Single qubit gates in frequency-crowded transmon systems
aa r X i v : . [ qu a n t - ph ] J un Single qubit gates in frequency-crowded transmon systems
R. Schutjens,
1, 2
F. Abu Dagga, D. J. Egger, and F. K. Wilhelm
2, 3 Quantum Transport, Delft University of Technology, 2628 CJ Delft, The Netherlands Theoretical Physics, Universit¨at des Saarlandes, 66123 Saarbr¨ucken, Germany IQC and Department of Physics and Astronomy,University of Waterloo, Ontario N2L 3G1, Canada (Dated: July 24, 2018)Recent experimental work on superconducting transmon qubits in 3D cavities show that theircoherence times are increased by an order of magnitude compared to their 2D cavity counterparts.However to take advantage of these coherence times while scaling up the number of qubits it isadvantageous to address individual qubits which are all coupled to the same 3D cavity fields. Thechallenge in controlling this system comes from spectral crowding, where leakage transition of qubitsare close to computational transitions in other. Here it is shown that fast pulses are possible whichaddress single qubits using two quadrature control of the pulse envelope while the DRAG method ofRefs. [1, 2] alone only gives marginal improvements over the conventional Gaussian pulse shape. Onthe other hand, a first order result using the Magnus expansion gives a fast analytical pulse shapewhich gives a high fidelity gate for a specific gate time, up to a phase factor on the second qubit.Further numerical analysis corroborates these results and yields to even faster gates, showing thatleakage state anharmonicity does not provide a fundamental quantum speed limit.
I. INTRODUCTION
Superconducting qubits are a promising candidate forthe realization of a quantum computer [3–7], owing inlarge parts to the success of circuit QED (CQED), wherethose qubits are coupled to microwave resonators [8–10].There is a multitude of designs of such qubits [4].A key challenge for implementing quantum computingin the solid state is decoherence from uncontrolled de-grees of freedom. Decoherence sources range from theelectromagnetic environment [11] to sources inherent tothe material [12]. Remarkably, many of the materialsources could be mitigated by changes in the circuit lay-out such as the optimum working point first embodied inthe Quantronium [13–16] and later in the Transmon [11]and the 3D-Transmon [17, 18]. Coherence times havebeen improved by going from the two dimensional im-plementation of a qubit interacting with a stripline res-onator [8] to a three dimensional system [17, 18]. Inthe latter, a single Josephson junction transmon qubit[11, 19] is placed inside a 3D cavity and addressed withthe surrounding microwave field. What is common tothese approaches is the trade-off of coherence against con-trol flexibility and ultimately operation speed. While thishas been studied in single Quantronium [16] the precisetrade-off is not fully understood in samples containingmultiple qubits let alone multiple 3D transmons.The gain in coherence times comes at a cost in control-lability. This is strongly felt when more than one qubitis in the cavity. To create single qubit operations eachqubit must be addressed individually requiring them tohave significantly different energy splitting between theground and first excited state. Spectral crowding refersto transitions coming too close to address them individ-ually. Now with the limited control, even if the logicaltransitions are well-spaced, crowding can occur betweenlogical and leakage transition, e.g., if the logical tran- sition of first qubit is close in frequency to the leakagetransition, the transition between a computational anda non-computational state, of the second qubit. Thuswhen performing, e.g., an ˆ X gate on first qubit leakageto second qubit’s | i state will occure. Although high fi-delity gates have