Implementation of the iFREDKIN gate in scalable superconducting architecture for the quantum simulation of Fermionic systems

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by  Per J. Liebermann

Implementation of the iFREDKIN gate in scalable superconducting architecture for
the quantum simulation of Fermionic systems
Per J. Liebermann,1 Pierre-Luc Dallaire-Demers,2 and Frank K. Wilhelm1
1Theoretical physics, Saarland University, 66123 Saarbrücken, Germany
2Department of Chemistry and Chemical Biology,
Harvard University, Cambridge, MA 02138, USA
(Dated: January 30, 2017)
WepresentaSuperconductingPlanarARchitectureforQuantumSimulations(SPARQS)intended
toimplementascalablequbitlayoutforquantumsimulators. Tothisend,wedescribetheiFRED-
KIN gate as a controlled entangler for the simulation of Fermionic systems that is advantageous
if it can be directly implemented. Using optimal control, we show that and how this gate can be
efficientlyimplementedintheSPARQScircuit,makingitapromisingplatformandcontrolscheme
7 forquantumsimulations. Suchaquantumsimulatorcanbebuiltwithcurrentquantumtechnologies
1 to advance the design of molecules and quantum materials.
0
2
I. INTRODUCTION finitelattice. FromtheRezQuliterature,weassumethat
n
single-qubit and two-qubit gates can be implemented
a
J straightforwardly based on known results [11, 13, 16].
Richard Feynman proposed building computers that
6 processesinformationbasedontherulesofquantumme- QuantumsimulationsalsobenefitfromiFREDKINgates
2 to efficiently implement the time evolution of Fermionic
chanicstosolvethedifficultproblemofsimulatingquan-
Hamiltonians. The iFREDKIN gate is a new entangling
tumsystems[1]. MotivatedbyShor’salgorithmtofactor
] gate in the family of three-qubit gates which includes
h largecompositenumbers[2]anditspotentialincryptog-
p raphy, considerable efforts have been invested in build- the TOFFOLI and FREDKIN gates. However, unlike
- the latter two, it has no classical analog. It performs
ing a universal fault-tolerant quantum computer [3–5].
t
n However, it was recently argued that such a universal an entangler, the iSWAP [17], on two target qubits con-
a ditioned on a control qubit (conditional iSWAP). This
machine may not be necessary for the purpose of sim-
u conditional evolution is naturally adapted to the need to
ulating Fermionic systems beyond the reach of modern
q
interfereanentangledmanybodystateinthetargetqubit
[ supercomputers [6–10] as long coherence times (quan-
tum memories) ought to be sufficient. Following ad- toareferencestateasakeystepinphaseestimation. We
1 expect it to be the most costly gate in quantum simula-
vances in quantum technologies from the past decades,
v tions,asitisusedbetweenchainsofqubitstoimplement
it can be suggested that Feynman’s machine could pos-
0 hopping and interaction terms of Fermionic Hamiltoni-
7 sibly be built in a near future. It is entirely possible
ans [15]. Therefore, the remaining challenge is to show
8 that quantum simulations of Fermionic systems such as
that the iFREDKIN gates can be implemented between
7 Fermi-Hubbard-likemodelsandmolecularmodelswillbe
the probe qubit P and neighboring system qubits S.
0 a main application of quantum computers in the coming
1. years. Insec.III,asaproofofprinciple,weuseGRadientAs-
cent Pulse Engineering (GRAPE) [18, 19] to show that
0 In this paper, we propose a superconducting circuit
7 architecture for simulating general Fermionic systems the iFREDKIN gate can be implemented in a time com-
1 parabletoasimpleiSWAPgatebetweenneighboringsys-
which can be cast in a second-quantized formulation.
