High-fidelity resonator-induced phase gate with single-mode squeezing

1.6 MB·

by  Shruti Puri

High-fidelity resonator-induced phase gate with single-mode squeezing
Shruti Puri1 and Alexandre Blais1,2
1D´epartment de Physique, Universit´e de Sherbrooke, Sherbrooke, Qu´ebec, Canada J1K 2R1
2Canadian Institute for Advanced Research, Toronto, Canada
(Dated: January 14, 2016)
Weproposetoincreasethefidelityoftwo-qubitresonator-inducedphasegatesincircuitQEDby
theuseofnarrowbandsingle-modesqueezeddrive. Weshowthatthereexistsanoptimalsqueezing
angleandstrengththaterasesqubit‘which-path’informationleakingoutofthecavityandthereby
minimizes qubit dephasing during these gates. Our analytical results for the gate fidelity are in
excellentagreementwithnumericalsimulationsofacascadedmasterequationthattakesintoaccount
thedynamicsofthesourceofsqueezedradiation. Withrealisticparameters,wefindthatitispossible
to realize a controlled-phase gate with a gate time of 200 ns and average infidelity of 10−5.
6
1 PACSnumbers: 03.67.-a,42.50.Pq,42.50.Dv,03.65.Vf
0
2
Taking advantage of pulse shaping techniques [1] and 01 10
n increasing coherence times [2], single-qubit gate fidelity 11 | |
a |
J exceeding 99% has been demonstrated with supercon- α] 00
3 ducting qubits [3, 4]. Similar fidelities have been re- m[ |
1 ported for two-qubit gates based on frequency tunable I
qubits[3]. Tuningthequbittransitionfrequencyis,how-

] ever, sometimes undesirable or difficult [2] and, for this
h
p reason,fixed-frequencytwo-qubitgatesarebeingactively Re[α]
- developed [5–11]. Unfortunately, the fidelity of these all-
t
n microwavegates[12]isstillbelowthatrequiredforfault-
FIG.1. (coloronline)Qubit-state-dependentevolutionofthe
a tolerant quantum computation [13].
outgoingresonatorfieldinphasespace(solidlines)inthero-
u
A promising all-microwave gate is the resonator- tating frame of the drive. Quantum fluctuations correspond-
q
[ induced phase gate [11]. This multi-qubit logical op- ing to a coherent (dashed circles) and squeezed drive (filled
ellipses) are represented at an extremum of the paths. By
eration is based on the dispersive regime of circuit
1 adjustingthesqueezingangle,quantumfluctuationscanhelp
QED where the qubits are far detuned from a cavity
v in erasing which-path information and thereby reduce qubit
5 mode [14, 15]. As schematically illustrated in Fig. 1, dephasing during the joint qubit-field evolution.
7 adiabatically turning on and off an off-resonant drive,
2 the cavity state evolves from its initial vacuum state by
3
following a qubit-state-dependent closed loop in phase Here we propose to use single-mode squeezing to ad-
0
. space. After this joint qubit-cavity evolution, the cavity dress the challenge of implementing resonator-induced
1 returns to vacuum state and the qubits are left unentan- phase gates with a gate error below the fault-tolerance
0
gled from the cavity but with an acquired a non-trivial threshold and with short gate times. We show that an
6
1 phase. By adjusting the drive amplitude, frequency, and optimal, and experimentally realistic, choice of squeez-
: duration, an entangling phase gate can be realized [11]. ing power and angle can dramatically improve the gate
v
i This is analogous to the geometric phase gate already fidelity. The intuition behind this improvement is
X demonstrated with ion-trap qubits [16] and theoretically schematically illustrated in Fig. 1: enhancing fluctua-
r studied in the context of circuit QED, quantum dots in tions in the appropriate quadrature erases the which-
a
a cavity, and trapped ions [6, 17–19]. path information while leaving the path area, and hence
In practice, the gate fidelity is limited by residual the accumulated phases, unchanged. This improvement
qubit-cavity entanglement and by photon loss. Indeed, in gate fidelity is the converse of the recent realization
during the adiabatic pulse, photons entangled with the that single-mode squeezed light is generally detrimental
qubit leave the cavity carrying ‘which-path’ information to dispersive qubit measurement [20, 21]. Using squeez-
about the two-qubit state, in turn causing dephasing. ing powers close to that already experimentally achieved
Thiscanbepartiallyavoidedbydrivingthecavitymany with superconducting circuits [22], we find average gate
linewidths from its resonance frequency. In this situ- errorsthataresuppressedbyanorderofmagnitudewith
ation, the cavity is only virtually populated and the respect to a coherent state input drive. In other words,
qubit-photon entanglement is small [6, 17–19]. Unfor- we suggest to use quantum-bath engineering to protect
tunately, this also leads to longer gate times, a problem the dynamics of a quantum system, going beyond the
that can be partially mitigated by using pulse shaping typical use of this approach, which is focussed on creat-
techniques [11]. ing or stabilizing certain steady-states [23–25].
