Ultrahigh-fidelity composite rotational quantum gates
aa r X i v : . [ qu a n t - ph ] D ec Ultrahigh-fidelity composite rotational quantum gates
Hayk Gevorgyan and Nikolay V. Vitanov
Faculty of Physics, St Kliment Ohridski University of Sofia, 5 James Bourchier blvd, 1164 Sofia, Bulgaria (Dated: January 1, 2021)Composite pulse sequences, which produce arbitrary pre-defined rotations of a qubit on the Blochsphere, are presented. The composite sequences contain up to 17 pulses and can compensate upto eight orders of experimental errors in the pulse amplitude and the pulse duration. Compositesequences for three basic quantum gates — X (NOT), Hadamard and arbitrary rotation — arederived. Three classes of composite sequences are presented — one symmetric and two asymmet-ric. They contain as their lowest members two well-known composite sequences — the three-pulsesymmetric SCROFULOUS pulse and the four-pulse asymmetric BB1 pulse, which compensate firstand second-order errors, respectively. The shorter sequences are derived analytically, and the longerones numerically (instead by nesting and concatenation, as mostly done hitherto). Consequently,the composite sequences derived here match or outperform the existing ones in terms of either speedor accuracy, or both. For example, we derive a second-order composite sequence, which is faster (byabout 13%) than the famous BB1 sequence. For higher-order sequences the speed-up becomes muchmore pronounced. This is important for quantum information processing as the sequences derivedhere provide more options for finding the sweet spot between ultrahigh fidelity and high speed.
I. INTRODUCTION
Quantum rotation gates, such as the Hadamard gateand the X (or NOT) gate are central elements in anyquantum circuit [1–3]. Traditionally, a general rotationat an angle θ is implemented by a resonant pulsed fieldwith a temporal area of θ , hence the name θ pulses. Inparticular, the Hadamard gate is implemented by a reso-nant π/ π pulse, which are the theoretically fastest meansfor producing these gates. However, resonant driving isprone to errors in the experimental parameters, e.g. thepulse amplitude, duration, and detuning.Various proposals have been made in order to generaterotation gates that are resilient to experimental errors,at the expense of being longer, and hence slower. Adi-abatic techniques are the traditional remedy for tack-ling such errors [4]. Ever since 1932 [5–8], adiabaticevolution via a level crossing is the ubiquitous methodto produce complete population inversion and hence theX gate. More recently, adiabatic evolution via a halfcrossing has gained popularity as a means for produc-ing half excitation, and hence the Hadamard gate [9–13]. This idea has been used in a technique known ashalf-SCRAP (Stark-chirped rapid adiabatic passage) [9]and the closely related two-state STIRAP (stimulatedRaman adiabatic passage) [10], which has been success-fully implemented in a trapped-ion experiment [11]. Inboth cases, pulse shaping and chirping are designed suchthat their time dependences resemble the delayed-pulseordering of conventional STIRAP [14]. In a variationof these, an adiabatic technique has been proposed [12]which generates arbitrary coherent superpositions of twostates, which is controlled by the initial and final ra-tios of the field’s amplitude and its detuning. An ex-tension of this half-crossing technique to three states hasbeen experimentally demonstrated in a trapped-ion ex-periment, with an error of about 1 . × − , i.e. close to the quantum computation benchmark level [13], whichwas achieved by using pulse shaping. Another proposalused a sequence of two half-crossing adiabatic pulses splitby a phase jump, which serves as a control parameter tothe created superposition state [15].In three-state Raman-coupled qubits, a very populartechnique is fractional STIRAP [16–18], in which theStokes pulse arrives before the pump pulse but the twopulses vanish simultaneously. This leads to the creationof a coherent superposition of the two end states of thechain. Tripod-STIRAP [19–21], an extension of STIRAPwherein a single state is coupled to three other states,has also been used for the generation of coherent su-perpositions of these three states or two of them. Wealso note a technique for creation of coherent superposi-tion states and for navigation between them by quantumHouseholder reflections [22, 23].While adiabatic techniques provide great robustnessto parameter errors, in general they struggle to deliverthe ultrahigh fidelity required in quantum computation.A powerful alternative to achieve ultrahigh fidelity whilefeaturing robustness to parameter errors is the techniqueof composite pulses [24–28]. The composite pulse se-quence is a finite train of pulses with well-defined relativephases between them. These phases are control parame-ters, which are determined by the desired excitation pro-file. Composite pulses can shape the excitation profile inessentially any desired manner, which is impossible witha single resonant pulse or adiabatic techniques. In par-ticular, one can create a broadband composite π pulse,which delivers transition probability of 1 not only for apulse area A = π and zero detuning ∆ = 0, as a singleresonant π pulse, but also in