Non-Abelian holonomic transformation in the presence of classical noise

0.19 MB·

by  Jun Jing

Non-Abelian holonomic transformation in the presence of classical noise
Jun Jing1,2,3,4, Chi-Hang Lam2, and Lian-Ao Wu3,5∗
1InstituteofAtomicandMolecularPhysicsandJilinProvincialKeyLaboratory ofAppliedAtomicandMolecularSpectroscopy,
Jilin University, Changchun 130012, Jilin, China
2Department of Applied Physics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
3Department of Theoretical Physics and History of Science,
The Basque Country University (EHU/UPV), PO Box 644, 48080 Bilbao, Spain
4Department of Physics, Zhejiang University, Hangzhou 310027, Zhejiang, China
5Ikerbasque, Basque Foundation for Science, 48011 Bilbao, Spain
(Dated: January 31, 2017)
It is proposed that high-speed universal quantum gates can be realized by using non-Abelian
7
holonomic transformation. A cyclic evolution path which brings the system periodically back to a
1
degenerate qubit subspace is crucial to holonomic quantum computing. The cyclic nature and the
0
resulting gate operations are fully dependent on the precise control of driving parameters, such as
2
themodulatedenvelopfunctionofRabifrequencyandthecontrolphases. Weinvestigatetheeffects

n offluctuationsinthesedrivingparametersonthetransformationfidelityofauniversalsetofsingle-
a qubit quantum gates. We compare the damage effects from different noise sources and determine
J the “sweet spots” in the driving parameter space. The nonadiabatic non-Abelian quantum gate is

8 found to be more susceptible to classical noises on the envelop function than that on the control
2 phases. Wealso extendour study to a two-qubitquantum gate.

]
h
I. INTRODUCTION of coherence can occur due to fluctuations in the control
p
parameters. Nonadiabaticnon-Abeliangeometric-phase-
-
nt Quantum geometrical phase [1–4], which is propor- based holonomic transformation[15, 26–33] has been re-
a tional to the area spanned in the parameter space but is cently proposed to demonstrate universal operations for
u insensitive to the trajectory followed by the system, has quantum computation in both theory and experiment.
q inspired more and more efforts in circuit-basedquantum As an all-geometric scheme, it still retains the advan-
[ computation and quantum control protocols [5–7]. An tages of the conventional HQC but the evolution speed
1 appealing modern application of quantum geometrical can be greatly accelerated. There have been studies on
v phase is non-Abelian holonomic quantum computation the reliability of the nonadiabatic non-Abelian quantum
4 (HQC) [8–15], in which one performs a universal set of gateuponconsiderationsofadverseeffectsincludinggate
3 unitarytransformations,i.e., quantumlogicgatesopera- decoherence and noise [34], influence from the Lindbla-
2 tionsviacyclicevolutioninadegeneratesubspace. More dian [35], systematical error [36, 37], rotating-wave ap-
8
generally,HQC belongsto the fieldofquantumstate en- proximation [29] and finite operational time [38]. The
0
. gineering [16, 17]. It is argued that the implementation robustness of HQC, in particular the performance of the
1 ofquantum gatesencoded in a degeneratesubspace sup- unitarytransformationovergeneralinputstates,against
0
presses the effect of dynamical phase around the given classical noise (fluctuations) in the control Hamiltonian
7
1 loops in the parameter space. Thus it is not surprising parameters is still under investigation [37]. And that
: to see that the conventional HQC schemes are based on would be the focus in this work.
v
adiabaticevolutionduetoitsresilienceagainstlocalfluc-
i Classical noise characterizing small system perturba-
X tuations. The adiabatic theorem [18, 19] asserts that at
tions canhavea dramaticimpact onthe cyclic time evo-
r any momenta quantum systemremainsclosely atanin-
a stantaneous eigenstate of a slowly varying Hamiltonian. lution of system as well as the performance of quantum
gateswhich requireprecisely controlledexternaldriving.
