aa r X i v : . [ qu a n t - ph ] A p r Composite pulses with errant phases
Boyan T. Torosov and Nikolay V. Vitanov Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tsarigradsko chauss´ee, 1784 Sofia, Bulgaria Department of Physics, St Kliment Ohridski University of Sofia, 5 James Bourchier blvd, 1164 Sofia, Bulgaria (Dated: May 1, 2019)Composite pulses — sequences of pulses with well defined relative phases — are an efficient,robust and flexible technique for coherent control of quantum systems. Composite sequences cancompensate a variety of experimental errors in the driving field (e.g. in the pulse amplitude, du-ration, detuning, chirp, etc.) or in the quantum system and its environment (e.g. inhomogeneousbroadening, stray electric or magnetic fields, unwanted couplings, etc.). The control parameters arethe relative phases between the constituent pulses in the composite sequence, an accurate controlover which is required in all composite sequences reported hitherto. In this paper, we introduce twotypes of composite pulse sequences which, in addition to error compensation in the basic experi-mental parameters, compensate systematic errors in the composite phases. In the first type of suchcomposite sequences, which compensate pulse area errors, relative phase errors of up to 10% can betolerated with reasonably short sequences while maintaining the fidelity above the 99.99% quantumcomputing benchmark. In the second type of composite sequences, which compensate pulse areaand detuning errors, relative phase errors of up to 5% can be compensated.
I. INTRODUCTION
Since their invention in nuclear magnetic resonance(NMR) 40 years ago [1–9] composite pulses have estab-lished themselves as one of the most accurate, robust andflexible technique for coherent control of quantum sys-tems. In recent years, they have enjoyed steadily grow-ing interest in quantum information [10–12] and quantumoptics [13–16]. Curiously, the concept of composite se-quences has been well known in polarization optics sincethe 1940s [17–25], where achromatic wave plates or po-larization filters can be constructed by a set of ordinarywave plates with their fast (or slow) polarization axesrotated at specific angles with respect to each other. Re-cently, the composite idea has been extended also to fre-quency conversion processes in nonlinear optics [26, 27].Composite pulses offer a unique combination of ultra-high accuracy, well below the error threshold (often re-ferred as 10 − ) in quantum computation, with robustnessto parameter errors similar to adiabatic passage tech-niques [28]. Moreover, they offer great flexibility in shap-ing the excitation profile, or even the propagator, in es-sentially any desired manner — a feature, which is notavailable in any other quantum control method.The composite pulse sequence is a finite train of pulseswith well-defined phases, which are used as control pa-rameters in order to compensate experimental errors orto shape the excitation profile in a desired manner. Themost ubiquitous composite sequences are the broadband π pulses, which produce unit transition probability notonly for a pulse area A = π and zero detuning ∆ = 0,as a single resonant π pulse, but also in some (broad)ranges around these values [2–4, 8, 9, 13, 14, 29]. Hencea composite π pulse can compensate the pulse area anddetuning errors of a single π pulse and make a sequenceof imperfect pulses look like an ideal π pulse. Among thebroadband composite π pulses, we note those, which com-pensate pulse area errors, detuning errors, and both pulse area and detuning errors. Recently, composite pulses,which compensate experimental errors in any experimen-tal parameter — universal composite pulses — have beenintroduced and experimentally demonstrated [16]. Com-posite θ pulses, which produce controlled partial excita-tion with probability sin ( θ/ π pulses, which are robust to systematic phase errors ofthe order of 5-10%. We present two types of such com-posite sequences: (i) sequences that deliver double com-pensation of simultaneous errors in the pulse area andthe composite phases, and (ii) sequences which producetriple compensation of simultaneous errors in the pulsearea, the detuning and the composite phases.This paper is organized as follows. In Sec. II we discussthe mathematical details of the derivation of these newcomposite pulses. In Sec. III the double-compensationsequences are introduced, and the triple-compensationsequences are presented in Sec. IV. Finally, Sec. V wrapsup the conclusions. II. DESCRIPTION OF THE METHOD FORCORRECTION OF PHASE ERRORS