some ranges around thesevalues [24–29, 31–33, 36–39]. Alternatively, narrowbandcomposite pulses [31, 33, 36, 40–45] squeeze the excita-tion profile around a certain point in the parameter space:they produce excitation that is more sensitive to param-eter variations than a single pulse, with interesting appli-cations to sensing, metrology and spatial localization inNMR spectroscopy. A third family of composite pulses— passband pulses — combine the features of broadbandand narrowband pulses: they provide highly accurate ex-citation inside a certain parameter range and negligiblysmall excitation outside it [33–35, 43, 46, 47].There are no universally applicable composite pulsesbecause the requirements in different applications are dif-ferent. For instance, in NMR, composite pulses whichcompensate errors in very broad parameter ranges withonly modest accuracy are ubiquitous. On the contrary,in quantum information, very high accuracy is requiredwithin some moderately large parameter ranges [48–53].In this paper, we present several sets of single-qubit ro-tation quantum gates constructed with composite pulsesequences. There are two classes of composite rotations,named variable and constant rotations [28, 29]. Variable-rotation composite pulses (sometimes called
Class B )compensate parameter errors only in the transition prob-ability p (or the population inversion w = 2 p − Constant-rotation , or phase-distortionless [55],composite pulses (sometimes called
Class A ) compensateparameter errors in both the transition probability andthe phases of the created superposition state (i.e., in theBloch vector coherences u and v ). The latter are obvi-ously more demanding and require longer sequences forthe same order of compensation. However, in quantuminformation processing wherein phase relations are essen-tial, constant rotations are clearly the ones to be used forquantum rotation gates [56].In this paper, we focus at the derivation of ultrahigh-fidelity composite rotation gates, including the X,Hadamard and general rotation, which compensatepulse-area errors up to eighth order. The X andHadamard gates are special cases of general rotations butthey are treated separately due to their importance inquantum information. Our results extend earlier resultson some of these gates using shorter pulse sequences. Thefirst phase-distortionless composite pulse was designed byTycko [40] which produces a composite X gate. It con-sists of three pulses of total nominal area of 3 π and pro-vides a first-order error compensation. A second-ordererror compensation composite pulse was constructed byWimperis, the well-known BB1 pulse [32, 33]. It consistsof four pulses with a total nominal pulse area of 4 π + θ and it produces a constant rotation at an arbitrary angle θ . More recently, Wimperis and co-workers developedseveral phase-distortionless anti-symmetric composite π pulses designed for rephasing of coherence [57–59]. Jonesand co-workers have devoted a great deal of attention tocomposite X gates, with an emphasis of geometric ap-proaches for derivation of such sequences, which work upto 5 and 7 pulses [56, 60–62].Composite rotation gates with a pulse area error com-pensation of third and higher order have been con-structed using nesting and concatenation of shorter com- posite sequences. For larger error order, this procedureproduces (impractical) composite sequences of extremelength. Here we use analytic approaches and brute-forcenumerics to derive three classes of composite sequencesfor X, Hadamard and rotation gates which achieve er-ror compensation of up to 8th order with much shortersequences than before.This paper is organized as follows. In Sec. II we ex-plain the derivation method. Composite π rotations, rep-resenting the X gate are presented in Sec. III. Compos-ite implementations of the Hadamard gate are given inSec. IV, and composite rotation gates in Sec. V. Finally,Sec. VI presents the conclusions. II. COMPOSITE ROTATION GATES:DERIVATIONA. Composite rotation gates
Our objective is to construct the qubit rotation gateˆ R y ( θ ) = e i ( θ/ σ y , where θ is the rotation angle and ˆ σ y isthe Pauli’s y matrix. In matrix form, R y ( θ ) = (cid:20) cos( θ/
2) sin( θ/ − sin( θ/
2) cos( θ/ (cid:21) . (1)The rotation gate (1) is equivalent to the rotation gateˆ R x ( θ ) = e i ( θ/ σ x , or in matrix form, R x ( θ ) = (cid:20) cos( θ/ i sin( θ/ i sin( θ/
2) cos( θ/ (cid:21) . (2)Indeed, ˆ R x ( θ ) can be obtained from ˆ R y ( θ ) by simplephase transformation, ˆ R x ( θ ) = ˆ F ( π/
