Quantum-control approach to realizing a Toffoli gate in circuit QED

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by  Vladimir M. Stojanovic

Quantum-control approach to realizing a Toffoli gate in circuit QED
Vladimir M. Stojanovi´c,1 A. Fedorov,2 A. Wallraff,2 and C. Bruder1
1Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
2Department of Physics, ETH Zu¨rich, CH-8093 Zu¨rich, Switzerland
(Dated: January 31, 2012)
We study the realization of a Toffoli gate with superconducting qubits in a circuit-QED setup
using quantum-control methods. Starting with optimized piecewise-constant control fields acting
on all qubits and typical strengths of XY-type coupling between the qubits, we demonstrate that
theoptimal gatefidelities areaffected onlyslightly bya“low-pass” filtering ofthesefieldswith the
typical cutoff frequencies of microwave driving. Restricting ourselves to the range of control-field
amplitudes for which the leakage to the non-computational states of a physical qubit is heavily
2 suppressed, we theoretically predict that in the absence of decoherence and leakage, within 75ns a
1
Toffoli gate can be realized with intrinsic fidelities higher than 90%, while fidelities above 99% can
0
bereached in about 140ns.
2
n PACSnumbers: 03.67.Lx,03.67.Ac
a
J
Superconducting(SC)qubits[1]havecomealongway
8
2 sincetherealizationthatJosephsonphysicsinSCcircuits
canbeutilizedtopreparewell-definedfew-levelquantum

] systems [2]. Coupling such qubits is essential for quan-
h
p tum computation. The most successful approaches up
- to now rely on coupling all the qubits in an array to an
t
n “interaction bus,” a central coupling element having the
a form of a transmission line with electromagnetic modes. FIG. 1: Lumped-element circuit diagram of three transmon
u
In the circuit quantum electrodynamics (circuit QED) qubits coupled to a superconducting transmission-line res-
q
regime strong coupling between the qubits and the con- onator. Theresonatorservesasacouplingbusforthequbits
[
finedphotons isrealized[3–5]; the cavityphotonsinduce and is also used for the readout of their states (red). Lo-

2 cal flux lines (blue) allow for individual control of the qubit
a long-rangecoupling between the qubits. The combina-
v frequencies on the nanosecond time-scale. Microwave lines
tionoftransmonqubits[6]andco-planarmicrowavecavi-
2 (green) are used to create control fields, each acting on its
4 tiesrepresentsthestate-of-the-artofmicrowavequantum corresponding qubit.
4 optics.
3
The environmental degrees of freedom limit the time
.
8 over which quantum coherence can be preserved. While study the realization of a Toffoli gate with SC qubits in
0 theyaresimilartochargequbits,transmonshaveamuch a circuit-QED setup (for an illustration, see Fig. 1) by
1
larger total capacitance such that the charging energy is applying quantum-control methods [16] to the effective
1
: significantly smaller than the Josephsonenergy. A small XY-type Hamiltonian of an interacting three-qubit ar-
v
charge dispersion of the energy eigenstates leads to a re- ray. Wedosoforrealisticqubit-qubitcouplingstrengths
i
X duced sensitivity to charge noise and longer dephasing and under typical experimental constraints on the qubit
r times (T2) [6]. Recently, a significant progress has been decoherence times.
a achieved[7,8],withT timesbeingincreasedbyanorder
2 We first determine optimal piecewise-constant control
of magnitude from T ∼1µs to T ∼20µs.
2 2 fields and evaluate the resulting gate fidelities. Then we
Given that two-qubit gates with SC qubits have been discuss how these fidelities are affected when the con-
demonstrated with fidelities higher than 90% [9], a trol pulses are smoothened by eliminating their high-
key challenge now is to realize three-qubit ones with frequencyFouriercomponents(spectral“low-pass”filter-
shortest possible gate times. An example is the Tof- ing). Our most important theoretical prediction is that
foli gate (controlled-controlled-NOT), which is relevant within 75ns a Toffoli gate can be realized with intrin-
for quantum-error correction [10] and has already been sic fidelities (in the absence of decoherence and leakage)
implemented with trapped ions [11] and photonic sys- higher than90%, while fidelities higher than 99%canbe
tems[12]withrespectivefidelitiesof71%and81%. Very obtained in approximately 140ns.
