Optimized cross-resonance gate for coupled transmon systems

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by  Susanna Kirchhoff

Optimized cross-resonance gate for coupled transmon systems
Susanna Kirchhoff,1 Torsten Keßler,1 Per J. Liebermann,1,∗ Elie
Assémat,1 Shai Machnes,1 Felix Motzoi,1,2 and Frank K. Wilhelm1
1Theoretical Physics, Saarland University, 66123 Saarbrücken, Germany
2Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark
(Dated: January 10, 2017)
The cross-resonant gate is an entangling gate for fixed frequency superconducting qubits intro-
duced for untunable qubits. While being simple and extensible, it suffers from long duration and
limited fidelity. Using two different optimal control algorithms, we probe the quantum speed limit
for a CNOT gate in this system. We show that the ability to approach this limit depends strongly
ontheansatzusedtodescribetheoptimalcontrolpulse. Apiecewiseconstantansatzwithasingle
carrierleadstoanexperimentallyfeasiblepulseshape,shorterthantheonecurrentlyusedinexper-
iments,butthatremainsrelativelyfarfromthespeedlimit. Ontheotherhand,anansatzbasedon
7
thetwodominantfrequenciesinvolvedintheoptimalcontrolproblemallowstogenerateanoptimal
1
solutionmorethantwiceasfast,inunder30ns. Thiscomesclosetothetheoreticalquantumspeed
0
limit, which we estimate at 15ns for typical circuit-QED parameters, which is more than an order
2
of magnitude faster than current experimental microwave-driven realizations, and more than twice

n as fast as tunable direct-coupling experimental realizations.
a
J

7 I. INTRODUCTION a deeply non adiabatic regime. Moreover, it is known
that a careful analysis with regard to parameterization

h] Circuit-QED is a promising technology for quantum of the control sequence is required to tailor efficiently
p computing. An important requirement for scalable oper- the pulse to the experimental constraints of the experi-
- ationofthearchitectureishigh-accuracyimplementation mental aparatus [12]. However, the QSL depends on the
nt of two-qubit gates. A leading candidate for resonator- constraints included in the optimization problem, and
a mediated interaction between a pair of superconducting thus may also depend on the chosen parameterization
u qubits is the so-called cross-resonance (CR) gate [1–4] of the control pulse. Therefore, we probe the relation-
q which has been implemented experimentally with over shipbetweentheQSLanddifferentphysicallymeaningful
[
99% average gate fidelity [5]. The CR gate functions by parameterizations, which leads to a new understanding

1 combiningthefrequencyaddressabilityofonequbitwith of the limitations of the original cross-resonant control
v the spatial addressability of a second, such that the the scheme and to propose alternative control strategies.
1 first qubit is driven by the control line of the second, This manuscript is organized as follows: Section II
4
i.e. “cross-resonantly". The design avoids the complexity presents the theoretical model of the system; Section III
8
1 andnoisesourcesthatarepresentvialow-frequencymag- introduces the numerical method; in Section IV we show
0 netic (flux) tuning of the qubit-qubit interaction [6]. It that the unconstrained quantum speed limit is much
. also aims to improve on methods for high-frequency ad- shorter in duration than the existing strategy; Section V
1
dressing of desired two-qubit transitions by utilizing the describes the first optimization results and point out the
0
7 additional spatial addressability that comes from dedi- role of higher frequency components; Section VI studies
1 cated control circuitry. the QSL with the Fourier parameterization of the drive
: The primary impediments to high-fidelity operation shape; Section VII explains the influence of the band-
v
i thusfarhavecomefromincoherenterrorssuchasT1 and width constraint and proposes two solutions; Finally, in
X T effects and unitary errors such as crosstalk [7] and Section VIII we make concluding remarks.