Here we describe the method for construction of com-posite pulses that produce excitation profiles which arerobust against simultaneous errors in the pulse area andthe composite phases (double compensation). Triplecompensation is derived similarly and the specifics areelaborated in Sec. IV.The propagator of a coherently driven two-state quan-tum system, described by the Hamiltonian H ( t ) = ~ [Ω( t ) σ x + ∆( t ) σ z ], is given by the SU(2) matrix U = (cid:20) a b − b ∗ a ∗ (cid:21) , (1)where a and b are the (complex) Cayley-Klein parametersobeying | a | + | b | = 1. For exact resonance (∆ = 0),we have a = cos( A / b = − ı sin( A / A is thetemporal pulse area A = R t f t i Ω( t )t.. For a system startingin state | i , the single-pulse transition probability is p = | b | = sin ( A / φ imposed on the driving field, Ω( t ) → Ω( t ) e ı φ , is imprinted onto the propagator as U φ = (cid:20) a be ı φ − b ∗ e − ı φ a ∗ (cid:21) . (2)Consider a train of N pulses, each with area A k and phase φ k , ( A ) φ ( A ) φ ( A ) φ · · · ( A N ) φ N . (3)In the presence of pulse area errors, we have to replacethe nominal pulse areas A k by the actual pulse areas A k = A k (1 + α ) ( k = 1 , , . . . , N ), where α is the relativepulse area error. In the presence of phase errors, eachnominal phase φ k should be replaced by the actual phase ϕ k = φ k (1 + ǫ ) ( k = 1 , , . . . , N ), where ǫ describes the(systematic) phase errors, the compensation of which isour primary concern here. In the presence of pulse areaand phase errors, the pulse sequence (3) produces thepropagator U ( N ) = U ϕ N ( A N ) · · · U ϕ ( A ) U ϕ ( A ) U ϕ ( A ) . (4)Yet, for the sake of brevity, in the notation of the com-posite pulse sequence (3) we shall use the nominal pulseareas A k and the nominal phases φ k . In this paper, based on numerical evidence, we considercomposite sequences of an odd number N = 2 n + 1 ( n =1 , , . . . ) of identical pulses, and nominal pulse area A k = π ( k = 1 , , . . . , N ). We also consider symmetric phases, φ k = φ N +1 − k ( k = 1 , , . . . , n ). Using the invarianceof the transition probability to the addition of the samephase shift to all phases (see Appendix A), we set φ = φ N = 0. Hence, the phase-error correcting compositesequences areΦ N = π π φ · · · π φ n π φ n +1 π φ n · · · π φ π , (5)and the total propagator turns into U ( N ) = U ( A ) U ϕ ( A ) · · · U ϕ n +1 ( A ) · · · U ϕ ( A ) U ( A ) , (6)with A = π (1 + α ) and U ϕ ( A ) = (cid:20) cos( A / − ı sin( A / e ı ϕ − ı sin( A / e − ı ϕ cos( A / (cid:21) . (7)We calculate the product in Eq. (6) and expand U ( N )11 vs α and ǫ at ( α = 0 , ǫ = 0). Then we set to zero as manyterms as possible in order to obtain a robust excitationprofile. If we denote the ( j, k )-th multivariate coefficientin the power series as s jk = α j ǫ k j ! k ! ∂ j + k U ( N )11 ∂α j ∂ǫ k ! (0 , , (8)one can easily verify that, because of the chosen sym-metry in the phases and pulse areas of the compositesequence, we have s jk ≡ j ) . (9)Hence the first nonzero derivatives with respect to thepulse area error α are the first derivatives ( j = 1), thenthe third derivatives ( j = 3), etc. With respect to thephase error ǫ all derivatives are generally nonzero. Be-cause the primary objective of the composite pulses isthe compensation of pulse area (and detuning) errors,we limit ourselves to the cancellation of the low-order(up to first or second) derivatives with respect to thephase error ǫ , which already provides significant improve-ment over the existing composite pulses. Generally, for N = 2 n + 1 pulses we have n different phases, with whichwe can nullify n different derivatives. III. COMPENSATION OF PULSE AREA ANDPHASE ERRORS
As a performance benchmark we use the broadbandcomposite pulses of Ref. [13], which compensate pulsearea errors. They contain an odd number of pulses N =2 n + 1 and symmetric (anagram) phases, which are givena simple analytic formula for arbitrary N . Below wepresent the new composite pulses which, in addition topulse area errors, compensate phase errors. - - - - - P ha s e E rr o r ϵ N = - - - - - - - - - - α P ha s e E rr o r ϵ - - - - - α FIG. 1: Transition probability vs pulse area deviation α and phase error ǫ for a single pulse (top left), the B3 se-quence A2a (top right), the B5a composite sequence (A3a)(bottom left) and the five-pulse phase-error compensating se-quence (12) (bottom right). The numbers m = 2 , , − − m . A. Composite sequences of three pulses
We first consider a sequence of three pulses. We cal-culate the product in Eq. (6) and expand U (3)11 at α = 0and ǫ = 0. We obtain for the first non-zero coefficient inthe expansion s = α π φ )] . (10)We can nullify this coefficient by setting φ = 2 π/
3. Theresulting composite sequence π π π π coincides with thewell-known broadband CP B3 of Eq. (A2a) [2, 13]. Hencewe do not obtain additional compensation in the phaseerror because of the absence of free phases. (Letting φ benonzero does not help annul s .) Longer sequences with N >
3, studied below, do allow for such compensation.