2
In the dispersive regime where the qubit-cavity fre- dependence of the squeeze parameter reflects the finite
quency detuning ∆ = ω ω is large with respect to bandwidth of the JPA around the drive frequency ω .
a r d
−
the coupling strength g, the system Hamiltonian in the Measurement-induced dephasing is caused by photon
presence of a cavity drive takes the form numberfluctuationsand,followingRefs.[15,30,32],can
ω ω be expressed as
Hˆ =ω aˆ aˆ+ aσˆ + aσˆ +χ(σˆ +σˆ )aˆ aˆ
r † z1 z2 z1 z2 †
2 2 (1) t
+(cid:15)(t)(aˆ†e−iωdt+h.c.). γφ(t)=2χ2 (cid:104)[nˆ(t)−n¯(t)][nˆ(t(cid:48))−n¯(t(cid:48)])(cid:105)dt(cid:48). (3)
(cid:90)0
In this expression, ω and ω are respectively the cav-
r a An approximate expression for this rate can be obtained
ity and qubit frequencies, χ = g2/∆ the dispersive cou-
in the limit of adiabatic evolution of the cavity where
pling strength, (cid:15) the drive amplitude, and ω the drive
d the fast dynamics can be neglected. In this situation,
frequency. This description is accurate for intra-cavity
this rate is given by [26]
photon number n n = ∆2/4g2 [15]. Although the
crit
resonator-induced(cid:28)phase gate is tolerant to large varia- 4χ2κ
γ (t)
tions in qubit frequencies and coupling strengths [26], to φ ≈ δ2
r (4)
simplify the discussion we assume the qubits to be iden- α(t)2
tical. N(ωr)+ | | e−2r(ωd)cos2Φ+e2r(ωd)sin2Φ ,
× 2
How the resonator-induced phase gate emerges from (cid:20) (cid:16) (cid:17)(cid:21)
evolution under Eq. (1) can be made clearer by per- where κ is the cavity decay rate and Φ = θ arg[α(t)]
−
forming the time-dependent polaron-like transformation is the angle of squeezing relative to the drive. We have
σDˆ (αˆ)α(cid:48))(t=) eoxnp(Hˆαˆ(cid:48)aˆ[2†7−,αˆ2(cid:48)8∗]aˆ.)Awsithshαˆo(cid:48)w(tn)=inαth(te)−Su(χp/pδler)m(σˆenz1ta+l aplospouilnattrioodnuacsesdocNia(tωedr)w=itshinthh2ers(qωure)e,ztehdeinthpeurtmfiaelldp.hCotroun-
z2
Material [26], this leads to the effective Hamiltonian cially, this quantity is evaluated at the cavity frequency.
This contribution to γ can be made negligible by work-
2χ2 α2 φ
Hˆ = 1[ω +2χnˆ(t)](σˆ +σˆ ) | | σˆ σˆ ,(2) ingatadetuningδ thatislargerthanthetypicallysmall
eff 2 a z1 z2 − δ z1 z2 r
r bandwidthofcurrentJPAs[22,31]. Moreover,intheab-
wherewehavedefined nˆ(t)=aˆ aˆ+ α(t)2, withtheam- sence of squeezing (r =0), the second term of Eq. (4) is
†
| |
plitude α(t) satisfying α˙ = iδ α i(cid:15)(t) and the drive- theusualexpressionformeasurement-induceddephasing
r
− −
cavity detuning δ = ω ω . The last term of Eq. (2) in a coherent field [30]. While N(ω ) in the first term
r r d r
−
representsthenonlinear,qubit-state-dependentphasein- is evaluated at the cavity frequency, the squeeze param-
ducedbythedrivencavity. Toavoidqubit-fieldentangle- eter in the second term is rather evaluated at the drive
ment after the gate, the cavity drive is chosen such that frequency ω . For δ (1/τ, κ), the cavity field closely
d r
(cid:29)
the field starts and ends in its vacuum state. For sim- follows α(t) (cid:15)(t)/δ such that Φ θ at all times. As a
r
∼ ∼
plicity, we consider the drive to have a Gaussian profile result,choosingthesqueezeangleθ =0leadstoanexpo-
(cid:15)(t)=(cid:15) e t2/τ2 fortimes t /2<t<t /2witht =5τ. nential reduction with increasing r(ω ) of the dephasing
0 − g g g d
−
As illustrated in Fig. 2(a), for δ 1/τ the cavity field rate in the adiabatic limit. This confirms the intuition
r
(cid:29)
evolves adiabatically and α(t) follows a closed path in presented in Fig. 1 that increasing the quantum fluctua-
phase space. With the cavity being only virtually pop- tions in the appropriate quadrature with respect to the
ulated, α(t) returns to the origin after the pulse. The field displacement leads to a reduction of qubit dephas-
qubit-state dependent phase acquired during this evolu- ing. Minimizing photon shot-noise by number-squeezed
tionisdeterminedbytheareainphase-spaceenclosedby radiation was also studied in the context of cavity spin
α(t) and is specified by the pulse amplitude (cid:15) , duration squeezing in Ref. [33].