4) ˆ R y ( θ ) ˆ F ( − π/ F ( φ ) = e iφ ˆ σ z , or in matrix form, F ( φ ) = R z ( φ ) = (cid:20) e iφ e − iφ (cid:21) . (3)We shall use the gate (1) because it is real and becauseit coincides with the ubiquitous definition of the rotationmatrix. Therefore, hereafter we drop the subscript y forthe sake of brevity.The propagator of a coherently driven qubit is the so-lution of the Schr¨odinger equation, i ~ ∂ t U ( t, t i ) = H ( t ) U ( t, t i ) , (4)subject to the initial condition U ( t i , t i ) = I , the identitymatrix. If the Hamiltonian is Hermitian, the propagatoris unitary. If the Hamiltonian is also traceless, then thepropagator has the SU(2) symmetry and can be repre-sented as U = (cid:20) a b − b ∗ a ∗ (cid:21) , (5)where a and b are the complex-valued Cayley-Klein pa-rameters satisfying | a | + | b | = 1. A traceless HermitianHamiltonian has the form ˆ H ( t ) = ~ [Ω( t ) cos( φ )ˆ σ x +Ω( t ) sin( φ )ˆ σ y + ∆ˆ σ z ], where Ω( t ) (assumed real and pos-itive) is the Rabi frequency quantifying the coupling, φ is its phase, and ∆ is the field-system detuning.On exact resonance (∆ = 0) and for φ = 0, we have a = cos( A / b = − i sin( A / A is the temporalpulse area A = R t f t i Ω( t )d t . For a system starting in state | i , the single-pulse transition probability is p = | b | =sin ( A / A = θ ǫ = θ (1 + ǫ ) produces the propagator ˆ R ( θ ǫ ) = e i [ θ (1+ ǫ ) / σ y =ˆ R ( θ )[1+ O ( ǫ )], i.e. it is accurate up to zeroth order O ( ǫ )in the pulse area error ǫ . Our approach is to replacethe single θ pulse with a composite sequence of pulses ofappropriate pulse areas and phases, such that the overallpropagator produces the rotation gate (1) with an errorof higher order, i.e. ˆ R ( θ )[1 + O ( ǫ n +1 )]. Then we say thatthe corresponding composite rotation gate is accurate upto, and including, order O ( ǫ n ). B. Derivation
The derivation of the composite rotation gates is donein the following manner. A phase shift φ imposed onthe driving field, Ω( t ) → Ω( t ) e i φ , is imprinted onto thepropagator (5) as U φ = (cid:20) a be i φ − b ∗ e − i φ a ∗ (cid:21) . (6)A train of N pulses, each with area A k and phase φ k (applied from left to right),( A ) φ ( A ) φ ( A ) φ · · · ( A N ) φ N , (7)produces the propagator (acting, as usual, from right toleft) U = U φ N ( A N ) · · · U φ ( A ) U φ ( A ) U φ ( A ) . (8)Let us assume that the nominal (i.e. for zero error) pulseareas A k have a systematic error ǫ , i.e. A k → A k (1+ ǫ ). Ifall nominal pulse areas are the same, as it is the case formany composite sequences, this is the natural assump-tion because the apparatus will produce possibly imper-fect but identical pulses. If the pulse areas are different,this is also a reasonable assumption in many cases. Forexample, if a trapped ion is addressed by an imperfectlypointed laser beam then it will “see” the same system-atic deviation from the perfect field amplitude (and hencepulse area) for any chosen target pulse area. Atoms inatomic clouds in magnetooptical or dipole traps or ionsin doped solids (e.g. for optical memories) addressed byelectromagnetic fields offer another example: they will“see” different field amplitude due to spatial inhomogene-ity depending on their position in the sample, but thisfield amplitude will deviate from the optimal one by thesame relative systematic error ǫ regardless of the value of the optimal amplitude if the atoms do not move muchduring the duration of the composite sequence.Under the assumption of a single systematic pulse areaerror ǫ , we can expand the composite propagator (8) ina Taylor series versus ǫ . Because of the SU(2) symmetryof the overall propagator, it suffices to expand only twoof its elements, say U ( ǫ ) and U ( ǫ ). We set their zero-error values to the target values, U (0) = cos( θ/ , U (0) = sin( θ/ , (9)and we set as many of their derivatives with respect to ǫ , in the increasing order, as possible, U ( m )11 (0) = 0 , U ( m )12 (0) = 0 , ( m = 1 , , . . . , n ) , (10)where U ( m ) jl = ∂ mǫ U jl denotes the m th derivative of U jl with respect to ǫ . The largest derivative order n satisfy-ing Eqs. (10) gives the order of the error compensation O ( ǫ n ).Equations (9) and (10) generate a system of 2( n + 1)algebraic equations for the nominal pulse areas A k andthe composite phases φ k ( k = 1 , , . . . , N ). The equa-tions are complex-valued and generally we have to solve4( n + 1) equations with the 2 N free parameters (nominalpulse areas and phases). Because of the normalizationcondition |U | + |U | = 1, an error compensation oforder n requires a composite sequence of N = 2 n + 1pulses (or N = 2 n in some lucky cases).As stated above, the derivation of the composite se-quences requires the solution of Eqs. (9) and (10). For asmall number of pulses (up to about five), the set of equa-tions can be solved analytically. For longer sequences,Eqs. (9) and the first two equations ( n = 1) of Eqs. (10)can still be solved analytically, but the higher orders inEqs. (10) they are solved numerically. We do this byusing standard routines in Mathematica © . C. Quantum gate fidelity
If Eqs. (9) and (10) are satisfied, then the overall prop-agator can be written as U ( ǫ ) = R ( θ ) + O ( ǫ n +1 ) , (11)with R ( θ ) = U (0). Then the Frobenius distance fidelity , F = 1 − k U ( ǫ ) − R ( θ ) k = 1 − r X j,k =1 |U jk − R jk | , (12)is of the same error order O ( ǫ n ) as the propagator, F =1 − O ( ǫ n +1 ). As shown by Jones and co-workers [3] forthe composite X gates, the trace fidelity , F T = Tr [ U ( ǫ ) R ( θ ) † ] , (13)has a factor of 2 higher error order O ( ǫ n ), i.e. F T =1 − O ( ǫ n +1 ). The reason is that in the Frobenius dis-tance, all information of the actual propagator is in-volved, while in the trace distance some of this infor-mation is lost. Therefore, throughout this paper we shalluse the Frobenius distance fidelity (12), which is a muchmore strict and unforgiving to errors fidelity measure;moreover, its error is of the same order as the propaga-tor error.We note here that for variable rotations, Eqs. (9) and(10) have to be satisfied for only one of the propaga-tor elements, say U . This means that with the samenumber of pulses one can achieve a factor of 2 higherorder of error compensation for variable rotations thanfor constant rotations. However, this error compensa-tion applies to the transition probability only, but not tothe propagator phases. For variable rotations the overallpropagator cannot be written in the form of Eq. (11),and consequently, neither of the fidelities (12) or (13) isof the form 1 − O ( ǫ n +1 ). D. Composite pulse sequences