been demonstrated with single junctiontransmsons in the 2D architecture [20] spectral crowdingwill limit the gate fidelity in 3D architectures. In orderto mitigate spectral overlap, the Derivative Removal byAdiabatic Gate (DRAG) technique has been developed[1, 2]. We will apply this technique to the problem athand and show that on its own it is of limited success.We will then combine DRAG with sideband drive to showa possibility to do these single-qubit gates fast.In this work we thus address the issue of spectralcrowding with optimal control theory methods. To betterillustrate the problem and show the effectiveness of theanalytical pulses we introduce specific gate fidelity func-tions in section III. In section IV we demonstrate thelimitations of the DRAG technique alone for this prob-lem. We then present an analytical pulse, found throughthe Magnus expansion [21], capable of minimizing leak-age out of the computational subspace of both qubits insection V. We then, in section VI, show pulses obtainednumerically that show similar characteristic but, with ad-ditional ingredients, improved fidelities. II. SYSTEM
Optimized superconducting qubits such as 3D trans-mons are well described by weakly anharmonic oscilla-tors [1, 22]. A realistic model of the qubit has to take atleast one extra non-computational level (a leakage level )into account [23–25]. This is reflected in the followingHamiltonian for two superconducting transmon qubitsin a common 3D cavityˆ H ( t ) = ˆ H + ˆ H C ( t )= X k =1 h ω k ˆ n k + ∆ k ˆΠ ( k )2 i + Ω ( t ) X j =1 h λ (1) j ˆ σ x (1) j,j − + λ (2) j ˆ σ x (2) j,j − i . (1)The 0 ↔ k are, respectively, ω k and ˆ n k = P j j | j i h j | ( k ) .We call the transition from the excited state | i to theextra state | i the leakage transition. It is detuned from ω k by the anharmonicity ∆ k . In the reminder of thiswork we assume ∆ = ∆ = ∆. The projectors on theenergy levels of transmon k are ˆΠ ( k ) k = | j i h j | ( k ) . Theterms coupling adjacent energy levels of qubit k areˆ σ x ( k ) j,j − = | j i h j − | ( k ) + | j − i h j | ( k ) and ˆ σ y ( k ) j,j − = i | j i h j − | ( k ) − i | j − i h j | ( k ) . Ω( t ) is the drive field and is applied simultaneously toboth qubits. The strength at which Ω( t ) drives the 1 ↔ ↔ λ ( k ) j . Table Ishow the variables and numerical values used in simula-tions. TABLE I. System parameters as shown in equation (1).Qubit 1 Qubit 2 ω k / π .
508 5 . / π − −
350 MHz λ ( k )1 λ ( k )2 √ √ Qubits are usually addressed by frequency selectionthrough pulses tuned to the respective qubit level split-ting. This is necessary whenever the control field cannotbe selectively focused on individual qubits as is the casefor multiple 3D transmons in the same cavity. An even-tual implementation of a quantum computer will consistof many such qubits, probably a whole register in onecavity. The problem to distinguish different qubits canthus be seen as a problem of spectral crowding. In trans-mon systems this can lead to the 0 ↔ ↔ δ . With δ/ π = 45MHz, the leakage These values were suggested to describe an experiment by LeoDiCarlo transition of qubit two is closer to the driving fields fre-quency than the leakage transition of qubit one detunedby ∆ / π = −
350 MHz. The situation is depicted in Fig.1.
