Arbitrarily accurate composite pulse sequences

Kenneth R. Brown,1 Aram W. Harrow,1,2 and Isaac L. Chuang1,2

1Center for Bits and Atoms, MIT, Cambridge, Massachusetts 02139, USA2Department of Physics, MIT, Cambridge, Massachusetts 02139, USA

(Received 10 May 2004; published 19 November 2004)

Systematic errors in quantum operations can be the dominating source of imperfection in achieving controlover quantum systems. This problem, which has been well studied in nuclear magnetic resonance, can beaddressed by replacing single operations with composite sequences of pulsed operations, which cause errors tocancel by symmetry. Remarkably, this can be achieved without knowledge of the amount of errore. Indepen-dent of the initial state of the system, current techniques allow the error to be reduced toOse3d. Here, weextend the composite pulse technique to cancel errors toOsend, for arbitraryn.

DOI: 10.1103/PhysRevA.70.052318 PACS number(s): 03.67.Pp, 32.80.Qk, 82.56.Jn

INTRODUCTION

Precise and complete control over closed quantum sys-tems is a long-sought goal in atomic physics, molecularchemistry, and condensed matter research, with fundamentalimplications for metrology[1,2] and computation[3,4].Achieving this goal will require careful compensation forerrors of both random and systematic nature. And while re-cent advances in quantum error correction[5–8] allow allsuch errors to be removed in principle, active error correctionrequires expanding the size of the quantum system, and feed-back measurements which may be unavailable. Furthermore,in many systems, errors may be dominated by those of sys-tematic nature, rather than random errors, as when the clas-sical control apparatus is miscalibrated or suffers from inho-mogeneities over the spatial extent of the target quantumsystem.

Of course, systematic errors can be reduced simply bycalibration, but that is often impractical, especially whencontrolling large systems, or when the required control errormagnitude is smaller than that easily measurable. Interest-ingly, however, systematic errors in controlling quantum sys-tems can be compensated without specific knowledge of themagnitude of the error. This fact is lore[9] in the art ofNMR, and is achieved using the method ofcomposite pulses,in which a single imperfect pulse with fractional errore isreplaced with a sequence of pulses, which reduces the errorto Osend.

Composite pulse sequences have been constructed to cor-rect for a wide variety of systematic errors[9–11]. Theseinclude pulse amplitude, phase, and frequency errors and canbe applied to any system with sufficient control. As systemcontrol increases, new uses for composite pulses emerge. Aremarkable example is the recent teleportation of an atomicstate in ion traps[12,13]. Barretet al. use a composite pulsefor individual addressing, while Reibeet al. use a compositepulse to perform two-qubit operations.

In the context of spectroscopy, the goal is often to maxi-mize the measurable signal from a system which starts in aspecific state. Thus, while composite sequences have beendeveloped[14] which can reduce errors toOsend for arbitraryn, these sequences are not general and do not apply, for

example, to quantum computation, where the initial state isarbitrary, and multiple operations must be cascaded to obtaindesired multiqubit transformations.

Only a few composite pulse sequences are known whichare fully compensating[15,16], meaning that they work onany initial state and can replace a single pulse without furthermodification of other pulses. As has been theoretically dis-cussed[17–19] and experimentally demonstrated in ion traps[12,13,20] and Josephson junctions[21], these sequences canbe valuable for precise single- and multiple-qubit control us-ing gate voltages or laser excitation.

Previously, the best fully compensating composite pulsesequence known[16–19] could only correct errors toOse3d.1

Here, we present a systematic technique for creating com-posite pulse sequences to correct errors toOsend, for arbitraryn. The technique presented is very general and can be used tocorrect a wide variety of systematic errors, though for largevalues ofn it can become quite long. Below, our technique isillustrated for the specific case of systematic amplitude er-rors, using two approaches. Also discussed is the number ofpulses required as a function ofn.

The problem of systematic amplitude errors is modeled byrepresenting single-qubit rotations as

Rfsud = expF− iu

2sfG , s1d

where u is the desired rotation angle about the axis thatmakes the anglef with the x axis and lies in thex-y plane,sf=cossfdX+sinsfdY, and X and Y are Pauli operators.Rfsud is the ideal operation, and due to errors, the actualoperation is, instead,Mfsud=Rf(us1+ed), where the angle ofrotation differs from the desiredu by the factor 1+e. Notethat f and u may be specified arbitrarily, but the errore isfixed for all operations, and unknown.