recently, several groups realized a Toffoli gate using su- Methods of quantum control [16] have been put for-
perconducting circuits [13–15]. The Toffoli gate was im-
ward in a number of theoretical proposals for realizing
plemented by a sequence of single- and two-qubit gates quantum logic gates with SC qubits [17–19], thus com-
(direct approach). plementing studies that solely involve time-independent
In this paper, we adopt an alternative approach and Hamiltonians [20]. More fundamentally, recent studies
2
in operator (state independent) control [21] have been aspects of the problem. Using the standard algorithm
focusing on interacting systems and employing the con- (see, for example, Ref. 16), it is straightforwardto check
cept of local control [22, 23]. The essential idea is that that the dynamical Lie algebra of the system, generated
systemssuchascoupledspin-1/2chains,modelsofinter- bytheskew-Hermitianoperators−iH ,−iσ ,and−iσ
0 ix iy
acting qubit arrays,can often be controlledby acting on (i = 1,2,3), has dimension 63 = d2−1 (d = 8 is the di-
asmallsubsystem. Forinstance,controllingonlyoneend mension of the Hilbert space of the system). This Lie
spinofanXXZ-Heisenbergchainensurescompletecon- algebra is isomorphic to su(d = 8) and the system is
trollability of the chain [22, 23]. Yet, in view of the cur- completely (operator) controllable. An arbitrary quan-
rentlyavailablefew-qubitexperimentalsetups[24],more tumgatecanthusinprinciple be realizedusingproperly
importantthanrestrictingcontroltoonlyonequbitisto designed control fields.
be able to carry out an optimal control-pulse sequence Ourgoalistofindthetimedependenceofcontrolfields
within times much shorter than T2. This is necessary in Ωx(i)(t) and Ωy(i)(t) for realizing a quantum Toffoli gate.
order to minimize the undesired decoherence effects [1]. Westartouranalysiswithsimplepiecewise-constantcon-
Under the condition of resonantdriving and assuming trolfields [23] acting onall three qubits in alternationin
that the qubits are in resonance with one another, the thex-andy directionswithcontrolamplitudesΩ(i) and
x,n
effective (time-independent) XY-type (flip-flop) qubit- Ω(i) (n=1,...,N /2; i=1,2,3).
y,n t
qubit interaction Hamiltonian is given by
At t = 0 control pulses are applied in the x direction
to all three qubits with constant amplitudes Ω(i) during
H = J (σ σ +σ σ ), (1) x,1
0 X ij ix jx iy jy the time interval 0 ≤ t ≤ T. The Hamiltonian of the
i<j system is then H ≡H + 3 Ω(i)σ . Then y con-
x,1 0 Pi=1 x,1 ix
where σix, σiy, and σiz are the Pauli matrices. The sys- trol pulses with amplitudes Ωy(i,)1 are applied during the
temisacteduponbytime-dependentZeeman-likecontrol interval T ≤ t ≤ 2T, whereby the system dynamics is
fields described by the Hamiltonian governed by H ≡ H + 3 Ω(i)σ . This sequence
y,1 0 Pi=1 y,1 iy
ofalternatingx andy controlpulses is repeateduntil N
3 t
H (t)= Ω(i)(t)σ +Ω(i)(t)σ , (2) pulses have been completed at the gate time tg ≡ NtT.
c Xh x ix y iyi The time-evolution operator U(t = t ) is then obtained
i=1 g
as a product of the consecutive U ≡ exp(−iH T)
x,n x,n
thusthesystemdynamicsisgovernedbythetotalHamil- and U ≡exp(−iH T), where n=1,...,N /2.
y,n y,n t
tonian H(t)=H +H (t). For varying choices of N and T, the 3N control am-
0 c t t
The control fields Ω(i)(t) and Ω(i)(t) can be imple- plitudes are determined so as to maximize the fidelity
x y
mented using arbitrary wave generators (recall Fig. 1),
1
which can produce an arbitrary signal with frequencies F(t )= tr U†(t )U , (4)
g 8(cid:12) (cid:2) g TOFF(cid:3)(cid:12)
up to 500MHz with minor distortions. In qubits based (cid:12) (cid:12)
on weakly-anharmonic oscillators, leakage from the two- where U is the Toffoli gate. The numerical
TOFF
dimensionalqubitHilbertspace(computationalstates)is maximization over these amplitudes is carried out us-
the leading source of errors at short gate times [19, 26]. ingtheBroyden-Fletcher-Goldfarb-Shanno(BFGS)algo-
This is especially pronounced if the control bandwidth rithm [27], a standard second-order quasi-Newton-type
is comparable to the anharmonicity [18]. We therefore procedure. Based on an initial guess for the control
imposeanadditionalconstraintonthecontrolfields,viz. amplitudes, the algorithm generates iteratively new se-
the condition that quencesofamplitudessuchthatineachiterationstepthe
fidelity is increased, terminating when the desired accu-
Ωmax =maxi,tq[Ω(xi)(t)]2+[Ω(yi)(t)]2 (3) racy is reached. This procedure is repeated for multiple
(∼ 200) initial guesses to avoid getting trapped in local
is smaller than some threshold value for the transmon (instead of the global) maxima of F(t ). The optimiza-
g
to be a well-defined two-level system. For typical an- tionisperformedunder the constraintΩ <130MHz.