: tem qubits, even when leakage is included in a SPARQS
v It is based on a superconducting circuit implementation
circuit. Thus,weconcludethatSPARQScircuitswithan
i called the RezQu architecture, as this has the tunability
X appropriateiFREDKINcontrolschemeprovideanatural
of its components and the simplicity of its implementa-
r platform for the simulation of Fermionic systems.
a tion [11–13]. The layout presented in sec. II is extensi-
ble and planar and it can be implemented with current
quantum technologies. The properties of this Supercon-
ducting Planar ARchitecture for Quantum Simulations II. CIRCUIT ARCHITECTURE
(SPARQS) are such that it can be used to prepare a
molecular or cluster state, measure its energy and corre- InRef.[15]wehighlightedthatadual-railqubitwitha
lation functions [14]. highly connected central qubit is well-suited for the pur-
It was also previously shown that for quantum simu- pose of simulating clusters of the Fermi-Hubbard model
lations, the number of gates that need to be tuned and and other Fermionic systems. As shown in fig. 1, the
benchmarked scales linearly with the size of the simu- layout consists of a register S which encodes a system
lated system [15]. For example, in the case of the Fermi- HamiltonianandabathregisterB usedintheprocedure
Hubbard model, the size of the system in a hybrid sim- of creating a Gibbs state of the system Hamiltonian in
ulation method [14? ? ] corresponds to the number of S. The Gibbs state preparation [20] or phase estimation
spinorbitalsinanexactlysolvedsub-latticeofthefullin- [21]digitalregisterR alsorequiresalineofqubitswhose
2
sizedependsonthedesiredprecisionofpreparedormea- times of superconducting qubits [24] is a good indication
sured energies. Interactions between registers S+B and thatthisarchitecturecouldbetestedwithminimalquan-
Randpossiblesubsequentcorrelationfunctionsmeasure- tum error correction. It is known that single-qubit and
mentsaremediatedthroughaprobequbitP betweenthe two-qubit gate can be efficiently implemented in RezQu-
digitalandanalogregisters. Linesbetweenregistersindi- like circuits [16]. However, iFREDKIN gates have not
cates where multi-qubit interactions are used in simula- been studied yet. In the next section, we show using
tionalgorithms. AnadvantageofusingamiddlequbitP optimal control that the iFREDKIN gate can be imple-
is that all-to-all connectivity is not required for the im- mented efficiently in a SPARQS processor for realistic
plementation of useful algorithms. The triangles formed circuit parameters.
by interaction lines between neighboring qubits in S and
register P are meant to indicate that iFREDKIN gates
have to be used with P as the control qubit.
Here, we will describe a possible superconducting cir-
cuit architecture for this purpose. Within this, we will
show in sec. III how to implement a fast and direct
iFREDKIN gate using optimal control methods.
FIG. 2. SPARQS: Superconducting Planar ARchitecture for
Quantum Simulations. It is a modified RezQu architecture.
Each frequency-tunable qubit (represented by a crossed box)
iscoupledtoacommontransmissionline. Notshownarethe
flux control lines of the qubits to change their detuning from
the bus.
FIG. 1. The S register is used to encode the system dynam-
ics, the B register contains bath qubits used to prepare a
simulated state in S. P is used to measure the correlation
functionsofthesimulatedsystemandtomediatetheinterac-
tion with the digital register R, which yields information on
the temperature of the Gibbs state prepared in S. (Figure
taken from [15])
The basic architecture for such a SPARQS circuit is
shown in in fig. 2, a modified RezQu architecture [11–
13].Qubitswithtunablefrequencyareconnectedthrough III. IFREDKIN
asuperconductingcavitywhichactsasabusforquantum
information[22,23]. Thequbitsdonotinteractwitheach
other unless they are brought in resonance with the cav- Analogous to the FREDKIN gate (conditional swap),
ity. This architecture can be fabricated in a planer way the iFREDKIN gate is an entangling three-qubit gate,
and is expeced to be extensible based on current quan- which performs an iSWAP operation on two qubits, de-
tum technologies. The exponential increase in coherence pending on the state of the first qubit, i.e., a conditional
3
iSWAP: decoherence limit compared to a gate decomposition of
multi-qubit gates.