1In Refs. [17–19], the distance measure used is one minus thefidelity, 1−iV†Ui (“the infidelity”) whereiAi is the norm ofA. Weuse instead the trace distanceiV−Ui following the NMR commu-nity. Thus, our composite pulses which arenth order in trace dis-tance are 2nth order in infidelity.

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TWO METHODS FOR CONSTRUCTINGCOMPOSITE PULSES

A composite pulse sequenceRffngsud is a sequence of op-

erationshMfsudj such thatRffngsud=Rfsud+Osen+1d, for un-

known error e. To constructRffngsud, we begin with two

simple observations: first,Rfs−uedMfsud=Rfsud and sec-ond,Mfs2kpd= ±Rfs2kped whenk is an integer. A compos-ite pulse sequence can thus be obtained by finding ways toapproximateRfs−ued by a product of operatorsRfl

s2klped.We obtain this using two approaches.

The first approach we call the Trotter-Suzuki(TS)method. Suzuki has developed a set of Trotter formulas thatwhen given a HamiltonianB and a series of HamiltonianshAlj such thatB=oAl there exists a set of real numbershpjnjsuch that

exps− iBtd = pj ,l

exps− ipjnAltd + Ostn+1d, s2d

and o jpjn=1 [22]. Without loss of generality, we may limitourselves to expansions where thepjn are rational numbers,and assume the goal is to approximateR0s−ued. Using Eq.(2), we set t=e and B=−su /2dX. Then we chooseA1=A3

=mpsX cosf+Y sinfd and A2=2mpsX cosf−Y sinfdwheref and m satisfy the conditions that 4mp cosf=u /2(i.e., A1+A2+A3=B) andqjn=pjnm is an integer. This yieldsan nth order correction sequence

Fn = pj

Mfs2pqjndM−fs4pqjndMfs2pqjnd

= R0s− ued + Osen+1d = R0fngs− ued s3d

and the associatednth order composite pulse sequenceFnM0sud=R0

fngs−uedR0suedR0sud=R0fngsud, thus giving a com-

posite pulse sequence of arbitrary accuracy.The second approach we refer to as the Solovay-Kitaev

(SK) method, as it uses elements of the proof of the Solovay-Kitaev theorem [23]. First, note that rotationsUksAd= I+Aek+Osek+1d can be constructed for arbitrary 232 Hermit-ian matricesA, and kù1, recursively. This is done usingan observation (from [23]) relating the commutatorfA,Bg=AB−BA to a sequence of operationsexps−iAeldexps−iBemdexpsiAeldexpsiBemd=expsfA,Bgel+md+Osel+m+1d. Thus to generateUksAd it suffices to generateUdk/2esBd and Ubk/2csCd such thatfB,Cg=A (choices of inte-gers other thandk/2e andbk/2c which sum tok also work, butgive worse performance).2

Next, we inductively construct a composite pulse se-quenceFn for R0sud. Note that the first order correction se-quence can be written asF1=Mfs2pdM−fs2pd=R0s−ued+Ose2d by selecting 4p cossfd=u. Assume we haveFn

=R0s−ued− iAn+1en+1+Osen+2d. We can then construct a se-

quence to correct for the next order, usingFn+1=Un+1sAn+1dFn, where Un+1sAn+1d is constructed as above.Iteratively applying this method fork=1, . . . ,n yields annth

order composite pulse sequenceFnM0sud=R0fngsud for any n.

This method, which appears to be unrelated to previous com-posite pulse techniques[9,14], gives an efficient algorithm tocalculate sequences for specificu andf but not necessarily ashort analytical description of the sequence. Furthermore, theSolovay-Kitaev technique relies on general properties ofHamiltonians and can be applied without modification toother systematic error models, e.g., frequency errors.

EXAMPLES

The TS and SK techniques described above are generaland apply to a wide variety of errors; explicit application ofthe techniques to generateR0

fngsud sequences for specificncan take advantage of symmetry arguments and compositionof techniques, and can relax some of our assumptions tominimize both the residual error and the sequence length.