Qubit 2 ω ω + δ ∆Qubit 1 ω d ω ∆ FIG. 1. Level diagram of the two qubits. The driving fieldis set to have the same frequency as the 0 ↔ δ with0 ↔ The second term in equation (1) is the control Hamil-tonian, described as a semiclassical dipolar interactionbetween the qubits and the classical cavity fieldΩ ( t ) = Ω X ( t ) cos ( ω d t ) + Ω Y ( t ) sin ( ω d t ) . (2)Both quadrature envelopes can be modulated separately.In the reminder of this work, we assume resonance be-tween the drive and qubit 1, i.e. ω d = ω . Single quadra-ture pulses employ Gaussian shapes Ω g due to their lim-ited bandwidth [2]. To remove fast oscillating terms wemove to another reference frame and invoke the rotatingwave approximation (RWA). The transformation into anappropriate frame is accomplished by the time-dependentunitary ˆ R that acts on the Hamiltonian asˆ H R = ˆ R ˆ H ˆ R † + i ˙ˆ R ˆ R † . (3)Here, ˆ R ( t ) = (cid:16)P j e − iω (1) j t ˆΠ (1) j (cid:17) ⊗ (cid:16)P j e − iω (2) j t ˆΠ (2) j (cid:17) .Transformations into this type of frame can lead to eitherthe rotating frame with respect to the drive ω d or the in-teraction frame by choosing ω ( l ) j = jω d , ω ( l ) j = jω ( l ) +∆ ( l ) j respectively. Here, we choose the former. In the rotat-ing frame, we use the RWA to neglect the fast oscillatingterms such as ± ω d , the system’s original Hamiltoniangiven by (1), isˆ H R = ∆ ˆΠ (1)2 + ( δ − ∆) ˆΠ (2)1 + δ ˆΠ (2)2 + Ω X ( t )2 X j =1 h λ (1) j ˆ σ x (1) j,j − + λ (2) j ˆ σ x (2) j,j − i + Ω Y ( t )2 X j =1 h λ (1) j ˆ σ y (1) j,j − + λ (2) j ˆ σ y (2) j,j − i . (4) III. SINGLE QUBIT GATES
We aim at applying, up to a global phase φ , a gate onthe first qubit without affecting the second oneˆ U F = e iφ ˆ U (1) ⊗ . (5)Unless otherwise specified ˆ U (1) is an ˆ X -gate. A specificcontrol pulse of duration t g results in a final gate givenby ˆ U ( t g ). The fidelity with which a control pulse meetsthe target gate is measured byΦ = 1 d (cid:12)(cid:12)(cid:12) Tr h ˆ U F † ˆ U ( t g ) i(cid:12)(cid:12)(cid:12) , (6)where d is the dimension of the Hilbert space of the sys-tem. The trace is taken over the computational subspaceconsisting of {| i , | i , | i , | i} . This takes leakageinto account since leaving this subspace diminishes thematrix elements of the projected unitary [2, 26].We will also investigate single-qubit gates that shiftthe phase of the second qubit. Such gates can be mademore efficiently and we later show how to correct thephase. Such gates can be studies using the reduced fi-delity functionsΦ |∗ ,i i = 12 (cid:12)(cid:12)(cid:12) Tr {| ,i i , | ,i i} h ˆ U F † ˆ U ( t g ) i(cid:12)(cid:12)(cid:12) . (7)The trace is taken over states where the second qubit isexclusively in the | i or | i . A gate producing a goodΦ |∗ ,i i has qubit 2 starting and ending in state | i i . Theaverage of the Φ |∗ ,i i ’s gives a fidelity function insensitiveto the phase of the second qubitΦ avg = 12 (cid:0) Φ |∗ , i + Φ |∗ , i (cid:1) . (8)In other words, Φ avg is maximal if ˆ U ( t g ) (in the compu-tational subspace of the two qubits) has the formˆ U ( t g ) = e iα " ⊗ " e i ( γ − α ) . (9)For a given gate time the phase error can be calculatedand subsequently corrected as this gate is not entangling.In fact, an entangling gate would be detected by deteri-orating Φ avg and given that the qubit controls are localand the two qubits are uncoupled, no entanglement isgenerated. IV. APPLYING DRAG
The DRAG method [1, 2, 27] negates leakage to the | i state with a two quadrature drive. Here we show thatthis method does not provide a sizable improvement overa single Gaussian envelope. We transform ˆ H R a second time along the lines of eq. (3) using the transformationmatrixˆ V ( t ) = exp − i Ω X β X j =1 h λ (1) j ˆ σ y (1) j,j − + λ (2) j ˆ σ y (2) j,j − i . (10)This is the two-qubit version of the DRAG transforma-tion [2, 27]. The parameter β selects which transition issuppressed. A first order expansion in η = Ω X ( t ) /β ≪ H V = ˆ H diag + ˆ H Y + ˆ H (1)X + ˆ H (2)X (11)The diagonal terms are of O ( η ), hence ˆ H diag is neglectedon our level of approximation. ˆ H Y contains a term gen-erated by the time-derivative in eq. (3) as well as the Y driveˆ H Y = Ω Y ( t )2 + ˙Ω X ( t )2 β ! X j =1 h λ (1) j ˆ σ y (1) j,j − + λ (2) j ˆ σ y (2) j,j − i . (12)ˆ H Y can be suppressed by choosing Ω Y ( t ) = − ˙Ω X ( t ) /β .This is the essence of the DRAG method [1]. The lasttwo