2
r frequency crowding [8]. The main method to counteract
a
the former is to shorten gate times as much as possible,
butthiscandrasticallyincreaseunitaryerrors,especially II. SYSTEM
from higher order corrections to the perturbative model
of the gate mechanism. Let us consider two transmon qubits coupled by a bus
Inthisarticlewesystematicallyoptimizethedesignof resonator. Eachtransmonisdescribedasananharmonic
the CR gate to reduce the gate time as much as possible oscillator and the coupling to the resonator by an ap-
andnumericallyestimatethequantumspeedlimit(QSL) propriately extended Jaynes-Cummings model [13–15].
for the gate [9–11]. We employ a full Jaynes-Cummings The qubits consist in the first two levels of these two
model which eliminates many of the analytic simplifi- anharmonic oscillators. As in [5], we assume that σ ,
x
cations that set bounds for an analysis of the gate in σ type controls are available. We further assume that
y
the qubit frequencies can be calibrated by a quasi-static
flux line. The aim of the latter is not to provide an-
other time dependent control function, but to statically
∗ [email protected] shift the qubit frequencies to a value more favorable to
2
the optimization process. With the notations of [14], the
Hamiltonian takes the form −6
2
(cid:88)(cid:16)
H(t)=ωra†a+ ωkb†kbk+αkb†kb†kbkbk+gk(ab†k+a†bk) −7
k=1
L (cid:33)
+(cid:88)l=1Ek,l(t)cos(cid:0)ωkd,lt+φk,l(cid:1)(bk+b†k) (1) Error) −8
g(
wherea†,b† andb† arethecreationoperatorsforthecav- lo −9
1 2
ity and the two transmons respectively; g are the cou-
k
plings between the resonator and the qubits; ω , ω and
r 1 −10
ω are the respective frequencies of the cavity and the
2
two transmons; are the low-frequency envelopes of
k,l
themicrowavedrEiveandωd arethecarrierfrequenciesof −11
0 10 20 30 40 50
the L control drives with phase offset φ . After mov-
k,l Gate duration (ns)
ing to the rotating frame at the frequency of the first
qubit and applying a rotating wave approximation, the
FIG. 1. Determination of the Quantum Speed Limit (QSL)
Hamiltonian takes the form
using arbitrary amplitude piecewise constant controls with
high sampling rate (≈0.1ns) and no filter.
H (t)=∆a†a
rwa
2
(cid:88)(cid:16)
+ g (ab† +a†b )+δ b†b +αb†b†b b
k k k k k k k k k k
k=1 vector based on this gradient. The first iteration starts
(cid:17)
+ x(t)(b +b†)+i y(t)(b† b )+f b†b , (2) with a guess of the control function. In many cases this
Ek k k Ek k− k k k k guess is extremely important, especially when the target
where ∆ is the detuning of the cavity from the principle gatetimeisclosetothequantumspeedlimit,wherelocal
(carrier) drive frequency of the controls and δ are the minima tend to appear in the optimal control landscape
k
detunings of the transmons. f are used to optimize the [18].
k
latter, and can be thought of as the static flux tuning of
the transmon frequencies and/or shifting of the princi-
pledrivefrequency. Thenumericalsimulationsandopti-
mizationspresentedbelowincludethethreefirstlevelsof
the resonator and the three first levels of each transmon.