0
τ, and detuning δ [29]. By appropriately choosing these Whiletheaboveargumentsuggestsanexponentialde-
r
parameters, the evolution under Eq. (2) can correspond creaseofthedephasingrateforarbitrarilylargesqueezing
to the two-qubit unitary U =Diag(1,1,1, 1). powers, in practice there exists an optimal r(ω ). In or-
zz d
−
Anotheradvantageofworkingwithalargedetuningδ der to understand this, it is useful to consider again the
r
is that measurement-induced dephasing γ of the qubits evolutionofthecavityfieldinphasespace. Asillustrated
φ
is small [15, 30]. On the other hand, the strength of the inFig.2(a)and(b),forlargedetuningsarg[α(t)]issmall
qubit-qubitinteractiongoesdownwithδ ,whichinturns at all times, and choosing a constant θ 0 minimizes
r
∼
leads to long gate times. As we now show, we solve the the dephasing rate. However, for large r(ω ), the anti-
d
challenge of minimizing γ and maintaining short gate squeezed quadrature enhances dephasing at short times
φ
timesbyusinganinputfieldthatisadisplacedsqueezed where arg[α(t)] fluctuates widely. This leads to an over-
field rather than a coherent state. The squeezed field allenhancementofγ . Figure2(c)showsthedependence
φ
is characterized by the squeeze parameter r(ω) and an- of the normalized instantaneous dephasing rate on r(ω )
d
gle θ, and, in practice, can be produced by a Joseph- with θ = 0 at the three times indicated by the dots on
son parametric amplifier (JPA) [22, 31]. The frequency Fig. 2(a). The existence of an optimal squeezing power
3
(a) -1.0 δr/2π=160MHz 0.1
0.04 t t3 -1.2 δr/2π=312 MHz
α] 2 -1.4 δr/2π=640 MHz 0.08
m[0.02 t
1 -1.6
I
-1.8 0.06
0.00 Error)-2.0 Error
0 0.5 1.0 1.5 2.0 2.5 g(-2.2 0.04
Re[α] lo
1.0 -2.4
(b)
-2.6 0.02
d] 0.5
a -2.8
[r (a) 0 (b)
α] 0.0 -3.00 10 20 0 0.1 0.2 0.3 0.4 0.5
arg[ t1 t2 t3 Squeezing power [dB] θ/π
-0.5
FIG.3. (coloronline)(a)Analytical(solidlines)andnumeri-
-2 -1 0 1 2 cal(symbols)gateerrorrateasafunctionofsqueezingpower
t/τ
for θ=0 and three detunings δ /2π=160 MHz (dark blue),
1.0 r
312MHz(blue)and640MHz(red). Thecorrespondingmax-
t
1
imum drive amplitudes are (cid:15) /2π = 278.5 MHz, 795.8 MHz,
0.8 t 0
0γφ t2 and 2.31 GHz. The gate time is tg = 200 ns, cavity decay
/ 0.6 3 κ/2π=10MHz,qubit-cavitycouplingg/2π=160MHz,and
φ
γ detuning∆/2π=3.2GHzcorrespondingtoadispersivecou-
0.4 plingofχ/2π=8MHzandn =100. (b)Analytical(solid
crit.
(c) lines) and numerical (symbols) gate error rate as a function
0.2 ofsqueezingangleθforafixedsqueezingpowerof5.7dBand
0 4 8 12 16
detuning δ /2π=320 MHz.