B. Composite sequences of five pulses
We now consider a sequence of five pulses. Similarly tothe N = 3 case, we calculate the product in Eq. (6) andexpand U (5)11 at ( α = 0 , ǫ = 0). We obtain for the firstnon-zero coefficients in the expansion the expressions s = − π α [1 + 2 cos( φ − φ ) + 2 cos(2 φ − φ )] , (11a) s = παǫ [( φ − φ ) sin( φ − φ )+(2 φ − φ ) sin(2 φ − φ )] . (11b) The set of equations s = 0 and s = 0 does not havean analytic solution because the latter of these is tran-scendental. The values of φ and φ , which nullify s and s , can be found numerically. One of the solutionsis (approximately) φ = 0 . π , φ = 0 . π , and hencethe composite sequence readsΦ5 = π π . π π . π π . π π . (12)This composite sequence eliminates errors in the pulsearea up to order O ( α ) and in the phases up to order O ( ǫ ).In Fig. 1 the transition probability is plotted as a func-tion of the pulse area error α and the systematic error ǫ in the phases for the composite sequences B3, B5a,and Φ5. We also plot the excitation profile of a singlepulse ( N = 1), which is insensitive to systematic phaseerror (as far as the transition probability is concerned),but it lacks compensation in the pulse area. Obviously,the phase-compensating composite pulse Φ5 provides anultra-accurate transition probability ( p > . C. Longer composite sequences
We continue with a sequence of seven pulses, whichpresents us three phases to be used as free control pa-rameters. We choose the non-zero coefficients, which wewant to nullify, to be s , s , and s . The explicit ex-pressions for these coefficients are too cumbersome to bepresented here, but their numeric cancellation is straight-forward. In such a way, we obtain numerous solutions forthe phases, one of which is φ = 0 . π , φ = − . π , φ = − . π . Hence the corresponding seven-pulsecomposite sequence readsΦ7 = π π . π π − . π π − . π π − . π π . π π . (13)Figure 2(top left) shows the excitation profiles of thisphase-compensating composite sequence. Clearly, thehigh-probability area is larger than for the Φ5 sequencein Fig. 1.For longer sequences we can proceed in a similar man-ner. The additional free phases allow us to cancel larger-order errors in the pulse area and the composite phases.The values of all phases, which are derived, as well as thecorresponding s ij terms, which are cancelled, are summa-rized in Table I. In Fig. 2 we plot the excitation profilesfor sequences of length N = 7 , , ,
13. A systematicimprovement of the excitation profile is observed as thelength of the composite sequences increases, with the tol-erance ranges exceeding 40% for pulse area errors and10% for phase errors. - - - - - P ha s e E rr o r ϵ - - - - - - - - - - α P ha s e E rr o r ϵ - - - - - α FIG. 2: Transition probability vs pulse area deviation α andphase error ǫ for phase-error compensating sequences with N = 7 , , ,
13 pulses. The phases are given in Table I. Thenumbers m = 2 , , − − m . D. Discussion
It is important to note that in the absence of phaseerror ( ǫ = 0) different composite pulses can produce thesame excitation profile due to the invariance of the transi-tion probability to various transformations of the phases,see Appendix A. However, in the presence of phase er-rors, the picture changes drastically. For example, wecannot add or subtract phases 2 π to/from any chosenphase because the phase error ǫ multiplies the phases φ k and φ k ± π differently and hence these different phaseswill lead to different excitation profiles.To this end, here we have restricted ourselves to so-lutions for the composite phases in the ranges [0 , π ] or[ − π, π ]. For some of the presented composite sequencesthere exist other sequences of the same length which pro-duce slightly better (i.e. broader) profiles which, how-ever, have phases lying outside these ranges. For exam-ple, the seven-pulse sequence with phases ( φ , φ , φ ) =(1 . , . , . π produces a slightly broader pro-file than the Φ7 sequence presented here. We have delib-erately omitted these other sequences because first, thepresented sequences already produce significant phase er-ror compensation, and second, in order to avoid ambigu-ity in experimental implementation.Indeed, if the phase shifts are created by electric ormagnetic fields as time-integrated Stark or Zeeman shiftsthen phases φ and φ ± π are physically different andit make sense to consider the respective composite se-quences as different. It is less obvious if phases φ and φ ± π can be physically different if created by other TABLE I: Nullified s jk terms and the corresponding nomi-nal phases for phase-error compensating composite sequenceswith different length N . All phases are in units π .