Specifically for a cyclic adiabatic process, a geometric
In this work, we introduce classical noises in the form of
phaseisacquiredoverthecourseofthecycle[20–23]. Ex-
stochastic fluctuations in the controlparameters [39, 40]
perimentalimplementationsofadiabaticHQChavebeen
ofadrivenHamiltonianofathree-levelatomorionform-
proposed in various physical systems, such as trapped
ing a Λ-configuration for realizing universal holonomic
ions[10],superconductingnanocircuits[24],semiconduc-
single-qubit and two-qubit gates. Such fluctuations are
tor quantum dots [25], to name a few.
often due to imprecise system controls and other un-
Despite the advantages such as robustness to fluctu-
knownenvironmentalinfluences. Theycanbeintroduced
ations in runtime and system energy, geometric opera-
during gate operations or during reading of the results.
tions in adiabatic HQC suffer from a dilemma between
Then, the output (resulting) quantum state under the
a long runtime and a good coherence, noting that a loss
operationbytheperturbedholonomicquantumgatewill
in general deviate from that under the noise-free uni-
tary transformation. We study a transformation fidelity
∗ Authortowhomanycorrespondenceshouldbeaddressed. Email for a general input state, which quantitatively measures
address: [email protected] these deviations. In particular, we estimate the robust-
2
ness of the nonadiabatic holonomic transformation and where ω and ω are the bare energies of e and 1 , re-
e 1
| i | i
determine the conditions in which the desired state pas- spectively, Ω(t) is the modulated Rabi frequency (pulse
sage can be reliably realized in the presence of classical enveloporamplitude),anda (t)andb (t)arethedriving
0 0
noise. We note that both the magnitude and the cor- coefficients assumed to satisfy a (t)2+ b (t)2 = 1 for
0 0
| | | |
relation of the classical noises are important factors in simplicity. To cancel the bare energy terms in the origi-
determining the transformation fidelity. nalHamiltonian(1),weturntotherotatingframebyap-
Therestofthisworkisorganizedasfollows. InSec.II, plying the unitary transformation U = exp[i(ω e e +
0 e
| ih |
weintroduceauniversalnonadiabaticnon-Abelianquan- ω 1 1)t]. Uponthisrotation,theHamiltonianbecomes
1
| ih |
tumgateimplementation. Thelogicalsubspaceandcon-
trol parameters including the envelop function and two H (t)=Ω(t)[a(t)e 0 +b(t)e 1 +h.c.]. (2)
1
| ih | | ih |
control phases are explained. In Sec. III, we consider
stochasticfluctuations ineachparameterandtheir dam- Here the coefficients are
age to the fidelity of the nonadiabatic holonomic trans-
formation. We analyze the fidelity as a function of or a(t)=a (t)eiωet, b(t)=b (t)ei(ωe−ω1)t,
0 0
minimized over the input (initial) states and the driving
parameters. In Sec IV, we extend our formalism to a which still satisfy the normalization condition a(t)2 +
two-qubitgate case. Aconclusionis presentedinSec. V. b(t)2 =1. Their time dependence canbe suppr|essed| by
|choo|sing a0(t) e−iωet and b0(t) e−i(ωe−ω1)t. In gen-
∝ ∝
eral,thetime-independentcoefficientsaandbcanfurther
II. CONSTRUCTING NONADIABATIC be parameterized by two control phases in the form
HOLONOMIC QUANTUM GATES
θ θ
a=sin eiφ, b=cos .
We first construct a universal holonomic single-qubit 2 2
gatebasedonadrivenΛ-configurationthree-levelsystem
as well as nonadiabatic non-Abelian geometric transfor- Then, we have three parameters Ω(t), θ, and φ control-
mations. The driving Hamiltonian, realized by system- lable via the two driving lasers. They are taken as real
laser interactions, admits classical noise originated from numbersintheidealcaseofstablecontrol. Inthefollow-
the control lasers and environmental disturbance. ing, we will show that the envelop function of the Rabi
The bare three-level system consists of two nondegen- frequencyΩ(t)determinesthecyclicperiodaswellasthe
erate ground states 0 and 1 , representing the logic speed of the quantum gate operation, while the control
statesintheencodedq|uiantum|giate,andoneexcitedstate phases θ and φ specify the type of the quantum gate.