Based on numerical evidence, we consider three typesof composite sequences, one symmetric and two asym-metric. • Each symmetric sequence consists of a sequence of2 n − π pulses, sandwiched by two pulsesof areas α , with symmetrically ordered phases, α φ π φ π φ · · · π φ n − π φ n π φ n − · · · π φ π φ α φ . (14)These sequences generalize the three-pulse SCRO-FULOUS sequence [56], which is of this type, tomore than three pulses. • The first type of asymmetric sequences consists ofa sequence of nominal π pulses, preceded (or super-seded) by a pulse of area θ , π φ π φ π φ · · · π φ N − θ φ N . (15)These sequences generalize the five-pulse BB1 se-quence [33], which is of this type, to more than fivepulses. • The second type of asymmetric sequences consistsof a sequence of N − π pulses, preceded(or superseded) by single pulses of areas α and β , α φ π φ π φ · · · π φ N − β φ N . (16)To the best of our knowledge, this type of compositesequences has not been reported in the literaturehitherto.Below we consider these three classes of composite se-quences and test their performance by using the Frobe-nius distance (12). We consider three figures of merit tobe essential. • The most important parameter is the order of errorcompensation O ( ǫ n ). The larger n , the broader thehigh-fidelity range and the larger the errors ǫ , whichcan be compensated. • The second most important parameter is the to-tal pulse area A tot = P Nk =1 |A k | . It determinesthe length of the sequences and hence the speed ofthe gates. Usually, the peak Rabi frequency is lim-ited either by the experimental apparatus or by thequbit properties, e.g., too large Rabi frequency cancause unwanted couplings to other levels or to otherqubits (cross-talk). Therefore, for a fixed peak Rabifrequency, the total pulse area determines the totalduration of the composite sequence. • Another consideration is the number of pulses N inthe sequence. Unless there are issues with the im-plementation of the phase jumps, this argument isof far less importance than the other two. However,if the phase jumps require some time to implementor cannot be implemented with high accuracy, thensequences of fewer pulses are preferable. For thisreason, we often give several different CPs for eacherror order. III. X (NOT) GATE
The X or NOT gate is defined as (cid:20) (cid:21) = ˆ σ x , (17)Because the determinant of this matrix is −
1, it is not ofSU(2) type. Instead, we shall construct the SU(2) gate X = (cid:20) − (cid:21) , (18)which is related to the gate (17) by a phase transfor-mation and it is equivalent to it. The gate (18) is alsoequivalent to the often used gate e i ( π/ σ x = (cid:20) ii (cid:21) , (19)which can be obtained from Eq. (18) by a phase trans-formation too. However, we prefer to use the gate (18)because it is real and also because it is a special case ofthe general rotation gate (1).As it is well known, such a gate can be produced by aresonant pulse of temporal area π . The propagator of a π pulse reads U = (cid:20) cos( π (1 + ǫ ) /
2) sin( π (1 + ǫ ) / − sin( π (1 + ǫ ) /
2) cos( π (1 + ǫ ) / (cid:21) , (20)where ǫ is the pulse area error. The Frobenius distancefidelity (12) reads F = 1 − √ (cid:12)(cid:12)(cid:12) sin πǫ (cid:12)(cid:12)(cid:12) . (21)For comparison, the trace fidelity is F T = 1 − πǫ πǫ . (22)Obviously the error stemming from the Frobenius dis-tance fidelity (21), which is of order O ( ǫ ), is far greaterthan the value of the error stemming from the trace fi-delity (22), which is of order O ( ǫ ), as noted by Jonesand co-workers [56].The three types of composite sequences (14), (15), and(16) coalesce into a single type, a sequence of π pulses.Below we consider these sequences, in the increasing or-der of error compensation. A. First-order error compensation
The careful analysis of Eqs. (9) and (10) shows that theshortest possible CP which can compensate first-ordererrors consists of three pulses, each with a pulse area of π , and symmetric phases, π φ π φ π φ . (23)Solving Eq. (9) along with Eq. (10) for the first deriva-tives gives two solutions for the phases, π π π π π π , (24a) π π π π π π . (24b)These two sequences generate the same propagator andhence the same fidelity.The Frobenius distance and trace distance fidelitiesread F = 1 − I , (25a) F T = 1 − I , (25b)where the Frobenius distance infidelity is I = r (cid:16) πǫ (cid:17) sin πǫ . (26)Obviously, the Frobenius distance infidelity I is of or-der O ( ǫ ) and it is much larger than the trace distanceinfidelity I , which is of order O ( ǫ ).The Frobenius distance fidelity and the trace fidelityare plotted in Figure 1 for X gates produced by a sin-gle pulse