Squeezing power [dB] r
FIG. 2. (color online) (a) Evolution in phase space of the
cavity field α(t) with δr/2π = 320 MHz, κ/2π = 10 MHz, i,j,k,l 0,1 , and where µij,kl are qubit-state de-
τ = 40 ns, and tg = 200 ns. The colored dots represent {pendent}p∈ha{ses a}nd γij,kl represents non-unitary evolu-
the field at three different times and the corresponding lines
tion due to the dephasing rate γ . In addition to two-
φ
represent the angles along which the quantum fluctuations
qubitphases,evolutionundertheHamiltonianofEq.(1)
shouldbereducedtominimizedephasing. (b)Timeevolution
leads to single-qubit z rotations. Since these rotations
of arg[α]. (c) Normalized dephasing rate evaluated at the
threeindicatedtimesvssqueezingpowerforafixedsqueezing can be eliminated by an echo sequence, these are not
angleθ=0andr(ω )=0. Thenormalizationγ0 corresponds considered in the error estimation [11].
r φ
to the situation without squeezing, i.e., r=0.
A prescription to evaluate µ and γ , and there-
ij,kl ij,kl
fore the error E, can be found in [26]. This corresponds
to the full lines in Fig. 3(a) that show the error as a
is clearly apparent. The finite bandwidth of the input function of squeezing strength for different detunings δ
r
squeezedstateisanotherreasonfortheexistenceofsuch and fixed θ =0. The symbols in this figure are obtained
an optimal point. Indeed, for a fixed squeezing band- bynumericalintegrationofthecascadedmasterequation
width Γ, an increase in the squeezing power at ωd will forthesystemdescribedbytheHamiltonianEq.(1)and
also lead to an increase in thermal photons at ωr with driven by a degenerate parametric amplifier acting as a
N(ωr) = N(ωd)/[(ω ωd)2+Γ2] [34]. This contributes sourceofsqueezedradiation[26,35]. Thenumericalsim-
−
to qubit dephasing via the first term of Eq. (4). ulations of the cascaded master equation are carried out
We now turn to a more quantitative description of the with an open source computational package [36, 37]. We
improvement of gate fidelity that can be obtained from have fixed τ = 40 ns which corresponds to a gate time
using squeezing. For this, we first compute the gate er- of t = 200 ns. The drive parameters are chosen such
g
rorE =1 ψ (ψ ψ )ψ forthepureinitialstate that the cavity is empty at t = t /2 and a maximum of
T 0 0 T g
−(cid:104) |E | (cid:105)(cid:104) | | (cid:105)
ψ = 1(00 + 01 + 10 + 11 ) at t = t /2. In this 6photonsareexcitedinthecavity,whichcorrespondsto
| 0(cid:105) 2 | (cid:105) | (cid:105) | (cid:105) | (cid:105) − g
expression, ψ = U ψ is the desired target state n/n =0.06. Thesqueezingspectrumiscentredatthe
T zz 0 crit
| (cid:105) | (cid:105)
and is the quantum channel representing the system drive frequency. Its linewidth Γ/2π = 32 MHz is cho-
E·
under evolution with the Hamiltonian of Eq. (1) in ad- sen to ensure that δ Γ, which minimizes the thermal
r
(cid:29)
dition to the dephasing in the presence of a displaced photon population at the cavity frequency. As expected
squeezed drive. Following the notation from Ref. [11], from the discussion above, the error goes down with in-
the action of the channel on the qubits’ density matrix creasingdetuning. Moreimportantly,theerrorisreduced
elements takes the form (ij kl) = eiµij,kl−γij,kl, with in the presence of a squeezing input field up to a opti-
E | (cid:105)(cid:104) |
4