e acting as the auxiliary state. In the presence of two Aspectralanalysisofthe HamiltonianinEq.(2)gives
| i
separable polarized laser pulses properly tuned to be at
resonance with transitions e 1 and e 0 , re- H1(t)=Ω(t)(ψb+ ψb+ ψb− ψb− )+0ψd ψd . (3)
| i ↔ | i | i ↔ | i | ih |−| ih | | ih |
spectively. Assuming the level 0 has an energy ω =0,
0
without loss of generality, the H| aimiltonian can be writ- In terms of the basis states 0 , 1 , e , ψb± =
{| i | i | i} | i
ten as (1/√2)[a,b, 1]′ represent two bright eigenstates, while
ψ =[b, a±,0]′ is a state playing no role in the dynam-
d
H0 =ωe e e +ω1 1 1 i|csiand th−e gate operation. The time-evolution operator
| ih | | ih |
+Ω(t)[a (t)e 0 +b (t)e 1 +h.c.], (1) resulting from H is found to be
0 0 1
| ih | | ih |
sin2 θ cosΩ¯ +cos2 θ sinθe−iφ(cosΩ¯ 1) isinθe−iφsinΩ¯
U(t)=e−iR0tdsH1(s) = sinθ2eiφ(cosΩ¯ 1)2 co2s2 θ cosΩ¯ +si−n2 θ ico2sθ sinΩ¯ , (4)
 2isinθeiφsin−Ω¯ i2cosθ sinΩ¯ 2 co2sΩ¯ 
2 2
 
where Ω¯ Ω¯(t) = tdsΩ(s). It is straightforward to Under the above conditions, the final time evolu-
≡ 0
see that when Ω¯(T) = π, i.e., T dtΩ(t) = π, the first tion operator U(T) is projected onto the qubit subspace
R 0 spanned by 0 and 1 and can be expressed as
two degrees of freedom, 0 and 1 , will be decoupled
| i | i
| i R | i
from the excited (ancillary) state e . It follows that the
| i cosθ sinθe−iφ
tqiumbeitesvpoalucetiospnaUnn(se)dibfyth|e0ilaasnedrs|1saitiissfiynvΩ¯a(rTia)n=tuπn.dIetrctahne Uh(T)= sinθeiφ − cosθ . (5)
(cid:18)− − (cid:19)
be verified that this evolution is purely geometric since
k U†(s)H (s)U(s)l = k H (s)l = 0 for k,l = 0,1 Itcanbeusedtorealizeanysingle-qubitrotation,i.e.,an
1 1
h | | i h | | i arbitraryunitary transformationfor a single qubit. This
and s [0,t].
∈ thus defines a universalholonomicsingle-qubitgate. For
3
examples,Eq.(5)canrealize(i)theHadamardgatewith where Ω¯′ = Ω¯ + tdsδ (s). It can be verified (see, e.g.,
0 Ω
θ = 3π and φ =0, (ii) the Pauli-X gate with θ = π and Ref.[39])thatfor anyclassicalGaussiannoiseδ (t) with
4 2 R ξ
φ = π, (iii) the Pauli-Z gate with θ = 0, and (iv) the a zeromean δ (t) =0 andanauto-correlationfunction
ξ
phase-shift gate with θ = 3π. In general, using Eq. (5), C (t,s)= δh(t)δ (is) ,
2 ξ h ξ ξ i
an input (initial) state ψ(0) = αeiη 0 +β 1 will be
transformed into | i | i | i M eimR0tdt1δξ(t1) =e−m2R0tdt1R0t1dt2Cξ(t1,t2),
|Ψ(T)i=(αcosθeiη−βsinθe−iφ)|0i h M eimδξ(t)i=e−m22Cξ(t,t),
(αsinθei(η+φ)+βcosθ)1 , (6)
h i
− | i
where m is a real constant. These results are helpful in
whereα,β andη areassumedtoberealnumberssatisfy-
evaluation of the fidelity in Eq. (7).
ingα2+β2 =1. Hereandinthefollowing,Ψdenotesthe
The classical noise sources in our work are associated
resulting state from an ideal noise-free holonomic trans-
with the three driving parameters, i.e., the amplitude-
formation.
envelop function Ω and two phases θ and φ in Rabi
The gate operation provided by Eq. (5) is in general
frequency. The shape and duration of the input laser
universal connecting any pair of pure states and there is
field are determined by the envelop function Ω, whereas
in principle no limit on the operation time T. The con-
the carrier-envelop phases of the two driving lasers are
trolparametersθ andφsetupthedesiredquantumgate.
described by θ and φ. Physically the amplitude and
The effective time evolution operator U in Eq. (5) thus
h the phases can be tuned by certain combinations of
provides a general protocol for quantum state engineer-
acousto-optical modulators and phase-modulation lock-
ing. It is therefore important to consider the reliability
ing achieved with low driving voltages, respectively. Re-
of this gate operation.
cently, techniques to separately modulate the amplitude
and phases of laser sources have been proposed and ex-
perimentally demonstrated [41–44]. In the following, we
III. RELIABILITY OF HOLONOMIC
assume that only one control parameter admits signifi-
TRANSFORMATION IN THE PRESENCE OF
cantfluctuations in eachcase and the fluctuations are of
CLASSICAL NOISE
Gaussian type. However, the conclusion is independent
ofthe spectralfunctionorthe correlationfunctionofthe
The ideal holonomic transformation specified by
noise.