and composite sequences of 3 and 5 (see below)pulses. The three-pulse composite X gate (24) producesmuch higher fidelity that the single-pulse X gate. Ob-viously, the trace distance fidelity is much higher thanthe Frobenius distance fidelity: compare the curves withlabels 1 and 1 T ; 3 and 3 T ; 5 and 5 T . In fact, asseen in the figure, the trace distance fidelity for a sin-gle pulse (label 1 T ) almost coincides with the Frobeniusdistance fidelity for the three-pulse composite sequence(label 3). With respect to the quantum computationbenchmark fidelity value of 1 − − , the Frobenius dis-tance fidelity (25a) for the three-pulse composite X gatesof Eqs. (24) remains above this value in the pulse areainterval (0 . π, . π ), i.e. for relative errors up to | ǫ | < . - - ϵ F i d e lit y T T T FIG. 1: Frobenius distance fidelity F (solid) and trace dis-tance fidelity F T (dashed) of composite X gates. The num-bers N on the curves refer to composite sequences X N listedin Table I. (25b) remains above this value in the pulse area interval(0 . π, . π ), i.e. for relative errors up to | ǫ | < . B. Second-order error compensation
For sequences of four pulses, it becomes possible toannul the second-order derivatives in Eq. (10). A numberof solutions exist, some of which are(2 π ) χ π π + χ π π π − χ , (27a) π π + χ (2 π ) χ π π + χ π π , (27b) π π π π + χ (2 π ) χ π π + χ , (27c) π − χ π π π π + χ (2 π ) χ , (27d)where χ = arcsin( ) ≈ . π . The second and third se-quences are related to the BB1 sequence of Wimperis [33].Note that all these sequences have a total nominal pulsearea of 5 π , and can be considered as five-pulse sequencesbecause the effect of (2 π ) χ is the same as π χ π χ .The Frobenius fidelity for all these sequences reads F = 1 − I , with the infidelity I = r πǫ πǫ (cid:12)(cid:12)(cid:12) sin πǫ (cid:12)(cid:12)(cid:12) . (28)Obviously, this fidelity is accurate up to order O ( ǫ ), asthe error is of order O ( ǫ ). The trace fidelity reads F T =1 −I . The trace fidelity is accurate up to order O ( ǫ ), asthe error is of order O ( ǫ ). Obviously, the trace infidelityis much smaller than the Frobenius distance infidelity, asfor the three-pulse composite sequences.The same second-order error compensation, and thesame fidelity, can be obtained by composite sequences of Name Pulses O ( ǫ n ) Phases φ , φ , . . . , φ n (in units π ) High-fidelity errorcorrection rangesingle 1 O ( ǫ ) [0 . π, . π ]X3 3 O ( ǫ ) , [0 . π, . π ]X5 5 O ( ǫ ) 0.0672, 0.3854, 1.1364 [0 , π, . π ]X7 7 O ( ǫ ) 0.2560, 1.6839, 0.5933, 0.8306 [0 . π, . π ]X9 9 O ( ǫ ) 0.3951, 1.2211, 0.7806, 1.9335, 0.4580 [0 . π, . π ]X11 11 O ( ǫ ) 0.2984, 1.8782, 1.1547, 0.0982, 0.6883, 0.8301 [0 . π, . π ]X13 13 O ( ǫ ) 0.8800, 0.6048, 1.4357, 0.9817, 0.0781, 0.5025, 1.8904 [0 . π, . π ]X15 15 O ( ǫ ) 0.5672, 1.4322, 0.9040, 0.2397, 0.9118, 0.5426, 1.6518, 0.1406 [0 . π, . π ]X17 17 O ( ǫ ) 0.3604, 1.1000, 0.7753, 1.6298, 1.2338, 0.2969, 0.6148, 1.9298, 0.4443 [0 . π, . π ]TABLE I: Phases of symmetric composite sequences of N = 2 n + 1 nominal π pulses, which produce the X gate with a pulsearea error compensation up to order O ( ǫ n ). The last column gives the high-fidelity range [ π (1 − ǫ ) , π (1 + ǫ )] of pulse areaerror compensation wherein the Frobenius distance fidelity is above the value 0 . − . five pulses of area π each, π φ π φ π φ π φ π φ . (29)Hence the total pulse area is 5 π , the same as the four-pulse sequences above. Because of the additional phasecompared to the four-pulse sequences, various phasechoices are possible. For example, an asymmetric se-quence of the kind (29) has the phases φ = 0, φ =arcsin (cid:16) √ (cid:17) ≈ . π , φ = π +arcsin (cid:16) √ − (cid:17) ≈ . π , φ = arcsin (cid:16) √ (cid:17) ≈ . π , φ =arcsin (cid:16) −√ (cid:17) ≈ . π .We have derived also the symmetric sequence π φ π φ π φ π φ π φ , (30)with φ = arcsin (cid:16) − p / (cid:17) ≈ . π , φ =arcsin (cid:0) (3 √ − / (cid:1) ≈ . π , φ = 2 φ − φ + π/ ≈ . π . For these five-pulse sequences the Frobeniusinfidelity I is given again by Eq. (28), and the traceinfidelity by I . The respective fidelities are plotted inFig. 1. Obviously, they are much larger than the re-spective fidelities for a single pulse and the three-pulsecomposite sequence (24).The Frobenius distance infidelity (28) remains belowthe quantum computation fidelity threshold 10 − in thepulse area interval (0 . π, . π ), i.e. for relative errorsup to | ǫ | < . I remains above this value in the pulse areainterval (0 . π, . π ), i.e. for relative errors up to | ǫ | < . C. Higher-order error compensation