IV. UNCONSTRAINED QUANTUM SPEED
LIMIT
III. NUMERICAL METHOD
We first look for the QSL by using the least possible
This work relies mainly on the optimal control algo- constrainedparameterization,i.e. noamplitudebounds
rithmGradientAscentPulseEngineering–GRAPE[16]. andhighresolution,with500timeslicesforeachgatedu-
It was developed to numerically determine the shape of ration t . Assuming that the parameterization is flexible
g
the control function that maximizes a goal function. In enough, if a QSL is observed it comes from the physics
the case of a gate generation problem, a possible goal of the system. One standard method to probe numer-
function is ically the QSL is to plot a measure of the gate fidelity
as function of the gate duration [15, 19]. For different
1 (cid:12) (cid:16) (cid:17)(cid:12)2
Φ = (cid:12)Tr U† U(t ) (cid:12) , (3) gates time we ran many GRAPE optimization with dif-
goal N2 (cid:12) target g (cid:12)
ferent random guesses. The average final gate error, de-
where t is the target gate duration, N the dimension of fined as 1 Φ are shown in fig. 1. We observe a
g goal
| − |
thecomputationalsubspaceandU theunitaryevolution clear jump between 10ns and 15ns gate time that indi-
operator. For other examples of goal functions and in- cates the presence of a QSL. Interestingly this is more
depthstudyofoptimalcontrolalgorithms,theinterested than an order of magnitude shorter than the gate time
reader may refer to [17]. presented in [2]. However, because no constraint is in-
The GRAPE algorithm considers a piecewise constant cluded, these optimal control shapes are far from what
approximation of the control function. The values of all can be experimentally implemented. Taking in account
slices are stored in a vector whose size defines the search the constraints of the experiments will push the QSL to-
space dimension. At each iteration, GRAPE computes wards a longer gate time, while remaining shorter than
thegradientofthegoalfunctionwithrespecttothecon- the original cross-resonance gate, as shown in the follow-
trol function vector, then updates the control function ing.
3
tions:
TABLE I. Values of the parameters of the Hamiltonian 4 in
GHz. x(t)=0 y(t)=0
E1 E1
(cid:18) (t µ)2(cid:19) t t
Parameter Value in GHz x(t)=0.4exp − µ= g, σ = g
E2 − 2σ2 2 4
∆/(2π) 0.4
1
gg12//((22ππ)) 00..11 E2y(t)= δ2E˙1x(t), f1 =0, f2 =0.1,
α/(2π) −0.32 (5)
δ /(2π) 0.0
1
δ2/(2π) −0.67 where the amplitude are given in GHz and f1 and f2
arerespectivelyconstantbuttunablefrequencyoffsetsof
qubit1and2fromthedrivefrequency. Here, weinclude
in the optimization a bound on the control amplitude
and a lower sampling rate (time slices of 0.2 ns). The
V. SEARCH FOR A MORE REALISTIC minimum gate time threshold for possible convergence
OPTIMAL CONTROL is found to be t = 27 ns. As expected, this is longer
g
than the physical QSL, but still much shorter than the
180nsoriginalCRgate. Theinitialandfinaldriveshapes
In this work, the initial guess of a trial function for x
E are shown in fig. 2. We observe that the strong cross-
is set here by following the cross resonant scheme imple-
resonant drive remains after the optimization, which in-
mentedin[2]basedontheeffectiveinteractionpresented
dicates that the CR scheme is still the main physical
in [1]. A cross resonant drive on a two transmon system
mechanism in play. Although the pulse involves a com-
is a drive applied on one qubit such that the drive fre-
plicated sequence of maneuvers to obtain high-fidelity at
quency is resonant with the second qubit. This scheme
a very short time, it is nonetheless very promising as it
leads to generating a CNOT gate. Indeed, let us con-
impliesthepossibilityofreducinggateoperationtimeby
sider the effective coupling J between the two qubits
eff anorderofmagnitude[5],reducingtheleadingsourcesof
that quantifies the effective interaction mediated by the
decoherence by a similar fraction, and even outperform-
resonator. In the case where J is small compared to
eff inggatetimeswithtunablequbitfrequencyarchitectures
the detuning δ between the qubits, a drive at frequency
2 [21].