terms respectively drive the first and second qubitaccording toˆ H (1)X ( t ) = Ω X ( t )ˆ σ x (1)10 + λ β − ∆2 β Ω X ( t )ˆ σ x (1)21 + λ ∆8 β Ω X ( t ) ˆ σ x (1)20 , ˆ H (2)X ( t ) = η β − δ + ∆2 β Ω X ( t )ˆ σ x (2)10 + ηλ β − δ β Ω X ( t )ˆ σ x (2)21 + η λ ∆8 β Ω X ( t )ˆ σ x (2)20 . Depending on the value of β a specific off resonanttransition can be suppressed. If β = δ the second qubitleakage transition is removed. However, since δ < ∆(by a factor > Y becomes large and strongly drives theother leakage transitions, i.e., introduces errors of a sizecomparable to what it is suppressing. Note, that for fastpulses with β = δ the perturbation expansion in [1, 2, 27].naturally breaks down. Selecting β = ∆ suppresses theleakage transition of the first qubit, but does not solvethe leading spectral crowding issue based on the small-ness of δ . We are explicitly highlighting this in figure2. It shows the fidelity, as a function of gate time, forthe single quadrature Gaussian (thin lines) and DRAG(thick lines) solutions with β = ∆.The difference between the fidelity function Φ, eq. (6),and the special fidelity functions Φ |∗ ,i i and Φ avg , eq. (7)and (8), show that while it is difficult to perform an X gate on qubit 1 without affecting qubit 2, we can imple-ment a high fidelity ˆ X gate with an additional phase shifton the other qubit for t g > -4 -3 -2 -1
10 20 30 40 50 60 70 80 A c h i e v e d E rr o r Gate Time [ns] ΦΦ |*,0> Φ |*,1> Φ avg FIG. 2. Error for a single control with a Gaussian pulseshape as a function of gate time and a single quadrature (thinlines) and for the DRAG method with β = ∆ (thick lines).The DRAG method gives only marginal improvements overthe single quadrature Gaussian pulse shape for Φ avg which isslightly lower at the dip around 42 ns. The DRAG solutionshown here is the optimal from picking β ǫ { ∆ , δ, δ − ∆ } . V. MAGNUS EXPANSION
Here we show how to find an improved pulse capa-ble of performing the desired gate faster and with betterfidelity. The full effect of system and Hamiltonian is de-scribed by the time evolution operatorˆ U ( t g ) = T exp − i t g Z d t ˆ H ( t ) (13)where T is the time-ordering operator. This can in gen-eral not be computed in closed form even for driven two-state systems with notable exceptions [28]. Still beingunitary, the solution of equation (13) can be written asthe exponential of an Hermitian matrix [21]. An expan-sion in this effective Hamiltonian gives the Magnus ex-pansion U ( t g ) = e − i P k ˆΘ k ( t g ) . (14)The equation above still requires exponentiating a ma-trix. However the absence of time ordering considerablysimplifies the derivation of an explicit expression for ˆ U .The Magnus expansion is asymptotic. Here, it convergesquickly as nested integrals lead to cancellations of fastoscillating terms. The constraints on the controls set bythe zeroth order in the expansion will thus be most im-portant. The first terms in the expansion are given by [21]ˆΘ ( t g ) = t g Z d t ˆ H ( t ) , ˆΘ ( t g ) = − i t g Z d t t Z d t h ˆ H ( t ) , ˆ H ( t ) i . (15)Here h ˆ H ( t ) , ˆ H ( t ) i is the commutator of the Hamilto-nian at different times. Higher order terms in the expan-sion can be worked out as nested commutators similar asthose shown above.We start with the system in the interaction frame (thetransformation is given in section II)ˆ H I = Ω C X j =1 h λ (1) j e − iδ (1) j t | j − i h j | (1) + λ (2) j e − iδ (2) j t | j − i h j | (2) i + h . c . (16)Here we have combined Ω C = Ω X + i Ω Y and set δ (1)1 =0 , δ (1)2 = ∆ , δ (2)1 = δ − ∆, and δ (2)2 = δ . In the inter-action frame, the Hamiltonian is purely off-diagonal andthe desired gate is changed by a phase on the | i stateof the second qubit. This phase is known since any uni-tary transformation ˆ V ( t ), transforms the time evolutionfollowing ˆ U V ( t g ) = ˆ V ( t g ) ˆ U ( t g ) ˆ V † (0). In equation (17) U F transforms in this way. If the zeroth order term is toimplement the gate, the control problem becomesˆ U F = e − i ˆΘ = e − i R tg d t ˆ H I ( t ) . (17)As an aside, this highlights why Θ /t g is often calledthe average Hamiltonian and P k ˆΘ k ( t g ) /t g the effectiveHamiltonian in NMR [21]. This and the form ˆ H I imposesrestrictions on the control Ω C Z t g d t Ω C = π (18)12 Z t g d t e − i ∆ t Ω C = 0 (19)12 Z t g d t e − iδt Ω C = 0 (20)12 Z t g d t e − i ( δ − ∆) t Ω C = 0 (21)These constraints are the Fourier transforms of the con-trol evaluated at the different detunings in the systemas is familiar from spectroscopy at weak drive [21, 29–31]- but here derived under intermediate-to-strong driveconditions. They state that the control should containno power at the off resonant frequencies. If Ω C is palin-dromic the complex conjugated equations are also sat-isfied. If equations (18-21) are met, the final unitaryevolution will be e iφ ˆ σ x ⊗ .So that the zeroth order implements the gate, higherorder terms have to be zero. Here is an example of thefirst order term ˆΘ . It only gives extra terms on the diag-onal and the 0 ↔ | , i h , | (neglecting terms oscillating faster than δ ) h | ˆΘ ( t g ) | i = 14 Z t g d t Z t d t Ω( t , t )[1 + cos ( δ ( t − t )) − sin ( δ ( t − t ))] , (22)with Ω( t , t ) = Ω X ( t ) Ω Y ( t ) − Ω X ( t ) Ω Y ( t ). Inthe spirit of the Magnus expansion, all slow oscillatingterms have the form above and are negligible if their in-tegral is small. This suggest a control pulse where Ω X ismodulated with a sinusoidal functionΩ X = A π e − σ (cid:16) t − tg (cid:17) (cid:18) − A cos (cid:20) ω x (cid:18) t − t g (cid:19)(cid:21)(cid:19) , Ω Y = − β ˙Ω X . (23)This is a Gaussian with added sideband modulation onthe in-phase part Ω x supplemented by DRAG on thequadrature Ω y . A frequency modulation with cos( ω x t )for a bandwidth of Ω g < ω x can be seen as adding an ef-fective drive at ω x proportional to Ω g . This added drivecan be used to counteract the population transfer of aspecific transition. The absolute errors of eqs. (19-21)are minimized by varying A, ω x , β yielding a pulse witha sideband modulation of δ/ X = A π e − t g (cid:16) t − tg (cid:17) (cid:18) − cos (cid:18) δ (cid:18) t − t g (cid:19)(cid:19)(cid:19) , Ω Y = −
12∆ ˙Ω X . (24)Here we chose σ = t g /
6. The factor of 2 in the denomi-nator of Ω Y comes from the absence of control over thequbit frequency [2]. This is shown experimentally in ref.[32, 33]. The pulse is shown in figure 3 for t g = 17ns andother parameters given by the values in table I. In orderfor the pulse to produce the X gate A π should be chosenso that relation (18) is satisfied. A. Sideband modulation
The black line in figure 4 shows the error of pulse (24)as function of gate time. Compared to the Gaussianand DRAG results, the error has a minimum ( 4%) ata shorter gate time, around 20ns. The reduced fidelityfunctions Φ |∗ ,i i (red and blue lines) and Φ avg (gray line)give additional insight by allowing a phase shift on qubit2. Comparing to figure 2, it is seen that the sidebandmodulated pulse attains a high fidelity ( > . -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 2 4 6 8 10 12 14 16 A m p lit ud e [ GH z ] Time [ns] Ω X Ω Y FIG. 3. Example of the control functions of equation (23) for t g = 17ns. The amplitude of Ω x is somewhat smaller than fora Gaussian only pulse (which has been used in figure 2). than half the time (17 ns compared to 42 ns) of the Gaus-sian or DRAG solutions. The 1 ↔ − Φ |∗ , i is always the biggest. Nonetheless for aspecific gate time a high fidelity is possible. -4 -3 -2 -1