For composite sequences of more than 5 pulses, theequations for the composite phases quickly become verycumbersome and impossible to solve analytically. Theyrepeat the pattern of the sequences of four and five pulsesabove: the composite sequences of 2 n and 2 n + 1 pulseshave a total pulse area of (2 n + 1) π , with all pulses inthe sequence being nominal π pulses, with the exceptionof one of the pulses in the 2 n -pulse sequence which hasa nominal pulse area of 2 π . Either sequences of 2 n and2 n + 1 pulses produce error compensation of the order O ( ǫ n ) and identical fidelity profiles.The 2 n + 1-pulse sequences have an additional freephase which can be used to make the composite sequence symmetric as in Eq. (14), viz. π φ π φ π φ · · · π φ n − π φ n π φ n − · · · π φ π φ π φ . (31)The propagators generated by the symmetric compositesequences (31) feature two important properties:1. All even-order derivatives U (2 k )11 (0) of the diagonalelements in Eq. (10) vanish, and so do all odd-orderderivatives U (2 k +1)12 (0) of the off-diagonal elements.2. The remaining nonzero derivatives in Eq. (10)are either real or imaginary: U (2 k +1)11 (0) are real,whereas U (2 k )12 (0) are imaginary.Therefore, Eqs. (9) and (10) reduce to a set of n + 1 realtrigonometric equations for n + 1 free phases. There aremultiple solutions for the phases for every (2 n + 1)-pulsecomposite sequence. - - ϵ F i d e lit y - - - - P(cid:0)(cid:1)(cid:2)(cid:3)
Area Error ϵ I n f i d e lit y FIG. 2: Frobenius distance fidelity F (top) and infidelity(bottom) of composite X gates. The infidelity is in logarith-mic scale in order to better visualize the high-fidelity (low-infidelity) range. The numbers N on the curves refer to com-posite sequences X N listed in Table I. Two of the phases can be found analytically. The so-lution of the zeroth-order Eqs. (9) reads φ n +1 = π φ n − φ n − + φ n − − φ n − + · · · + ( − ) n φ ] . (32)Given this relation, the equation U (1)11 (0) = 0 reduces to2 n X k =1 sin(Φ k ) = ( − ) n +1 , (33)withΦ k = 2 k − X j =1 ( − ) j +1 φ j + ( − ) k +1 φ k = 2[ φ − φ + φ + · · · + ( − ) k φ k − ] + ( − ) k +1 φ k , (34)from where we can find φ n . For example, for 3, 5, and 7 pulses we have, respectively,sin( φ ) + sin(2 φ − φ ) = − , (35a)sin( φ ) + sin(2 φ − φ ) + sin(2 φ − φ + φ ) = , (35b)sin( φ ) + sin(2 φ − φ ) + sin(2 φ − φ + φ )+ sin(2 φ − φ + 2 φ − φ ) = − . (35c)From each of these we can find two solutions for the phasewith the largest subscript.The remaining n − φ , φ , . . . , φ n − can bedetermined numerically.We have derived numerically the composite phasesof symmetric sequences of an odd number of pulses,Eq. (31). They are presented in Table I. The fidelityof these composite X gates is plotted in Fig. 2. It is clearfrom the table and the figure that a single pulse has verylittle room for errors as the high-fidelity X gate allowsfor pulses area errors of less than 0.01%. The three-pulsecomposite X gate offers some leeway, with the admissi-ble error of 0.8%. The real pulse area error correctioneffect is achieved with the composite sequences of 5 to9 pulses, for which the high-fidelity range of admissibleerrors increases from 3.6% to 11.7%. Quite remarkably,errors of up to 25% can be eliminated, and ultrahigh fi-delity maintained, with the 17-pulse composite X gate.Note that these error ranges are calculated by using therather tough Frobenius distance fidelity (12). Had we usethe much more relaxed trace distance fidelity (13), theseranges would be much broader, see the numbers for 1, 3and 5 pulses above.That said, very long sequences are barely practical be-cause the gate is much slower. Moreover, it is hard toimagine a quantum computer operating with 25% pulsearea error. Therefore, the composite sequences of 5, 7and 9 pulses seems to offer the best fidelity-to-speed ra-tio. IV. HADAMARD GATE
We shall use the following form of the Hadamard gate(known as pseudo-Hadamard form), H = R y ( π/
2) = e i ( π/ σ y = √ (cid:20) − (cid:21) . (36)It is SU(2) symmetric and it is equivalent to the morecommon Walsh-Hadamard form √ (cid:20) − (cid:21) , (37)which is not SU(2) symmetric. The gate (36) is equiva-lent to the often used SU(2) symmetric gate (known asthe Splitter gate) H x = e i ( π/ σ x = √ (cid:20) ii (cid:21) , (38)which is related to it by a phase transformation.The Hadamard gate can be generated by an ideal reso-nant π/ A. First-order error correction
The shortest pulse sequence that can provide a first-order error compensated Hadamard gate consists of threepulses, α φ π φ α φ . (39)Equations (9) result in the equations − sin( α ) cos( φ − φ ) = √ , (40a) e − iφ [sin( φ − φ ) − i cos( α ) cos( φ − φ )] = √ . (40b)The first-derivatives of Eqs. (10) are annulled by the sin-gle equation 2 α cos( φ − φ ) + 1 = 0 . (40c)From Eqs. (40a) and (40c) we findsin αα = √ . (41)Therefore the value of the pulse area α is given by aninverse sinc function of √
2, which gives α ≈ . π .Given α , we can find φ − φ from Eq. (40a) or (40c),and then φ from √ φ − φ ) = cos( φ ) , (42)which is the real part of Eq. (40b). The values are φ ≈ . π and φ ≈ . π . Therefore, this compositepulse reads(0 . π ) . π π . π (0 . π ) . π . (43)In term of degrees, it reads 115 ◦ ◦ ◦ ◦ ◦ ◦ .This composite sequence is related to the well-known se-quence SCROFULOUS [56]. B. Second-order error correction
Second-order error compensation is obtained by a com-posite sequence of at least 4 pulses. A popular CP is theBB1 pulse of Wimperis [33],BB1 = ( π/ π χ (2 π ) χ π χ , (44) which produces the gate (38), with a total pulse areaof 4 . π . It can be viewed as identical to the five-pulsesequence ( π/ π χ π χ π χ π χ . (45)We have derived a different, asymmetric four-pulseCP, H4a = α φ π φ π φ β φ , (46)where α = 0 . π , β = 1 . π , φ = 1 . π , φ =0 . π , φ = 1 . π , φ = 0 . π . This pulse has atotal area of about 4 . π , i.e. it is faster than the BB1pulse. It is accurate up to the same order O ( ǫ ) andproduces essentially the same fidelity profile as BB1.We have also derived a five-pulse composite Hadamardgate by using the symmetric sequenceH5s = α φ π φ π φ π φ α φ , (47)with α = 0 . π , φ = 1 . π , φ = 0 . π , φ =1 . π . It delivers again the second-order error com-pensation O ( ǫ ), however, with a total pulse area of justabout 3 . π . Therefore it is considerably faster than theBB1 pulse, by over 13%, while having a similar perfor-mance. C. Higher-order error correction