δr κ χ (cid:15) F0 FSqz. Squeezing
2π 2π 0 avg avg
(MHz) (MHz) (MHz) (GHz) % % Power (dB)
320 10 8 0.796 98.16 99.89 16
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Supplemental material for “High-fidelity resonator-induced phase gate with
single-mode squeezing”
Shruti Puri1 and Alexandre Blais1,2
1. D´epartment de Physique, Universit´e de Sherbrooke, Sherbrooke, Qu´ebec, Canada J1K 2R1 and
2. Canadian Institute for Advanced Research, Toronto, Canada
EFFECTIVE HAMILTONIAN
In the rotating frame of the drive the Hamiltonian of the two qubits dispersively coupled to a driven resonator
reads,
δ δ
Hˆ =δ aˆ aˆ+ q,1σˆ + q,2σˆ +(χ σˆ +χ σˆ )aˆ aˆ+(cid:15)(t)(aˆ+aˆ ) (S1)
r † 1z 2z 1 1z 2 2z † †
2 2
with δ = ω ω and χ the dispersive shift for i = 1,2. On this Hamiltonian we apply the time-dependent
qi ai d i
−
displacement transformation Hˆ˜ = D†(αˆ0)HˆD(αˆ0) iD†(αˆ0)D˙(αˆ0), where D(αˆ0) = exp(αˆ0aˆ† αˆ∗0aˆ) and αˆ0 = α
− − −
(χ σˆ +χ σˆ )α/δ . This transforms the resonator operator to aˆ αˆ +aˆ, where αˆ is the qubit-state-dependent
1 1z 2 1z r 0 0
→
coherent amplitude of the resonator mode and the quantum fluctuations are contained in the operator aˆ. Under this
transformation with αˆ˙ = iδ αˆ i(χ σˆ +χ σˆ )αˆ i(cid:15)(t) the Hamiltonian reduces to
0 r 0 1 1z 2 1z 0
− − −
δ δ χ χ α2
Hˆeff =δraˆ†aˆ+ q1σˆ1z+ q2σˆ2z+(χ1σˆ1z+χ2σˆ2z) nˆ 2 1 2| | σˆ1zσˆ2z. (S2)
2 2 h i− δ
r
To simplify the above expression, we have taken the expression for αˆ to zeroth order in χ, in other words α˙
0
∼
iδ α i(cid:15)(t) is a c-number in Eq. (S2). Taking nˆ = α2 +aˆ aˆ and simplifying to χ = χ = χ, we recover the
r † 1 2
− − | |
Hamiltonian of Eq. (2) of the main paper. It is important to note that although thermal resonator population
(i.e. aˆ aˆ )contributestothephaseacquiredbythequbitsduetothedispersiveinteraction,acoherentresonatorfield
†
h i
(α)ishoweverrequiredtomediateatwo-qubitinteractionbetweenthequbits. Singlequbitrotationsarisingfromthe
fourthtermoftheaboveequationcanbeeliminatedbyspinechoandbychoosingthedrivepulseparametersi.e., the
detuning,durationandpower,itispossibletoimplementagategivenbytheunitaryoperationU =Diag(1,1,1, 1).
zz
−
THEORETICAL ESTIMATION OF DEPHASING RATE
As described in the main paper, the dephasing rate is given by
t
γ =2χ2 (nˆ(t) n¯(t)) (nˆ(t) n¯(t)) dt (S3)
φ 0 0 0
h − h − i
Z0
where nˆ(t)=a (t)a(t) and n¯(t)= a (t)a(t) . Under the displacement transformation D(α) given above
† †
h i
δn(t)δn(t) =α (t)α (t) aˆ(t)aˆ(t) +α(t)α(t) aˆ (t)aˆ (t) +α(t)α (t) aˆ (t)aˆ(t) +α (t)α(t) aˆ(t)aˆ (t)
0 ∗ ∗ 0 0 0 † † 0 ∗ 0 † 0 ∗ 0 † 0
h i h i h i h i h i
+α (t) aˆ(t)aˆ (t)aˆ(t) +α(t) aˆ (t)aˆ (t)aˆ(t) +α (t) aˆ (t)aˆ(t)aˆ(t) +α(t) aˆ (t)aˆ(t)aˆ (t)
∗ † 0 0 † † 0 0 ∗ 0 † 0 0 † † 0
h i h i h i h i
+ aˆ (t)aˆ(t)aˆ (t)aˆ(t) aˆ (t)aˆ(t) aˆ (t)aˆ(t) ,
† † 0 0 † † 0 0
h i−h ih i
(S4)
where δn(t)=nˆ(t) n¯(t). In this displaced and rotating frame, the Langevin equations read [S1]
−
d κ
a= (iδ + )aˆ iˆ eiωdt (S5)
r in
dt − 2 − F
dα κ
= (iδ + )α i(cid:15)(t), (S6)
r
dt − 2 −
where ˆ corresponds to the input quantum noise coupled to the resonator. In the presence of the squeezer, the
in
correlatFion functions of this field are as usual: hFˆi†n(ω)Fˆin(ω0)i = N(ω)κδ(ω − ω0), hFˆin(ω)Fˆi†n(ω0)i = (N(ω) +
2
1)κδ(ω−ω0), hFˆin(ω)Fˆin(ω0)i = M(ω)κδ(ω+ω0), and hFˆi†n(ω)Fˆi†n(ω0)i = M∗(ω)κδ(ω+ω0), with M = |M|e2iθ and
M N(N +1) [S2].