ω˜ =ω (J )2/δ generates dynamics that can be de-
2 2 eff 2
−
scribed by an effective Hamiltonian of the form
VI. QUANTUM SPEED LIMIT WITH THE
FOURIER ANSATZ
Heff =ue1ffσ1z⊗σ2x+ue2ffσ1x⊗1, (4) Another important feature of quantum control se-
quences is that they can be parametrized with as small
as possible a number of controls. This simplicity can
where σ are the Pauli operators and ueff denote the rel-
i i greatlyaidinanalytical,numerical,andexperimentalre-
ative scaling of the effective interaction. The ZX inter-
producibility of the solution, and avoid getting stuck in
action present in this effective Hamiltonian can generate
local minima due to miscallibration. For these purposes
directly a CNOT gate.
weuseaparameterizationbasedonthenumberofFourier
The performance of the initial guess is further im- componentsofthepulse, thoughanyothersetofparam-
provedbylimitingtheeffectofthesecondterminEq.4. eters more appropriate for the given control task could
Hereweassumethisspectralconstraint(suppressingthe have been substituted. Each control is parametrized by
secondterm)canbesatisfiedbyusingasecondoff-phase a truncated Fourier series that is fed to a sum of two
quadrature y of the control envelope set proportional sigmoids to enforce the global bound and a smooth start
to the derivaEtive of x and inversely proportional to the and finish. The amplitude coefficients, frequencies and
qubit frequency sepEaration δ , as in the Derivative Re- phases are optimized with the GOAT algorithm [12], a
2
moval by Adiabatic Gate (DRAG) method [7, 20]. recent gradient base algorithm that allows to handle an-
alytic ansatzes more efficiently than GRAPE.
These analytical techniques would not be sufficient to
We first start with a large number of Fourier compo-
obtain high fidelity gate with the model at hand due to
nents and probe the QSL. The number of components is
the complexity of the levels structure. However, the ini-
chosentoofferroughlythesamenumberofparametersto
tial guess should be relevant enough to be located in the
describe the drive shape as we had in the piecewise con-
basins of attraction of a higher fidelity solution, which
stant case of Fig. 1. We observe in Fig. 3 that the QSL
allows the GRAPE algorithm to converge.
is less sharp than what was observed with the piecewise
Followingthisdirection,thefirstoptimizationattempt constantdescription,whichillustratestheclearinfluence
was carried out with the following initial control func- of the choice of the control representation on the control
4
E1x E1y E2x E2y f1 f2 −2
0.4
−3
0.3 −4
z]
H
[G or)−5
ude 0.2 Err
plit og(−6
m l
A 0.1
−7
0 −8
0 5 10 15 20 25 −9
Gatedurationtg[ns] 10 20 30 40 50 60 70
Gate duration (ns)
E1x E1y E2x E2y f1 f2
FIG. 3. Determination of the Quantum Speed Limit (QSL)
0.4 using 167 Fourier components per control. The choice of pa-
rameterizationofthedriveshapehasaclearinfluenceonthe
QSL, when compared to Fig. 1.
0.3
z]
H
G
[ 0.2
de can expect to do in this typical regime of control pa-
u
t rameters, regardless of the chosen control parameteriza-
pli
m 0.1 tion. We find relatively short gate times even with just
A
9 Fourier components per control, which is far fewer pa-
0 rameters than the approximately 200 time steps used in
thepiecewise-constantparameterization. Thisisasignif-
icantreductioninthenumberofparameters,butsuchop-
-0.1
0 5 10 15 20 25 timalcontrolshapesrequireincreasedexperimentalover-
Gatedurationtg[ns] head compared to current state-of-the-art experimental
setups, which usually only employ at most a handful of
FIG. 2. Preliminary, high-bandwidth optimization of the carrier frequencies at a time.
drive functions. Top: initial control guesses. Bottom: op-
timized controls with Φ = 0.998. The average structure
goal
oftheinitialguessremainsaftertheoptimizationwhichvali-
VII. INFLUENCE OF THE BANDWIDTH
dates the CR scheme.