Eq. (6) is not always attainable once nonideal driving in
the original Hamiltonian (1) is taken into account. We
nowconsiderthestochastictime-evolutionoperatorU (t)
ξ A. Fidelity under noisy envelop function Ω
which deviates from U(t) in Eq. (4) under the effect of a
single noisy controlparameter ξ Ω,θ,φ . To measure
∈{ } The fidelity in the presence of fluctuations in Ω(t) can
the robustness of the holonomic quantum gate for HQC,
be obtainedfromthe overlapbetweenthe idealandnon-
we study a transformation fidelity defined by
ideal wave functions,
=M[ Ψ(T)ψ (T) ψ (T)Ψ(T) ]. (7)
Fξ h | ξ ih ξ | i Ψ(T)ψ (T) =[1+cosΩ¯′(T)]f(θ,φ) cosΩ¯′(T),
Ω
h | i −
Here M[] means the ensemble average over all random
realizatio·nsof fluctuations inthe controlparameter,and where
ψ (t) U (t)ψ denotes the nonideal output state
ξ ξ 0 θ θ
o|f thein≡oisy qua|ntuim gate. The transformation fidelity f =f(θ,φ) α2cos2 +β2sin2 αβsinθcos(φ+η).
≡ 2 2 −
indicates the leakage of the output state out of the logic
subspace. For example, putting ξ Ω, U (t) can be Note that the tight upper and lower bounds of f(θ,φ)
Ω
≡
obtained after letting Ω(t) Ω′(t) = Ω(t) + δ (t) in are limited by αcos(θ/2) βsin(θ/2)2 or αcos(θ/2)+
Ω
Eq. (4). It yields → βsin(θ/2)2. Ei|ther situati−onsatisfies0| f| 1. Substi-
tuting the|above results into Eq. (7) wit≤h Ω¯(≤T)=π and
θ sinθ
ψ (t) = cosΩ¯′(αsin2 eiη+β e−iφ) performingtheensembleaverageforthefluctuations,the
Ω
| i 2 2 fidelity is obtained as
(cid:20)
θ sinθ
+αcos2 eiη β e−iφ 0 1+e−4C¯Ω(T)
2 − 2 | i = + 2e−C¯Ω(T) e−4C¯Ω(T) 1 f
(cid:21) FΩ 2 − −
θ sinθ
+ cosΩ¯′(βcos2 +α ei(φ+η)) 3 4e−C¯Ω(T)+eh−4C¯Ω(T) i
2 2 + − f2, (8)
(cid:20) 2
θ sinθ
+βsin2 α ei(φ+η) 1
2 − 2 | i where C¯ (T) T dt t1dt C (t ,t ). Here, e−C¯Ω(T)
(cid:21) Ω ≡ 0 1 0 2 Ω 1 2
θ θ canbe consideredasadecayfunction,whichisclearlyin
+isinΩ¯′(βcos +αsin ei(φ+η))e , the range (0,1]. R R
2 2 | i
4
1 0.9 4 0.9
3.5
0.8 0.8 0.8
3
0.7
0.6 2.5 0.7
)
θφf(,0.4 0.6 γΓ/ 2 0.6
1.5
0.5
1 0.5
0.2
0.4 0.5
0.4
00 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10
x(T) Γ T
FIG. 1. (Color online) Landscape of the transformation fi- FIG.2. (Coloronline)AveragetransformationfidelityF¯ un-
Ω
delity F in the parameter space of decay function x(T) ≡ der non-Markovian processing caused by δ (t), as a function
Ω Ω
e−C¯Ω(T) and f ≡f(θ,φ). of dimensionless cyclic period ΓT , and the memory param-
eter of noise γ/Γ. Here the correlation function is supposed
to be C (t,s) = Γγe−γ|t−s|/2, so that the decay function
Ω
x=x(T)≡e−C¯Ω(T) =exp[−Γ(e−γT +γT −1)/(2γ)]. When
Evidently, this transformation fidelity depends on the
γ →∞,C (t,s)reducestoadeltafunctionΓδ(t−s)implying
correlation function of the classical noise δ , the initial Ω
Ω apurelyMarkoviannoiseandthenxreducestotheexponen-
state characterized by α, β, and η, and the control pa- tialdecayfunction. Ontheotherhand,forγ →0,thedecayis
rametersθandφ. InFig.1,weprovidealandscapeof Ω strongly suppressed and thefunction x(T)approaches unity.