| |≤
Using these expressions, we can evaluate the field correlation functions entering the dephasing rate. For example,
p
t t0
haˆ†(t)aˆ(t0)i= hFˆi†n(s)Fˆin(s0)ie−iωd(s−s0)+iδr(t−s)−κ(t−s)/2)−iδr(t0−s0)−κ(t0−s0)/2dsds0
Z−tg/2Z−tg/2
= t t0 ∞ N(ω)ei(ω−ωd)(s−s0)+iδr(t−s)−κ(t−s)/2)−iδr(t0−s0)−κ(t0−s0)/2dωdsds0
Z−tg/2Z−tg/2Z−∞
∞ ei(ω−ωd)(t−t0)
N(ω) dω (S7)
∼ (ω ω )2+κ2/4
Z−∞ − r
The last expressions is obtained by eliminating fast oscillating and decaying terms. Similarly,
∞ ei(ω−ωd)(t−t0)
aˆ(t)aˆ(t) M(ω) dω (S8)
0
h i∼ [i(ω ω)+κ/2][i(ω +ω 2ω )+κ/2]
Z−∞ r− r − d
Inordertoevaluatetheaboveexpressionsweassumethatthesqueezingiscentredatfrequencyω withbandwidth
d
Γ such that N(ω)=N(ω )Γ2/[(ω ω )2+Γ2] and M(ω)=M(ω )Γ2/[(ω ω )2+Γ2]. If δ Γ,κ then the product
d d d d r
− − (cid:29)
of Lorentizians in Eq. (S7) and (S8) can be written as a sum of Lorentzians and the two equations reduce to:
κΓ
haˆ†(t)aˆ(t0)i∼N(ωd)2(δ2+κ2/4)e−Γ(t−t0), (S9)
r
κΓ
haˆ(t)aˆ(t0)i∼M(ωd)2(iδ +κ/2)2e−Γ(t−t0). (S10)
r
Following the same procedure, we find
κΓ
haˆ(t)aˆ†(t0)i∼N(ωd)2(δ2+κ2/4)e−Γ(t−t0)+e−iδr(t−t0)−κ(t−t0)/2 (S11)
r
κΓ
haˆ†(t)aˆ†(t0)i∼M(ωd)∗2( iδ +κ/2)2e−Γ(t−t0). (S12)
r
−
Assuming the output field of the squeezer to be Gaussian, aˆ (t)aˆ (t)aˆ(t) = aˆ(t)aˆ (t)aˆ(t) = aˆ (t)aˆ(t)aˆ (t) =
† † 0 0 † 0 0 † † 0
h i h i h i
aˆ (t)aˆ(t)aˆ(t) =0 and using Wick’s theorem the remaining terms in Eq. (S4) can be evaluate to be
† 0
h i
aˆ (t)aˆ(t)aˆ (t)aˆ(t) aˆ (t)aˆ(t) aˆ (t)aˆ(t) = aˆ (t)aˆ(t) aˆ(t)aˆ (t) + aˆ(t)aˆ(t) aˆ (t)aˆ (t) (S13)
† † 0 0 † † 0 0 † 0 † 0 0 † † 0
h i−h ih i h ih i h ih i
The above expression can be evaluated using Eq. (S9)-(S12) and since we assume δ Γ κ it simplifies to,
r
(cid:29) (cid:29)
N(ω )Γ2κ N(ω )κ
d r
aˆ (t)aˆ(t)aˆ (t)aˆ(t) aˆ (t)aˆ(t) aˆ (t)aˆ(t) (S14)
† † 0 0 † † 0 0
h i−h ih i∼ δ4 ∼ δ2
r r
Theexpressionsforthecorrelationfunctions,Eq.(S9)-(S14)cannowbesubstitutedinEq.(S4)toobtainanexplicit
expression for the dephasing rate. A compact approximate expression for this rate can be obtained under an ideal
adiabatic evolution of the resonator field i.e., when the duration of the drive pulse τ 1/δ . Under this assumption,
r
(cid:29)
the field amplitude evolves slowly compared to other relevant energy scales in the system, so that α(t) α(t). In
0
∼
this situation we find
2χ2κα(t)2 4χ2κ
γφ(t)∼ δ2+|κ2/4| e−2r(ωd)cos2Φ+e2r(ωd)sin2Φ + δ2 sinh2r(ωr) (S15)
(cid:18) r (cid:19)h i r
with Φ = θ arg[α(t)], N(ω ) = sinh2r(ω ) and M(ω ) = sinhr(ω )coshr(ω ). Under ideal adiabatic evolution of
d d d d d
−
the cavity field amplitude with δ κ, arg[α(t)] 0 during the whole gate operation. Thus a squeezed input field
r
(cid:29) ∼
with a constant squeezing angle θ =0 leads to an exponential decrease of the dephasing rate.