10 20 30 40 50 60 70 80 A c h i e v e d E rr o r Gate Time [ns] ΦΦ |*,1> Φ |*,0> Φ avg FIG. 4. Error as a function of gate time for the pulse withsideband modulation. The target gate is ˆ σ x ⊗ . At t g ∼ avg reaches a maximum. The gate fidelity functionsare defined in equations (6), (7) and (8) respectively. The state populations during the pulse reveal the un-derlying mechanism. Figure 5 shows the populations forgate times 17 and 20 ns. In the latter there is still a netpopulation in the | i state of the qubit 2 after the gate.For the former, there is no net change to the second qubitat the end. This suggest that the the drive on the secondqubit makes it perform a closed transition cycle in the( | i , | i ) subspace, thus acquiring a local phase.Finally we note in this section that the method workedout here is not the only way to determine new analyticalresults for pulse shapes. In general, the different termsof equation (14) need to combine into the correct gate in E xp ec t a ti on V a l u e Time [ns](a) <01|U + ρ U|01><02|U + ρ U|02><11|U + ρ U|11><12|U + ρ U|12> E xp ec t a ti on V a l u e Time [ns](b)
FIG. 5. Populations of the states during the pulse sequence ofequation (24) for gate time of 17ns (a), and 20ns (b). At 20nsthe pulse sequence clearly leaves part of the excitation in the {| i , | i} subspace of qubit two, while at 17ns the trajectory isoptimal in the sense that no net population transfer is presenton qubit two. some manner, whereas we have enforced that this com-bination consists of all terms beyond the lowest one tovanish. Our approach has the advantage that it producesan intuitive result, providing frequency selectivity crite-ria eqs. (18,19,20,21) in the form of the Fourier transformof the driving pulse. B. Phase correction
The average reduced fidelity (8) is insensitive to thephase of the second qubit and leads to a gate of the formof eq. (9). This phase error does not influence populationmeasurements after the gate; only the X and Y compo-nent have different contributions. The global phase α and the phase error γ for specific gate times are plottedin fig. 6. One can correct for this error in multiple ways.If there is a Z control available on the separate qubits [17] one can simply compensate the phase following π Z Z ( t ) d tα ( t g ) = Z Z ( t ) d t (25)Instead of compensating the qubit phase, one can adjustthe phase of the next gate in the XY -plane accordingly.This is possible because the phase error is constant givena set gate time, as shown in figure 6. In essence this is thesame as changing the frame in the XY plane accordingto X ′ = cos ( α ( t g )) X + sin ( α ( t g )) YY ′ = − sin ( α ( t g )) X + cos ( α ( t g )) Y. (26)This technique is analogous to phase ramping as de-scribed in Refs. [1, 2] The phases in the leakage statesare irrelevant, it is thus sufficient to correct the compu-tational subspaces of the qubits individually. -3-2.5-2-1.5-1-0.5 0 10 20 30 40 50 60 70 80 ph a s e Gate Time [ns] αγ FIG. 6. Phases as defined in equation (9) of the gate withthe control sequence from equation (24). It is by these phasesthat the qubits or the subsequent gates need to be corrected.
C. Experimental protocol
The procedure to implement the pulse on an actualexperiment is • Use spectroscopy to determine the qubit frequen-cies, yielding δ and ∆. • Equation (24) gives the shape of the pulses for allpossible gate times t g . The normalization parame-ter A π is chosen so that the area theorem, equation(18), is satisfied, which in general requires numeri-cal root finding. • The gate time t g is chosen so that the pulse se-quence optimizes the reduced average fidelity de-fined by equation (8). • With the gate time known, the phase offset α ( t g )is computed, so that it can be corrected accordingto the procedures given in section V B. VI. NUMERICAL OPTIMIZED CONTROLS
By using numerical methods one can go beyond theanalytic methods discussed in the last sections. Here isdiscussed how further improvements can be made withthe GRAPE algorithm.