Φ N nullified terms and phases Φ5 ( s , s )(0 , . , . , . , s , s , s )(0 , . , − . , − . , − . , . , a ( s , s , s , s )(0 , . , . , . , . , . , . , . , b ( s , s , s , s )(0 , . , . , . , . , . , . , . , a ( s , s , s , s , s )(0 , . , . , . , . , . , . , . , . , . , b ( s , s , s , s , s )(0 , . , . , . , . , . , . , . , . , . , a ( s , s , s , s , s , s )(0 , . , . , − . , − . , − . , . , − . , − . , − . , . , . , b ( s , s , s , s , s , s )(0 , . , . , . , − . , − . , . − . , − . , . , . , . , c ( s , s , s , s , s , s )(0 , . , . , − . , − . , − . , . − . , − . , − . , . , . , mechanisms, e.g. by a microwave generator or an acous-tooptical generator. Therefore, in order to avoid ambi-guity, we have presented only composite sequences withphases in ranges of length 2 π , i.e. (0 , π ) or ( − π, π ). Itis important that any experimental realization of our se-quences should consider this argument and should usethe phases as reported here. IV. COMPENSATION OF PULSE AREA,DETUNING AND PHASE ERRORS
We can apply the idea of phase-error compensatingcomposite pulses to produce sequences which compen-sate errors in more than one parameter. The derivationof the phases is done in a way much similar to the onedescribed in Sec. II. For instance, in order to derive se-quences, which are insensitive to errors in the Rabi fre-quency, the detuning, and the composite phases, we pro-ceed as follows. First, we should specify the pulse shapein order to obtain the explicit formula for the single-pulsepropagator. In our derivation, we assume rectangularpulses. Then we calculate the total propagator by tak-ing the product of the constituent propagators. Next,we calculate the multivariate coefficients in the expan- - - R ab i F r equen cy ( un i t s o f π / T ) - - T90 % - - R ab i F r equen cy ( un i t s o f π / T ) U92 % - - T92 % - - R ab i F r equen cy ( un i t s o f π / T ) U95 % - - T95 % - - ( units of π / T ) R ab i F r equen cy ( un i t s o f π / T ) U910 % - - ( units of π / T ) T910 % FIG. 3: Transition probability vs Rabi frequency and detun-ing for composite sequences of nine rectangular pulses. Theleft column shows the excitation profile for the universal se-quence U9 (14) with the phases of Eq. (16) [16], while theright column shows the profiles for the phase-error-correctedsequence T9 with the phases of Eq. (15). The error in thephases is 0%, 2%, 5%, and 10%, from top frames to bottomframes. sion of the total propagator vs the Rabi frequency, thedetuning, and the phase error, near the point of perfectpopulation transfer. Finally, we try to cancel as many ofthese derivative terms as possible.We call these composite pulses triple compensating anddenote them with T N . For instance, the anagram com- posite sequence of nine rectangular pulses, π π φ π φ π φ π φ π φ π φ π φ π , (14)with phases φ = 1 . π, φ = 1 . π, φ = 0 . π, φ = 0 . π. (15)is names as T9 and it is robust against errors in the Rabifrequency, the detuning and the composite phases.In Fig. 3 we compare the transition probabilities ofthis composite pulse (right column) with the transitionprobability, produced by a nine-pulse universal compositesequence U9, derived in Ref. [16] (left column), which hasthe same form as Eq. (14) but with the phases φ = 0 . π, φ = 1 . π, φ = 0 . π, φ = 0 . π. (16)As seen in the figure, with attention to the 99.99% con-tour (label 4), the universal sequence U9 produces amore robust excitation profile in the absence of phaseerrors (top frames), but in the case when phase errorsare present, the phase-compensating sequence T9 beatsthe universal one (middle and bottom frames). V. DISCUSSION AND CONCLUSIONS