Note that the above expressions is derived for an ideal squeezed state satisfying M(ω ) = N(ω )[N(ω )+1].
d d d
However this need not be the case and the above equation can be generalized to a thermal squeezed drive as,
p
2χ2κα(t)2 4χ2κ
γφ(t)= δ2+|κ2/4| 2N0(ωd)+e−2r(ωd)cos2Φ+e2r(ωd)sin2Φ + δ2 sinh2r(ωr) (S16)
(cid:18) r (cid:19)h i r
whereN(ω )=N (ω )+N (ω ),N (ω )=sinh2r(ω )andM(ω )=sinhr(ω )coshr(ω ). Thus,atΦ=0,r(ω )=0
d 0 d 1 d 1 d d d d d r
the dephasing rate decreases exponentially with effective squeezing strength r = ln[exp( 2r(ω ))+2N (ω )]/2.
eff d 0 d
− −
AslongastheunsqueezedthermalpopulationN (ω )issmall(forr >0),anexponentialdecreaseinγ isobtained.
0 d eff φ
3
DPA
FIG.S1. Illustrationofthesetupproposedforimplementingthetwo-qubitresonator-inducedphasegate. Asourceofsqueezing
(DPA)drivesaresonatorthatisdispersivelycoupledtotwotransmonqubits. Thedrivenresonatorinducesaπ-phasegateon
the qubits.
NUMERICAL ESTIMATION OF GATE FIDELITY
ThesymbolsinFig.3andtheentriesofTable1ofthemainpaperareobtainedbynumericallysolvingthecascaded
master equation for the system shown in Fig. S1. This corresponds to a resonator coupled to two qubits and driven
by an ideal degenerate parametric amplifier (DPA). In frame rotating at ω , the cascaded master equation reads [S2]
d
dρˆ
= i[Hˆ,ρˆ]+ [√Γˆb+√κaˆ]ρ, (S17)
dt − D
with
√Γκ δ +∆
Hˆ =δ aˆ aˆ+i(cid:15) (t)(ˆb 2eiθ ˆb2e iθ)+(cid:15)(t)(aˆ+aˆ )+i (ˆb aˆ aˆ ˆb)+ r (σˆ +σˆ )+χ(σˆ +σˆ )aˆ aˆ,
r † s † − † † † 1z 2z 1z 2z †
− 2 − 2
(S18)
whereˆb( )aretheannihilation(creation)operatorfortheDPA,ΓlinewidthoftheDPAandθdeterminesthesqueezing
†
angle at the output of the DPA. The DPA is resonantly driven with a parametric drive (cid:15) (t) of equal frequency to the
s
coherent drive (cid:15)(t).
Asdiscussedinthemaintext,forsimplicityweuseGaussianpulseshapesforboththedrives(cid:15)(t)=(cid:15) (t)exp( t2/τ2)
0
−
and (cid:15) (t)=(cid:15) (t)exp( (t t )2/τ2). It takes finite time for squeezing to develop in the DPA. As a result the para-
s 0,s 0
− −
metric drive is displaced in time by a small amount (t 8 ns) to ensure that the peak of the coherent drive and
0
∼
that of the squeezing strength r(t) coincide. At t= t /2, the DPA and the resonator are initialized to their vacuum
g
−
state and the qubits to the separable state ψ = (00 + 01 + 10 + 11 )/2. The cascaded master equation is
0
| i | i | i | i | i
simulated using the computational package Qutip [S3],[S4]. Figure S2 shows the numerical estimate of the gate error
for different squeezing powers when κ χ. From the figure we see that the qualitative dependence of the error on
(cid:28)
the squeezing power remains the same as when χ<κ (Fig. 3 in the main paper).