CONSTRAINT AND WORKAROUND
Thepiecewiseconstantparameterizationwithhighres-
landscape. Then, we iteratively remove the component
olution controls provides a theoretic lower bound to gate
withthesmallestamplitudeandreoptimize,downtoonly
times, but for time resolved optimization we are in prac-
9 components. However, this reduction is at the cost of
ticelimitedbytheAWG’stimeresolution. Furthermore,
an increased gate time of 70ns and the appearance of
the applied control from the AWG usually undergoes
some very high frequencies. This offers a hint to explain
some filtering. Here we optimize the pulses for an AWG
why the unconstrained piecewise optimization manages
with finite time resolution of 1ns, fine steps of 0.2ns,
toconvergetoagateerrorof10−10,whereasthespectral buffers of 4ns duration at the beginning and the end,
optimization with a smaller frequency range converged
and filtering of the signal, see [22] for details. The filter-
only to 10−3. It seems that high frequency components ingisappliedviaaGaussianwindowfunctionwithstan-
arenecessaryforthefinetuningneededtoachieveahigh
dard deviation σ =0.4ns, i.e., a bandwidth of 331MHz.
accuracy. Moreover, this could also be a sign that the
For a gate time of 70ns, a GRAPE optimization gener-
Fourier ansatz is not the most efficient for this system,
ates the controls and spectrum shown in Fig. 4 with a
andonemaywishtotryafewotheranalyticansatz. The
fidelity of 99.9%. The filtering process leads to slower
GOAT package would be well suited for such study.
optimization and forces the use of longer gate times to
Nonetheless, using 167 Fourier components we find obtain higher fidelities. Nonetheless, this optimal drive
that the minimum time is around 15ns to obtain 0.001 shape includes all relevant constraints and could be di-
error, which is consistent with the piecewise-constant rectly implemented in an experiment. A remaining step
case, and thus indicates this is probably as fast as we would be to tune it to the unknown exact values of the
5
stuysntienmg opraorapmtimetiezrast,iownit[h23p]o.ssibly additional closed-loop E1x0 E1y0 E2x0 E2y0 f1 f2
0.4
0.3
x y x y Hz]
0.15 E1 E1 E2 E2 G
[ 0.2
e
d
u
t
0.1 pli
m 0.1
A
z] 0.05
H
G 0
[
e
d 0
u
plit -0.1 0 5 10 15 20 25
m
A -0.05 Gatedurationtg[ns]
-0.1 0.1 E1x00 E1y00 E2x00 E2y00
-0.15
0 10 20 30 40 50 60 70
0.05
Gatedurationtg[ns]
z]
H
x y x y G
100 E1 E1 E2 E2 e[
d 0
u
t
pli
10−2 Am
-0.05
r
e
w
po 10−4
d
e
ormaliz 10−6 -0.1 0 5 G1a0teduratio1n5tg[ns] 20 25
N
10−8 FIG. 5. Final optimization of the control functions. Top:
controls at qubit 1 frequency. Bottom: controls at qubit 2
frequency.
10−10
0.25 0.5 0.75 1 1.25 1.5 1.75 2
Frequencyf [GHz]
FIG.4. Pulseshapesandspectrumoftheoptimaldriveshape
x(t)= x(cid:48)(t)+cos((δ+2g)t) x(cid:48)(cid:48)(t)
for filtered PWC ansatz with gate time of 70ns and 99.9% E1 E1 E1
fidelity. The flux is not optimized here. y(t)= y(cid:48)(t)+cos((δ+2g)t) y(cid:48)(cid:48)(t)
E1 E1 E1
x(t)= x(cid:48)(t)+cos(δt) x(cid:48)(cid:48)(t)
E2 E2 E2
y(t)= y(cid:48)(t)+cos(δt) y(cid:48)(cid:48)(t) (6)
To further decrease the experimental overhead and fa- E2 E2 E2
cilitateitscalibration,wefurtherconstrainthewaveform where the quotes symbols denote the new control func-
basedontheprimaryfrequenciesfoundintheinitialopti- tionsandδ =δ f isthequbitseparation. Thisdoubles
2 2
−
mizationandlimittotheprimarythreefrequencycompo- the number of functions to optimize but their smoother
nents,withtime-domaincontrolofeachwaveformwithin behaviour improves the convergence speed nonetheless.