First, we explicitly write out the TS composite pulses andconnect them to the well-known pulse sequences of Wimp-eris [16]. We choose to use the TS formulas that are sym-metric under reversal of pulses, i.e, an anagram. These for-mulas remove all even-ordered errors by symmetry, and thusyield only even-order composite pulse sequences. For con-venience, we introduce the notationS1sf1,f2,md=Mf1

smpdMf2s2mpdMf1

smpd and Snsf1,f2,md=Sn−1sf1,f2,md4n−1

Sn−1sf1,f2,2mdSn−1sf1,f2,md4n−1. We

can now define a series ofn order composite pulses Pn as

P0 =M0sud, s4d

P2 =Mf1s2pdM−f1

s4pdMf1s2pdP0, s5d

P2j = Sjsf j,− f j,2dP0, s6d

wheref j =cos−1−u /8pf j and f j =s2s2j−1d−2df j−1 when f1=1.P2 is exactly the passband sequence PB1 described by Wim-

2Here we definebxc to be the largest integer less than or equal tox,and dxe to the smallest integer that is greater than or equal tox.

FIG. 1. Comparison of the narrowband and broadband compos-ite pulse sequences generated by the TS method. The WimperisBB1, PB1, and NB1 sequences are included in this family, and areequivalent to B2, P2, and N2.

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peris[16]. Figure 1 compares the performance of these high-order passband pulse sequences.

Wimperis also proposes a similar broadband sequenceBB1=S1sfB1,3fB1,1dP0 where fB1=cos−1s−u /4pd. Thebroadband sequence corrects over a wider range ofe byminimizing the first order commutator and thus the leadingorder errors. Furthermore, although BB1 and PB1 appeardifferent when written as imperfect rotations, a transforma-tion to true rotations shows that they have the same form,

PB1 =Mf1s2pdM−f1

s4pdMf1s2pdP0

= Rf1s2pedR−f1

s4pedRf1s2pedP0, s7d

BB1 = MfB1spdM3fB1

s2pdMfB1spdP0

= RfB1spedR−fB1

s2pedRfB1spedP0. s8d

This “toggled” frame suggests a way to create higher-orderbroadband pulses. One simply takes a higher-order passbandsequence and replaces each elementS1sf j ,−f j ,md withS1(fBj ,−fBj+4fBjsm/2 mod 2d ,m/2) where fBj satisfiesthe condition cossfBjd=2 cossf jd. Applying this to Pn cre-ates a family of broadband composite pulses Bn.

Similar extensions allow creation of another kind of com-posite pulses(useful, for example, in magnetic resonanceimaging), which increaseerror so as to perform the desiredoperation for only a small window of error. Such “narrow-band” pulse sequences Nn may be obtained starting with apassband sequence Pn, and dividing the angles of the correc-tive pulses by 2. These higher-order narrowband pulses maybe compared with the Wimperis sequence NB1[16], asshown in Fig. 1.

The SK method yields a third set ofnth order compositepulses SKn, and for concreteness, we present an explicit for-mulation of this method. It is convenient to letUnXsad= I− iansX/2den+Osen+1d, such that one can then generateUnZsad=M90s−p /2dUnXsadM90sp /2d and UnYsad=M45sp /2dUnXsadM45s−p /2d.3 Using the first order rota-tions

U1Xsad = MfS2pd a

4peDM−fS2pd a

4peD , s9d

where f=cos−1ha/ f4pda/ s4pde g j, as described above,we may recursively construct UnXsad=Ubn/2cYsadUdn/2eZsadUbn/2cYs−adUdn/2eZs−ad, and anyn.1 andany a.

With these definitions, the first order SK composite pulsefor R0

fngsud is simply

SK1 =U1XsudM0sud = R0sud − iA2

2e2 + Ose3d. s10d

From the 232 matrix A2, we can then calculate the normiA2i and the planar rotationRA2

that performsRA2s−A2dRA2

−1

=iA2iX. The second order SK composite pulse is then

SK2 =MA2

−1U2XsiA2i1/2dMA2SK1 s11d

=R0sud − iA3

2e3 + Ose4d, s12d

where MA2is the imperfect rotation corresponding to the

perfect rotationRA2.

The nth order SK composite pulse family is thus

SKn = MAn

−1UnXsiAni1/ndMAnSKsn − 1d s13d

=R0sud − iAn+1

2en+1 + Osen+2d. s14d

A nice feature of the SK method is that when given a com-posite pulse of ordern described by any method one cancompose a pulse of ordern+1. The “pure” SK method SKnis outperformed in terms of both error reduction and pulsenumber by the TS method Bn for nø4. Therefore, we applythe SK method for ordersn.4 using B4 as our base com-posite pulse. We label these pulses SBn.