In the present work we presented an approach to buildcomposite pulses, which are insensitive to systematic er-rors in the composite phases. We have shown explicitresults for sequences of up to 13 pulses, with simulta-neous compensation of pulse area and phase errors, butthese results can be readily extended to larger numberof pulses. We have also presented triple compensation oferrors in the pulse area, the frequency detuning and thephases. These phase-error-corrected composite sequenceshave the potential to eliminate the main limitation of thecomposite pulses technique: the requirement for accuratephase control. They should make it possible to extendthe application of the powerful and flexible concept ofcomposite pulses to physical platforms wherein the accu-rate phase control is difficult or impossible.One possible way to implement the proposed phase-compensating composite pulses is to use a recently de-rived generalization of the composite-pulses method todetuning pulses [33]. In that work, a sequence of detun-ing pulses is used, with the areas of these pulses beingthe control parameters. In the limit of very short pulsesthe sequence can be seen as a composite pulse, wherethe composite phases are proportional to the area of thedetuning pulse. Therefore, a systematic error naturallyoccurs.Another application of these phase-error resilient com-posite sequences could be for achromatic devices for fre-quency conversion in nonlinear optics [26, 27]. There thecomposite approach is implemented by using nonlinearcrystals of different materials and different thicknesses:alternating thick slabs of one material used as analogs - - - - - P ha s e E rr o r ϵ - - - - - - - - - - α P ha s e E rr o r ϵ - - - - - α FIG. 4: Transition probability vs pulse area deviation α andphase error ǫ for the four different B5 sequences of Eq. (A3)(a, b, c, and d). The numbers m = 2 , , − − m . of π pulses, and thin slabs of another material used forthe phase jumps (via controlled phase mismatch). Sys-tematic errors in the composite phases occurs naturallybecause the phases are proportional to the thickness ofthe corresponding slabs and to a scale differently for dif-ferent frequencies. Acknowledgments
The authors acknowledge useful discussions with An-don Rangelov. This work is supported by the EuropeanCommission’s Horizon-2020 Flagship on Quantum Tech-nologies project 820314 (MicroQC).
Appendix A: Non-equivalence of composite pulses inthe presence of phase errors
In the absence of phase error ( ǫ = 0) different compos-ite pulses can produce the same excitation profile dueto the invariance of the transition probability to var-ious transformations of the phases [30]. Examples ofpopulation-preserving transformations are: (i) the simul-taneous sign flip of all composite phases, {− φ , − φ , . . . − φ N } ; (ii) the addition/subtraction of arbitrary integermultiples of 2 π to any composite phase, { φ + 2 k π, φ +2 k π, . . . φ N +2 k N π } , where k j are arbitrary integers; (iii)the application of the composite sequence in the reverseorder, { φ N , φ N − , . . . φ } ; (iv) the addition of the samephase shift, e.g. φ , to all phases in the sequence. Givena composite pulse sequence, these four features allow one to construct other composite sequences, which deliverthe same transition probability. In addition, because thecomposite phases are derived from a set of trigonometricequations, there are multiple solutions which cannot beobtained from each other by using the above operationsbut still deliver the same transition probability.For example, the symmetric composite sequences B n of Ref. [13], which are of the type (5), with phases φ k = k ( k − nN π ( k = 1 , , . . . , N ) , (A1a)produce the same excitation profiles as the pulse sequencewith the phases φ k = k ( k − N π ( k = 1 , , . . . , N ) . (A1b)For N = 3 pulses, both Eqs. (A1a) and (A1b) give thewell-known sequence [2, 13]B3 = π π π π , (A2a)By inverting the sign of the second phase and adding 2 π we find another equivalent sequence, π π π π . (A2b)For N = 5 pulses, Eqs. (A1a) and (A1b) produce thetwo sequences [13]B5 a = π π π π π π π π , (A3a)B5 b = π π π π π π π π . (A3b)Using the transformations described above we can gen-erate two other sequences,B5 c = π π π π π π π π , (A3c)B5 d = π π π π π π π π . 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