitytoo-resonance eects. Thisisconrmedforsimultaneouserrors byFigure4.

VI I. CONCLUSIONS

Comp ositepulsesshowgreatpromiseforreducingdata errorsinNMRquantumcomputers. Moregenerally,any

implementation of a quantum computer must b e con-cernedonsomelevelwithrotationsontheBlo chsphere, andso comp ositepulse techniques mayndvery broad application in quantum computing. Comp osite pulses arenot,however,apanacea,and somecautionmustb e exercisedintheiruse.

The corpse pulse sequence app ears to b e the b est approachfortacklingsmallo-resonanceerrors(forlarge known o-resonanceeectstheresonanceosettailored, orrotten, scheme[21] is preferable). For pulselength errors variations on the BB1 scheme of Wimp eris [20] give the b est results; the scrofulous familyof pulses is less eective, but do es have the advantage of b eing considerablyshorter.

Acknowledgments

HKC thanks NSERC (Canada) and the TMR pro-gramme(EU) fortheirnancialassistance. JAJ thanks theRoyalSo cietyofLondonfornancialsupp ort.

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-1

0

1

-1

0

1

f

g

(a)

(b)

(c)

(d)

FIG.4: Fidelityof(a)plain,(b)corpse,(c)scrofulousand(d)BB1180 Æ

pulsesasafunctionofsimultaneousoresonance eects,f,andfractionalpulselength error,g . Contoursareplotted at5%intervals.