A. GRAPE
To handle our system numerically we use the GRadi-ent Ascent Pulse Engineering (GRAPE) algorithm [34].GRAPE maximizes the fidelity eq. (6) by changing thecontrol amplitudes at discrete times. In discrete timethe evolution operator is given by ˆ U ( t g ) = Q j ˆ U j , withˆ U j = exp[ − i ˆ H ( j ∆ t ) ∆ t ]. The fidelity is increased byupdating the controls in the direction of the gradientΩ l ( j ) = Ω jl → Ω jl + ǫ∂ Φ /∂ Ω jl . An analytic expressionfor the gradient is given in ref. [35]. B. Numerical results
The system of equation (4) is numerically optimizedusing the parameters in table I. Figure 7 is an exampleof a short gate (4 ns) high fidelity (99 . t g ≪ π/δ and therefore the small-est spectral crowding frequency scale δ does not imposea quantum speed limit. The limit rather seems to beset by the number of control parameters available. E.g.,we have verified that if the size of a time step is 1 nsas in current experimental equipment, the shortest pos-sible time is 8ns. From numerical results we have notobserved a quantum speed limit. By decreasing the gatetime the pulse can be shortened at the expense of higheramplitudes. The pulse in figure 7 has large amplitudesat t = 0 and t = t g . These can be removed by addingpenalties to the fidelity used by GRAPE [36]. Only asmall increase in gate time is usually needed to enforcepulse sequences to start and end at zero amplitude. Thenumerical results show that no speed limit is set by theoverlap of the control field in the frequency domain withdifferent qubit transitions. Additionally, numerical pulsesequences don’t leave a phase error on the second qubit,eliminating the need for post-processing.To get insight for the shape of the solutions we runthe GRAPE algorithm for short time steps and longergate times to increase the resolution of the discrete timeFourier transform (DTFT). These solutions show rapidoscillations, figure 8. The DTFT of the pulse sequenceshows that both quadrature components have contri-butions at the energy splittings δ, δ − ∆ , ∆ , δ − ∆.This shows that the numerical solution augments the one −1−0.50 0.51 0 0.5 1 1.5 2 2.5 3 3.5 4 A m p lit ud e [ GH z ] Time [ns] Ω x Ω y FIG. 7. Example of a numerically optimized pulse for gatetime t g = 4ns and ∆ t = 0 . Y control is usually notproportional to the derivative of Ω X . based on the Magnus expansion by adding small furthersideband drives. -0.02-0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0 20 40 60 80 100 120 C on t r o l a m p lit ud e [ GH z ] Time [ns] Ω x (t) Ω y (t)3,5*d/dt Ω x (t) FIG. 8. Solution found by GRAPE for a long gate time, here∆ t = 0 . t g = 130ns. The dotted line shows a rescaledversion of the derivative of the Ω X control. When one goes to shorter gate times however Fourieranalysis shows that the contribution of the higher fre-quency components increases, making the Fourier trans-form less useful due to the lower frequency resolution.For faster pulses one could suggest that adding more side-band modulations could improve the results further.
VII. CONCLUSION
We have found numerical as well as analytical pulseshapes implementing single qubit gates in a 3D cavitycoupled to two single junction Transmons. Such qubitsare typically hindered by spectral crowding whereby leak-age transitions lie close in frequency to main qubit 0 ↔ N o r m a li ze d P o w e r Frequency [Ghz] Ω x ( ω ) Ω y ( ω ) FIG. 9. Fourier transform of the pulse shown in 8 found byGRAPE. transitions. We combine average Hamiltonian theory forarbitrary waveforms with the DRAG methodology, showsthat it is possible to find better controls using a sidebandmodulation.Numerically optimized pulses support this conclusionand provide greater improvements in fidelity. They showthat qubits can still be addressed individually with shortgate times. Faster control pulses require more bandwidthand amplitude, therefore the limiting factor is the capa-bilities of the arbitrary waveform generator. 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