U = 0 0 1 + 1 1 iSWAP
±iFREDKIN 4
| (cid:105)(cid:104) |⊗ | (cid:105)(cid:104) |⊗±
1 0 0 0 0 0 0 0
2
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0 z]
=  . (1) H 1.5
0 0 0 0 1 0 0 0 G
  [
00 00 00 00 00 ±0i ±0i 00 tuning 1 δδP((tt))
0 0 0 0 0 0 0 1 e S1
  D δS2(t)
  s
u
It is used in the context of quantum simulations to per- B 0.5
form the time evolution of Fermionic Hamiltonians [15] bit-
u
inaJordan-Wignerbasis[25],whichmapsindistinguish- Q
able particles with antisymmetric exchange properties 0
to a register of distinguishable qubits. The iFRED-
KIN gate is entangling since it will map a separable 0 10 20 30 40 50
state (001 + 101 )/√2 to a generalized GHZ state Time[ns]
| (cid:105) | (cid:105)
(001 +i 110 )/√2 [26]. Specifically, it executes a two-
| (cid:105) | (cid:105)
qubitiSWAPgate,whichisaperfecttwo-qubitentangler FIG.4. The3Z-controls,includinga4nsbufferoneachside
[27], conditional on a control qubit. When the control andaGaussianfilterwithstandarddeviationσ=0.4ns. The
qubit is in a superposition, it accumulates a phase i if coupling between the qubits is mediated by a bus.
and only if the state of the system qubits are different.
Hence, the iFREDKIN gate can be used to characterize The following Hamiltonian describes the architecture,
the interaction between qubits as if they where indistin- where the coupling between each qubit is mediated by a
guishable particles with antisymmetric exchange proper- common bus
ties. In a SPARQS circuit, the control qubit is always P
∆ ∆ 2
and the conditional iSWAP is performed between neigh- H =ω a†a+ ω (t) i b†b + i b†b
B i − 2 i i 2 i i
boring S qubits. (cid:88)i (cid:18) (cid:19) (cid:16) (cid:17)
+ g a†b +ab† . (2)
i i i
(cid:88)i (cid:16) (cid:17)
a and b are the bus and qubit annihilation opera-
i
tors, respectively. We pick realistic parameters for
the architecture in order to proceed with the proof-
of-principle. With a bus at ω /2π = 6.5GHz, the
B
off-resonant frequencies are set to ω /2π = 7.5GHz,
P
ω /2π = 8.0GHz and ω /2π = 8.5GHz, with anhar-
S1 S2
monicities ∆ /2π = 200MHz, ∆ /2π = 300MHz
P S1
− −
and ∆ /2π = 400MHz. The coupling strengths are
S2
−
g /2π = 30MHz, g /2π = 45MHz and g /2π =
BP BS1 BS2
60MHz, keeping the ratio g /∆ = 0.15 fixed. In all
i i
−
runsthecontrolshaveatimeresolutionof1ns,typicalfor
arbitrary waveform generators (AWGs), with fine steps
of 0.1ns for the simulated time evolution. Additionally,
the pulse shapes are filtered by a Gaussian window with
FIG. 3. Unit cell for the optimal control problem. It is as- abandwidthof331MHz(standarddeviationσ =0.4ns).