ESTIMATION OF SQUEEZING POWER FROM DPA
Following Ref. [S2], the equation of motion for the DPA field reads
dˆb Γ
= ˆb+2(cid:15) (t)ˆb √Γˆb . (S19)
s † in
dt −2 −
Taking the Fourier transform of the above equation yields,
Γ
iωˆb(ω)=2 (cid:15) (ω ω )ˆb(ω )dω ˆb(ω) √Γˆb (ω). (S20)
s 0 0 0 in
− − − 2 −
Z
For a broad Gaussian pulse with duration τ > 1/Γ, we can approximate the drive to be time-independent so that
(cid:15) (ω ω ) 2(cid:15) δ(ω ω ). In this case the correlations in the output field of a one-sided resonator are given
s − 0 ∼ π 0,s − 0
q
4
−2.6
−2.8
−3
−3.2
or)
Err−3.4
g(
o
l
−3.6
−3.8
−4
−4.2
0 5 10 15 20
Squeezing Power (dB)
FIG. S2. Numerical result of gate error rate as a function of squeezing power for θ = 0, δ /2π = 56 MHz, (cid:15) /2π = 104 MHz
r 0
(red)and111MHz,(cid:15) /2π=295MHz(blue). Thegatetimeist =200ns,τ =40ns,κ/2π=50kHz,Γ=32MHz,t =8ns
0 g 0
and χ/2π=4.5 MHz
by [S2],
hˆb†out(t)ˆbout(t0)i= λ2−4 µ2 "e−µ2(µt−t0) − e−λ2(λt−t0)# (S21)
hˆb†out(t)ˆbout(t0)i= λ2−4 µ2 "e−µ2(µt−t0) + e−λ2(λt−t0)#, (S22)
with λ= Γ +2(cid:15) and µ= Γ 2(cid:15) so that,
2 0,s 2 − 0,s
(λ2 µ2)2 λ4 µ4
N(ω )= − , M(ω )= − . (S23)
d 4λ2µ2 d 4λ2µ2
Thus the squeezing power at the DPA resonator (or drive) frequency becomes
Power= 10log(2N(ω )+1 2M(ω ))=20r(ω )log(e). (S24)
d d d
− − | |
ANALYTICAL GATE FIDELITY
As discussed in the main text, the full lines in Fig. 3 of the main text are obtained by analytically evaluating the
dephasing rate. In this evaluation, we take advantage of the fact that the action of the gate on each element of the
density matrix is (ij kl) = eiµij,kl−γij,kl, i,j,k,l = 0,1, where µij,kl = 2χ2(ij kl) α(t)2dt/δr are the qubit-
E | ih | − − | |
state dependent phases and γ represents the non-unitary decay, γ =γ =γ =γ = tg/2 γ (t)dt,
ij,kl 01,00 10,00 10,R11 01,11 −tg/2 φ
γ01,10 =0(foridenticalqubits)andγ11,00 =4 −tgt/g2/2γφ(t)dt. γij,kl isobtainedbysubstitutingEq.(S6)aRnd(S9)-(S14)
inEq.(S4)andintegratingnumericallytheresultingexpressionforγ (t). Wehaveneglectedthesinglequbitrotations
R φ
in the estimation of µ as they can be eliminated by an echo sequence.
ij,kl
As described in the main article, the gate fidelity is F = Tr[ψ ψ (ψ ψ )]. Rather than evaluating the
2T 2T 0 0
| ih |E | ih |
fidelity for a specific initial state, a more useful quantity is the entanglement fidelity [S5],
1
= Tr[Q (Q) ] (S25)
FEnt. d3 U†E U
Q
X
where Q is the set of d d Pauli operators (with d the dimension of the Hilbert space). In case of two qubits, this is
×
the set of 16 operators = ,σ ,σ ,...σ σ ,.... (Q) represents the action of the gate on each of these operators.
x1 y1 x1 x1
I E
We obtain the action of the gate on each element kl mn, (k,l m,n) = eiφkl,mn−γkl,mn both numerically and
| ih | E | ih |
5
analytically following the above described approach. The average gate fidelity quoted in Table 1 of the main text is
then obtained from [S6],
d +1
Ent.
= F (S26)
avg
F d+1
such that the average fidelity reads
1 1 1
avg = + cos(φ01,00)e−γ01,00 cos(φ11,01)e−γ11,01 e−γ11,00cosφ11,00. (S27)
F 2 5 − − 10
(cid:2) (cid:3)
[S1] D. F. Walls and G. J. Milburn, Quantum optics (Springer Science & Business Media, 2007).
[S2] C.GardinerandP.Zoller,Quantumnoise: ahandbookofMarkovianandnon-Markovianquantumstochasticmethodswith
applications to quantum optics, Vol. 56 (Springer Science & Business Media, 2004).
[S3] J. Johansson, P. Nation, and F. Nori, Computer Physics Communications 183, 1760 (2012).
[S4] J. Johansson, P. Nation, and F. Nori, Computer Physics Communications 184, 1234 (2013).
[S5] M. A. Nielsen, Physics Letters A 303, 249 (2002).
[S6] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge university press, 2010).