max
harmonicities of transmon qubits (300−400MHz) and Whilepiecewise-constantcontrolpulsesareconvenient
values of Ω of 100 − 130MHz the error associated as a starting point for a theoretical analysis, the actual
max
withthisleakageshouldnotexceedafewpercent. While pulse-shaping hardware cannot generate such fields with
controlschemesexplicitly involvingthe higherlevelsofa arbitrarily-highfrequencycomponents. Turningthetime
physicalqubitareinprincipleconceivable,inthepresent course of piecewise-constant control pulses into an opti-
study we aim for simplicity and will in the following de- mized shape can be considered the central problem of
terminethecontrolpulsesforasystemofqubitsthatare numerical optimal control [28].
genuine two-level systems. We therefore perform spectral filtering of our opti-
Before evaluating numerically the optimal control mal control fields. Quite generally, after acting with a
pulses we would like to comment on the controllability frequency-filter function f(ω) on the Fourier transforms
3
F[Ω(i)(t)] of the optimal fields Ω(i)(t) (j = x,y), one
j j TABLEI:Examplesofcalculated intrinsicToffoli-gate fideli-
switches back to the time domain via inverse Fourier
tiesforoptimalpiecewise-constantcontrolfields(F)andtheir
transformation to obtain the filtered fields Ωj(t): low-pass filtered versions (Fe), both corresponding to gate
Ωj(i)(t)=F−1(cid:2)f(ω)F[Ω(ji)(t)](cid:3) (j =ex,y). (5) tfeimreensttvgagluiveesnsihnouwnnitfsoorfFeJ−a1ndwhΩeemreaxJc=orr3e0spMoHndz.tTohreestpweoctdivife-
e high-frequency cutoffs of ω0 = 500MHz and ω0 = 450MHz
In particular, we consider an ideal low-pass filter, which (in brackets).
removesfrequenciesabovethe cut-offω andbelow−ω .
Inotherwords,f(ω)=θ(ω+ω0)−θ(ω−0ω0),whereθ(x0) Nt tg[ns] F [%] Fe[%] Ωemax[MHz]
istheHeavisidefunction. Whenappliedtoourpiecewise- 14 75.0 92.92 92.08 (91.43) 102.7 (96.0)
constantcontrolfields,thetransformationsinEq.(5)can 12 76.0 91.74 91.38 (91.20) 96.5 (96.2)
be carried out semi-analytically. They lead to 10 81.3 91.91 91.39 (91.35) 107.5 (104.7)
20 139.2 99.72 99.29 (99.28) 111.2 (112.2)
Nt/2
1 18 165.0 99.72 99.45 (99.35) 102.9 (94.9)
Ω(i)(t)= Ω(i) a (t)−a (t) ,
x π X x,n(cid:2) 2n−1 2n−2 (cid:3) 16 180.0 99.00 98.79 (98.77) 119.1 (116.0)
e n=1
(6)
18 180.0 99.70 99.52 (99.40) 116.4 (107.2)
Nt/2
1
Ω(i)(t)= Ω(i) a (t)−a (t) , 30 195.0 99.99 99.57 (99.15) 94.8 (83.9)
ey π nX=1 y,n(cid:2) 2n 2n−1 (cid:3) 28 198.9 99.99 99.23 (98.68) 126.4 (119.0)
18 215.0 99.78 99.61 (99.59) 102.7 (100.8)
where a (t) ≡ Si ω (mT −t) (m ∈ N) and Si(x) ≡
m 0 20 213.3 99.96 99.70 (99.70) 105.8 (102.0)
x (cid:2) (cid:3)
(sint/t)dt stands for the sine integral. Based on
R0 24 205.0 99.99 99.84 (99.72) 129.0 (122.9)
Eq.(6),wenumericallydeterminethetime-evolutionop-
22 207.2 99.98 99.88 (99.78) 118.7 (113.4)
eratorscorrespondingtothefilteredcontrolfieldsusinga
22 210.5 99.99 99.89 (99.86) 119.8 (114.7)
product-formula approach(for details, see the Appendix
24 215.0 99.99 99.61 (99.47) 129.6 (126.9)
in Ref. 23). We then obtain the fidelities F(t ) corre-
g 22 224.6 99.99 99.91 (99.79) 108.9 (101.0)
sponding to the filtered fields from an analogeof Eq. (4).