PERFORMANCE AND EFFICIENCY

Two important issues with composite pulses are the actualamount of error reduction as a function of pulse error, andthe time required to achieve a desired amount of error reduc-tion. These performance metrics are shown in Fig. 2, com-paring the SK, broadband, and passband composite pulsesfor varying errore, and f=0, and using as the composite

3The optimal way to generateUnY is to shift the phasesf of theunderlyingMfs0d that generateUnx by 90°.

FIG. 2. Composite pulse errorE as a function of base errore fora variety of composite pulse sequences. Pn, Bn, SKn, and SBn arethe nth order passband, broadband, SK, and combined B4-SK se-quences, respectively. The number in the brackets refers to the num-ber of imperfect 2p rotations in the correction sequence. Note howpulses of the same order(such as P6, B6, SK6, SB6) have the sameslope(asymptotic scaling) for low values ofe, but can have widelyvarying performance whene is large. The inset plots the scaling ofthis sequence length with ordern for SKn (SBn is very similar) fornø30 and compares it with the upper bound obtained with numeri-cal methods.

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pulse errorE=iRfsud−Rffngsudi. We find that for practical

values of error reduction,n,30, the number ofp pulsesrequired to reduce error toOsend grows as,n3.09, which isclose to the lower bound of,n3 which can be analyticallyderived [24]. In contrast, the TS sequence Bn requiresO(expsn2d) pulses.

For a wide range of base errors«, the TS formulationoutperforms the SK method in achieving a low compositepulse errorE. The recursive nature of the TS method buildsoff elements that remove lower-order errors, resulting in arapid increase of pulse number and a monotonic decrease ineffective error at every order for any value of the base error.However, the SK approach is superior to the TS method forapplications requiring incredible precision,Eø10−12, fromrelatively precise controls,e,10−2.

The SK and TS pulse sequences presented here are con-ceptually simple but may not be optimal. Integrating ideasfrom both methods, we can develop additional families ofcomposite pulses. As an example, the SK method relies oncancellation of error order by order by building up sequencesof 2p pulses. However, there is no reason that the basic unitshould be a single pulse. Instead, one can build a sequencefrom TS (B2) style pulse triplets,Gsf1d=S1sf1,3f1,1d. Byusing an additional symmetry that the TrfYGs−f1dGsf1dg=0, the leading order error is guaranteed to be proportionalto X at the cost of doubling the pulse sequence. The resultingpulses are of length expsnd [compared to expsn2d for TS],broadband compared to SK sequences, and described in de-tail in [24].

CONCLUSIONS

We have presented a set of tools that allows one to gen-erate arbitrarily accurate composite pulse sequences for sys-tematic, but unknown, error. As an example, we have con-structed explicit composite pulse sequences for errors in

rotation angle. These can be constructed withOsn3d pulses,for n&30. For high-precision applications such as quantumcomputation, these pulses allow one to perform accurate op-erations even with large errors. Practically, the B4 and B2=BB1 pulse sequences seem most useful, depending on themagnitude of error.

While we have focused on composite pulse sequences forrotation errors, we emphasize that these methods also applyto correcting systematic errors in control phase and fre-quency[24]. For example, a frequency error can be repre-sented for an expected rotationR0sud as M08sud=expf−isu /2X+ uu /2udZdg. Note that M08su /2dM08s−udM08su /2d yields to first order in d the phase shiftU1Zs2udd. Starting with any fully compensating compositepulse sequence that corrects frequency errors to orderd 2,e.g., CORPSE[17], and the basic operationU1Zs2udd, onecan then apply the SK technique to create a pulse sequenceof Osd nd [24].

Furthermore, the TS and SK approaches can be extendedto any set of operations that has a subgroup isomorphic torotations of a spin. For example, Jones has used this isomor-phism to create reliable two-qubit gates based on an Isinginteraction to accuracyOse3d [18]. Similarly, the techniquesoutlined here can immediately be applied to gain arbitraryaccuracy multiqubit gates. Interestingly, the TS formula canbe directly applied to any set of operations, if the operationssuffer from proportional systematic timing errors. Therefore,this control method could also be applied to classical sys-tems.

ACKNOWLEDGMENTS

We are grateful to A. Childs and R. Cleve for stimulatingdiscussions. This work was supported in part by ARDA Con-tract No. DAAD19-03-1-0075 and A.W.H. was supported bythe NSA and ARDA under ARO Contract No. DAAD19-01-1-06.

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