Similarly to the second order, the third-order errorcompensation is obtained in several different manners,requiring at least 6 pulses. The 6-pulse sequence withthe minimal pulse area of about 5 . π readsH6a = α φ π φ π φ π φ π φ β φ , (48)with α = 0 . π , β = 1 . π , and the phases givenin Table II. The same error correction order is achievedwith the symmetric seven-pulse sequenceH7s = α φ π φ π φ π φ π φ π φ α φ , (49)with α = 0 . π , and the phases given in Table II. Itproduces the same fidelity profile as the 6-pulse sequencebut it is a little faster as its pulse area is about 5 . π .Another seven-pulse composite sequence is built similarlyto the BB1 sequence (44),H7w = ( π/ π/ π φ π φ π φ π φ π φ π φ , (50)with the phases given in Table II. It achieves the sameerror order compensation O ( ǫ ), however, with a largertotal pulse area of 6 . π compared to the previous twoCPs. Fourth-order error compensation is obtained by atleast 8 pulses. The 8-pulse sequence with the minimalpulse area of about 7 . π readsH8a = α φ π φ π φ π φ π φ π φ π φ β φ , (51) Symmetric sequences α φ π φ · · · π φ n π φ n +1 π φ n · · · π φ α φ notation N O ( ǫ n ) α φ , φ , . . . , φ n (in units π ) A tot RangeH3s 3 O ( ǫ ) 0.6399 1.8442, 1.0587 2 . π [0 . , . π H5s 5 O ( ǫ ) 0.45 1.9494, 0.5106, 1.3179 3 . π [0 . , . π H7s 7 O ( ǫ ) 0.2769 1.6803, 0.2724, 0.8255, 1.6624 5 . π [0 . , . π H9s 9 O ( ǫ ) 0.2947 0.2711, 1.1069, 1.5283, 0.1283, 0.9884 7 . π [0 . , . π H11s 11 O ( ǫ ) 0.2985 1.7377, 0.1651, 0.9147, 0.1510, 0.9331, 1.6415 9 . π [0 . , . π H13s 13 O ( ǫ ) 0.5065 0.0065, 1.7755, 0.7155, 0.5188, 0.2662, 1.2251, 1.3189 12 . π [0 . , . π H15s 15 O ( ǫ ) 0.3213 1.2316, 0.9204, 0.2043, 1.9199, 0.8910, 0.7381, 1.9612, 1.3649 13 . π [0 . , . π Asymmetric sequences ( π/ φ π φ π φ · · · π φ N − π φ N notation N O ( ǫ n ) α , β φ , φ , . . . , φ N (in units π ) A tot RangeH5w 5 O ( ǫ ) 0.5, 1.0 0.5, 1.0399, 0.1197, 0.1197, 1.0399 4 . π [0 . , . π H7w 7 O ( ǫ ) 0.5, 1.0 0.5, 1.4581, 0.7153, 0.1495, 1.3738, 0.2568, 0.7752 6 . π [0 . , . π H9w 9 O ( ǫ ) 0.5, 1.0 0.5, 1.1990, 0.3622, 0.6007, 1.6773, 1.7779, 0.6773, 04124, 1.2732 8 . π [0 . , . π H11w 11 O ( ǫ ) 0.5, 1.0 0.5, 0.7807, 0.1769, 1.4678, 0.1085, 1.0174, 0.2988, 0.8883,1.2697, 0.3773, 1.6775 10 . π [0 . , . π H13w 13 O ( ǫ ) 0.5, 1.0 0.5, 1.3795, 0.5435, 0.5111, 1.3032, 0.4295, 1.7578, 1.4181,0.3340, 0.4403, 1.7563, 0.6708, 1.1544 12 . π [0 . , . π Asymmetric sequences α φ π φ π φ · · · π φ N − β φ N notation N O ( ǫ n ) α , β φ , φ , . . . , φ N (in units π ) A tot RangeH4a 4 O ( ǫ ) 0.7821, 1.3914 1.8226, 0.6492, 1.2131, 0.3071 4 . π [0 . , . π H6a 6 O ( ǫ ) 0.5917, 1.1305 1.5943, 0.2860, 0.8435, 1.6553, 0.7962, 0.2523 5 . π [0 . , . π H8a 8 O ( ǫ ) 0.4954, 0.9028 1.5971, 0.7674, 0.5721, 1.8487, 1.0592, 1.9512, 0.3824, 0.9846 7 . π [0 . , . π H10a 10 O ( ǫ ) 0.6041, 1.1819 1.3480, 0.9259, 0.0292, 0.7288, 0.0996, 1.3909, 0.0183, 0.9322,0.2169, 0.7975 9 . π [0 . , . π H12a 12 O ( ǫ ) 0.4168, 0.8841 1.5817, 1.1160, 0.3751, 0.9583, 0.1333, 1.9445, 1.0381, 1.6293,0.4845, 0.0046, 0.8278, 0.7416 11 . π [0 . , . π TABLE II: Phases of three types of composite sequences, which produce the Hadamard gate with a pulse area error compensationup to order O ( ǫ n ). The total pulse area A tot and the high-fidelity range [ π − ǫ , π + ǫ ] wherein the Frobenius distance infidelityremains below 10 − are listed in the last two columns. with α = 0 . π , β = 0 . π , and the phases are givenin Table II. The same error correction order is achievedwith the symmetric nine-pulse sequenceH9s = α φ π φ π φ π φ π φ π φ π φ π φ α φ , (52)with α = 0 . . π . The BB1-like nine-pulse compositesequence,H9w = ( π/ π/ π φ π φ π φ π φ π φ π φ π φ π φ , (53)with the phases in Table II, achieves the same fourth-order error compensation O ( ǫ ), however, with thelargest total pulse area of 8 . π compared to the previ-ous two CPs.The same pattern is repeated for the longer pulse se-quences presented in Table II: for the same order of pulsearea error compensation, the fastest sequences, with thesmallest total pulse area are either the asymmetric H N aor symmetric H N s sequences, and the BB1-like sequencesH N w are the slowest ones.The fidelity and the infidelity of the compositeHadamard gates of up to seventh-order error compen-sation are plotted in Fig. 3. Obviously, as the number ofpulses in the composite sequences, and hence the com-pensated error order, increase the fidelity and infidelityprofiles improve and get broader. V. GENERAL ROTATION GATEA. First-order error correction