22 230.0 99.99 99.96 (99.93) 126.6 (120.4)
Our numerical results are summarized in Table I. It
shows examples of calculated Toffoli-gate fidelities for
Inreality,thefidelitylossresultingfromdecoherenceis
optimal piecewise-constant control fields (F) and their
inextricably linked to the particular experimental setup
low-pass filtered versions (F), for two different high-
and noise sources present in it. The errors due to de-
frequency cutoffs (ω = 500eMHz and ω = 450MHz).
0 0
coherence certainly depend sensitively on the total gate
The last column of the table shows the maximum Ω
max time,whichisminimizedinourapproachbytheinterplay
of q[Ω˜x(i)(t)]2+[Ω˜(yi)(t)]2 over all qubits and all teimes of always-on interactions between the qubits and time-
and obeys the constraint discussed after Eq. (3). The dependentcontrolpulsesactingonallqubits. Quitegen-
fidelities corresponding to the piecewise-constant fields erally,thesuppressionofthegatefidelityduetodecoher-
arevirtuallyunaffectedbythefilteringprocess,sincethe ence is approximately given by the factor exp(−t /T ),
g 2
highest frequencies achievable by current pulse-shaping determinedbytheratioofthegatetimet andthedeco-
g
hardware are much larger than the qubit-qubit coupling herencetimeT [9]. Itisthereforequiteencouragingthat
2
strengths. the required times we find for high-fidelity (F > 90%)
To make contact with possible experiments, we now realizations of the Toffoli gate (t ∼ 75ns) represent a
g
assume J = 30MHz and J12 = J23 = 6J13 = J [25]. rather small fraction of the newly achieved decoherence
As can be inferred from the table, a Toffoli gate can be times (T ∼ 5−20µs). Remarkably, this is better than
2
realized in 2.25J−1 = 75ns with a fidelity higher than achieved experimentally (with T ∼ 1µs) for two-qubit
2
90%, while fidelities larger than 99% can be reached for gates (t ∼30−60ns).
g
gate times of around 4.18J−1 = 140ns. Examples of To summarize, employing methods of quantum opera-
optimal x and y control fields on all three qubits are torcontrolwehaveinvestigatedthefeasibilityofrealizing
shown in Fig. 2. a quantum Toffoligate with superconducting qubits in a
For fixed total time tg = NtT higher fidelities are ob- circuitQEDsetup. Ourcalculationsindicatethatwithin
tained for larger Nt (i.e., smaller T), and the same is 75ns a Toffoli gate can be realized with intrinsic fideli-
true of the robustness of these fidelities to random er- ties higher than 90%, while fidelities larger than 99%
rors in the control-field amplitudes [23]. However, for require gate times of about 140ns. A particularly ap-
T smaller than some (nonuniversal) threshold value, it pealing feature of our approach is that it does not make
becomes impossible to reach high fidelities (F > 90%) aprincipaldifferencebetweentwo-andthreequbitgates,
without violating the constraint Ω <130MHz. incontrasttothemoreconventionalapproachesinwhich
max
e
4
4 X control on the first qubit 2 Y control on the first qubit
control field ( units of J )−01231 ( a ) control field ( units of J )−−−01321 ( b ) [[12]] FPRNJL.oehearMtvyttua.s...rMr5MeTe5ovo(a,dLidre1.toawi5ynPn,4dih53ssoey,8n(esM,1).Y497.428.3H55M(,3).2.a3,0Dk501e7h50vl)3(io;n12rJ,0e(.Gt02,C10.a)l0S;na8crdJ)hk..¨oJeQn.a,C.naYldnaordFuk.AeaK,.nP.SdhWhyFnisi.lr.hNmReolaemrnvi,,,.
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3
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financially supported by EU project SOLID, the Swiss (2011).
NSF, the NCCR Nanoscience, and the NCCR Quantum
Science and Technology.