sumed that the other qubits are decoupled from interacting Fortheoptimization,weworkintherotatingframewith
witheachotherduringthecontrolsequencebydetuningthem angular frequency ω =ω . Therefore the implemented
R B
from the bus. Hamiltonian reads
∆ ∆ 2
Here we want to implement the iFREDKIN gate on a H = δ (t) i b†a + i b†b
clusteroftheSPARQSprocessorasshowninfig.3. Pulse i − 2 i i 2 i i
(cid:88)i (cid:18) (cid:19) (cid:16) (cid:17)
shapes found by numerical methods, such as GRAPE
+ g(i) a†b +ab† . (3)
[18], have proven to be faster than analytical control i i
pulses on this architecture [16], and optimal control (cid:88)i (cid:16) (cid:17)
methods have been demonstrated successfully on three δ =ω ω isthedetuningofqubitifromthebus. For
i i B
−
qubit gates [28]. Additionally, we avoid to reach the each qubit, the first three energy levels are taken into
4
account. Since we are only interested in the correct evo- 100
lution of the computational subspace, the fidelity func- iSwap
tiononlymeasurestheoverlapoftheprojectedtotaltime iFredkin
evolution with the target gate [29] 10 1
−
Φ= 1 tr U†P U(t )P 2 , (4) Φ
4 F Q g Q −
and global phases a(cid:12)(cid:12)re(cid:110)omitted. (cid:111)(cid:12)(cid:12) 1ty10−2
In fig. 4 we show(cid:12) the optimized qub(cid:12)it-bus detuning deli
fi
parameter for the implementation of a iFREDKIN gate. In10−3
Thegate canbeimplemented in56nsfor thechosenpa-
rameters with a realistic control sequence with 99.99%
fidelity. The time evolution of the populations is shown
10 4
−
infig.5. Ascanbeseen,thecontrolqubitgetsde-excited
and re-excited during the process, hence allowing for the 20 24 28 32 36 40 44 48 52 56 60
Gatedurationt [ns]
g
1.2 FIG. 6. Speed limit for the 3 Z-controls for the fidelity
0110 Φ = 0.9999, compared to the iSWAP-gate with only two Z-
| | i
0101
1 |1|100i controls, S1 ans S2, and P is set to a parking frequency of
| | i 10GHz. The target fidelity in both cases is Φ=0.9999.
0δ
i| | i,2i
1
0.8 P−
dynamics to be reenacted a two-excitation interference
0.6
experiment, being consistent with the speed limit corre-
sponding for a small multiple of the periods induced by
0.4 the various g couplings. Leakage into the second level
of the qubits plays an important role in the gate im-
0.2 plementation of the numerical pulse. Also, the pulse
shapes are highly symmetric, as are the resulting time
0 evolutions of the populations. The speed limit shown in
0 10 20 30 40 50 fig. 6 proves that the iFREDKIN gate can easily be im-
Time[ns] plementedbelowagatedurationoft =55ns. Thistime
g
scale is typical for analytic two-qubit pulse shapes, i.e.,
1.2
0110 the simultaneous version of the Strauch sequence[16, 30]
| | i
0101 and compares to the implementation of a traditional
1 |1|100i
| | i iSWAP on the same architecture as shown in fig. 6, set-
0δ
i| | i,2i ting the P qubit to a off-resonant parking frequency of
1
0.8 P− ω /2π =10GHz.
P
0.6
IV. CONCLUSION
0.4
Quantum simulations could be one of the main appli-
0.2
cations for future quantum computers where they could
outperform their classical counterparts. We outlined the
0
SPARQS circuit as an explicit superconducting imple-
0 10 20 30 40 50
mentation for a quantum simulator. We showed how the
Time[ns]
mostexpensivegateinquantumsimulationsofFermionic
systems, the three-qubit iFREDKIN, can be efficiently
FIG. 5. Time evolution in the subspace with two excita-
implemented in such a device using GRAPE, a standard
tions. Shown are the population of the states |0|110(cid:105) and
optimal control method. For reasonable parameters of
|0|101(cid:105), which are swapped, the population of the intermedi-
atestate|1|100(cid:105),thesumofpopulationsoftheleakagelevels theSPARQSqubits,theiFREDKINcanbeimplemented
(cid:80) |0|δ (cid:105)=|0200(cid:105)+|0020(cid:105)+|0002(cid:105)andthesumofpopula- in a time slightly longer than a typical iSWAP gate. As
i i,2
tionsinthefourotherstates1−=|0|011(cid:105)+|1|010(cid:105)+|1|001(cid:105)+ coherencetimesforsuperconductingqubitskeepincreas-
|2|000(cid:105). Thereissomeexcursionintheleakagelevelswhichis ingtodate,itisrealistictoexpectthatalargenumberof
canceled by the end of the pulse. The inclusion of this extra thosegatescouldbereliablyusedforthepurposeofper-
Hilbert space may be beneficial for the efficiency of the gate. forming quantum simulations beyond what can be done
5
on classical computers.
ACKNOWLEDGMENTS
WeacknowledgesupportfromtheEuropeanUnionun-
der the ScaleQIT integrated project.
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