the bandwidth limit of each of the three frequencies. Moreover, a Gaussian filter with a bandwith of 331MHz
is added to account for the distortion of the piecewise-
The spectra of the controls are dominated by the fre- constant control functions by waveform generators [22].
quencies 0.47GHz, 0.57GHz and 0.67GHz. The 0.57GHz Inanexperiment,thiswouldrequiretwocarrierfrequen-
corresponds to the frequency of the second qubit shifted cies, one for each qubit. The first one would shape the
byf2, while0.47GHzand0.67GHzcanbeinterpretedas termsEin(cid:48) andthesecondonewouldshapethetermsEin(cid:48)(cid:48).
theinfluenceoftheRabisplittingonthecontrolprocess. Thisshouldbefeasiblewithcurrenttechnology, theonly
Thus, we choose remaining issue would be the calibration of the optimal
6
the transmons, the maximum is also taken over the two
0 0 transmons: M = log[max (p (t))] and M =
r t r Transmons
log[max (p (t))]. Thepopulationofthefourthlevel
t;i∈1,2 ri
-1 -1 oftheresonatorandtransmonsremainbelow10−3 while
the transmons populate their fourth level up to 10−2.
n
atio-2 -2 Thohwiseviesr,sufuffirctiheenrtofpotrimthizeataicocnuroancythaeimtreadnsimnotnhipsowpourlak-,
ul
pop-3 -3 tions would be necessary for high accuracy applications.
m
u
m
-4 -4
xi VIII. CONCLUSIONS
a
M
-5 -5
We have explored numerically the quantum speed
limit of the Cross-Resonance gate using the algorithms
-60 2 4 -60 2 4 GRAPE and GOAT. We showed that the physical quan-
Resonator levels Transmon Levels tum speed limit is more than an order of magnitude
smaller than the gate duration currently used, approx-
FIG. 6. Maximum transient Log(population) leakage in the imately 10 15ns for typical circuit-QED parameters.
higherlevelsoftheresonator(left)andthetransmons(right) However,to−reachthislimitahighresolutionisrequired.
during a dynamics driven by the optimal controls.
Twointermediatesolutionswereproposed. Thefirstuses
asinglefrequencyresonantwiththedriventransitionand
generates a realistic control shape with a gate duration
pulse in the experiment, see [12, 23] for an in depth dis- of 70ns, far from the quantum speed limit but that still
cussion of this matter. represent a significant improvement over existing control
The optimization is carried out with coarse pixels of shapes. The second implies to shape independently two
1ns and a fine time step of 0.2 ns and reaches a final different carrier frequencies resonant respectively with
fidelity of 0.99. The control functions optimized i(cid:48), i(cid:48)(cid:48) the two qubits. This should be feasible with existing
Ej Ej
andfj areshowninfigure5. Notetheapproximatetime hardwareanditwouldallowtoreachagatetimeof27ns,
symmetry of the control function x(cid:48), reminiscent of the or just above the quantum speed limit.
E2
Hahn echo pulses used in nuclear magnetic resonance.
The constant value of the frequency detuning is also op-
timized but its value appears to be stuck in a local min- ACKNOWLEDGMENTS
imum and does not evolve significantly during the opti-
mizations. E.A. acknowledges support from the Alexander von
To test the validity of the truncation of the Hilbert Humboldt Foundation. S.K. and S.M. acknowledges
space, the optimal control functions were applied to a funding from the IARPA through the LogiQ grant No.
larger space including 5 levels of the resonator and each W911NF-16-1-0114. P.J.L. and F.K.M. acknowledges
transmon. The maximum population reached during supportfromtheEuropeanUnionthroughtheScaleQIT
the dynamics by each level is plotted in Fig. 6. For project.
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