The shortest pulse sequence that can provide a first-order error compensation, as for the X and Hadamardgates, consists of three pulses, α φ π φ α φ . (54)Equations (9) result in the equations − sin( α ) cos( φ − φ ) = cos( θ/ , (55a) e − iφ [sin( φ − φ ) − i cos( α ) cos( φ − φ )] = sin( θ/ . (55b)The first-derivatives of Eqs. (10) are annulled by the sin-gle equation 2 α cos( φ − φ ) + 1 = 0 . (55c)From Eqs. (55a) and (55c) we findsin( α ) α = 2 cos( θ/ . (56)Therefore the value of the pulse area α is given by aninverse sinc function of 2 cos( θ/ α , we can find φ − φ from Eq. (55a) or (55c), and then φ fromsin( φ − φ ) = sin( θ/
2) cos( φ ) , (57)0 O ( ǫ ) 5 pulses, O ( ǫ ) 7 pulses, O ( ǫ ) 9 pulses, O ( ǫ ) α φ π φ α φ α φ π φ π φ π φ α φ α φ π φ π φ π φ π φ π φ α φ α φ π φ π φ π φ π φ π φ π φ π φ α φ θ α ; φ , φ α ; φ , φ , φ α ; φ , φ , φ , φ α ; φ , φ , φ , φ , φ TABLE III: Pulse area α and phases of composite pulse sequences which produce rotation gates of angle θ . The area α and allphases are given in units π . The case of θ = π repeats the symmetric Hadamard gates already presented in Sec. IV; they aregiven here for the sake of comparison and completeness. - - Pulse Area Error ϵ F i d e lit y - - - - Pulse Area Error ϵ I n f i d e lit y FIG. 3: Frobenius distance fidelity (top) and infidelity (bot-tom) of composite Hadamard gates produced by using thesymmetric composite sequences H N s from Table II. which is obtained from Eq. (55b).This composite sequence is related to the SCROFU-LOUS composite pulse [56], as mentioned above. Thevalues of the pulse area and the composite phases are given in Table III. B. More than three pulses
The five-pulse sequence, α φ π φ π φ π φ α φ , (58)provides a second-order error compensation. The se-quences with 7, 9, etc. pulses have the same structureand deliver an error compensation of order 3, 4, etc. Gen-erally, a 2 n + 1-pulse symmetric sequence of this struc-ture delivers an error compensation up to order O ( ǫ n ).Unfortunately, analytic expressions for the composite pa-rameters for more than three pulses are hard to obtain, ifpossible at all. Hence we have derived them numericallyand their values are listed in Table III. The fidelity ofthese sequences behave similarly to the ones for the Xand Hadamard gates. VI. COMMENTS AND CONCLUSIONS
In this work we presented a number of composite pulsesequences for three basic quantum gates — the X gate,the Hadamard gate and arbitrary rotation gates. Thecomposite sequences contain up to 17 pulses and cancompensate up to eight orders of experimental errors inthe pulse amplitude and duration. The short compositesequences are calculated analytically and the longer onesnumerically.Three classes of composite sequences have been de-rived — one symmetric and two asymmetric. For the Xgate, the three classes coalesce into a single set of sym-metric sequences of nominal π pulses presented in Ta-ble I. For the Hadamard gate, cf. Table II, two of theclasses contain as their lowest members two well-known1composite sequences: the three-pulse symmetric SCRO-FULOUS pulse [56] and the four-pulse asymmetric BB1pulse [33], which compensate first and second-order pulsearea errors, respectively. The third, asymmetric classof composite sequences, does not contain members pub-lished before. All three classes produce essentially iden-tical fidelity profiles for the same order of error compen-sation. In general, the SCROFULOUS-like symmetricsequences H N s and the asymmetric sequences H N a re-quire the least total pulse area and hence are the fastest,whereas the asymmetric BB1-like sequences H N w are theslowest. For the general rotation gates, the three classesbehave similarly, although we have presented only thesymmetric sequences in Table III for the sake of brevity.The composite rotations derived here outperform theexisting composite rotations in terms of either speed,or accuracy, or both. Although we could not improvethe first-order SCROFULOUS sequence, we have de-rived second-order composite sequences which are faster(by over 13%) than the famous BB1 sequence [33]: oursecond-order error compensated Hadamard gate has a to-tal nominal pulse area of about 3 . π , which is substantialimprovement over the BB1 pulse, which delivers the sameerror order with a total pulse area of 4 . π [33]. The longercomposite sequences are derived by brute numerics andthey are much shorter than previous sequences with thesame order of error compensation obtained by nestingand concatenation of short sequences. For example, our n th order error-compensated X gates are constructed by2 n + 1 nominal π pulses, which is much shorter than theconcatenated composite sequences. For example, the 5thorder error compensation is produced by a concatenated15-pulse sequence, whereas we achieve this by an 11-pulsesequence. Similar scaling applies to the Hadamard andthe rotation gates.The results presented in this work demonstrate theremarkable flexibility of composite pulses accompaniedby extreme accuracy and robustness to errors — threefeatures that cannot be achieved together by any othercoherent control technique. We expect these compos-ite sequences, in particular the X and Hadamard gates,to be very useful quantum control tools in quantum in-formation applications because they provide a variety ofoptions to find the optimal balance between ultrahigh fi-delity, error range and speed, which may be different indifferent physical systems. Acknowledgments
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