plotted against the decay function x(T) = e−C¯Ω(T) aFnd
f(θ,φ) according to Eq. (8). Note that the impacts of
the noise correlation function are already considered via follow uniform probability distributions. We then find
x(T)whilethoseofthe controlphasesθ andφandinput that the averageof f(θ,φ) is 1/2 and
stateparametersα, β andη areincludedviaf(θ,φ). We
3+4e−C¯Ω(T)+e−4C¯Ω(T)
thus have considered all possible regimes. ¯ = , (9)
Ω
It is interesting to note that when f(θ,φ) is close to F 8
unity, Ω is maintained at a high level for the whole which has a minimum of 3/8. In Fig. 2, we plot the
range oFf the decay function e−C¯Ω(T), which depends on average fidelity assuming an Ornstein-Uhlenbeck noise
both T (the cyclic period for constructing the logic sub- with a correlation function
space)andtheformofthenoisecorrelationfunction. For
Γγ
those particular combinations of quantum gates and in- C (t,s)= e−γ|t−s|,
Ω
2
put states, the gate operation is then robust against the
classical noise even for a long runtime, noting that x(T) where Γ is the correlation intensity of the noise and γ
often decreases with T especially after coarse graining is the memory parameter and is inversely proportional
over the time domain. Similarly, when x(T) is close to to the memory retention time of the classical noise δ .
Ω
unity (larger than about 0.96), corresponding to a short In a strongly non-Markovianregime with γ/Γ 0.1, the
∼
runtime, the transformationis found to be fault tolerant transformationfidelitycanbemaintainedbeyond0.99for
inthewholerangeoff(θ,φ). Therefore,thereareclearly T .6/Γ,alimitwhichisalmost12timesaslongasthat
“sweet spots” at x(T) 1 and f(θ,φ) 1. in a near-Markovian case with γ/Γ 4. Figure 2 thus
≃ ≃ ∼
A perfect “sweet spot” for this control problem (in imposesanexplicitdemandontheruntimeT forHQCin
view of quantum state engineering) emerges only when the presence of the classicalnoise with different memory
f(θ,φ) = 1. From Eq. (8), this gives rise to = 1 capabilities. As the correlation function of the noise ap-
Ω
irrespectiveoftheexistenceofthestochasticflucFtuations proaches a delta function, the holonomic quantum gate
overthe envelopfunction Ω. The condition for f(θ,φ)= runtime must become ever shorter.
1 to holdis cos(φ+η)=1 when αβ 0,or cos(φ+η)= We now calculate the minimal value of the fidelity for
1 when αβ 0 and α = cos(≤θ/2). Note that in various initial states. From Eq. (8), we get
− ≥ ±
the derivation, we apply the fact that f(θ,φ) 1. For ∂
example, if the input state is chosen as cos(3π≤/8)0 + FΩ =(3 4x+x4)f (x4 2x+1),
| i ∂f − − −
sin(3π/8)1 ,theHadamardgateisalwayserrorfreeeven
| i ∂2
if Ω(t) is noisy. FΩ =3 4x+x4,
Anaveragefidelity ¯ overallinputstatesisalsostud- ∂f2 −
Ω
F
ied. We adopt the parametrization α = cos(ϕ/2) and where we have simplified the notation by writing x
β = sin(ϕ/2) and assume that ϕ [0,π] and η [0,2π] e−C¯Ω(T). Recall that x (0,1], where the two bound≡s
∈ ∈ ∈
5
correspond to an infinitely large T (or a strongly cor- 1
related noise) and a vanishing T (or a Markovian
0.95
noise) respectively. Consequently the second derivative
0.8 0.9
∂2 /∂f2 is always positive. Thus attains its mini-
Ω Ω
muFmwhen∂ /∂f =0. Since the fiFrsttermin∂ /∂f 0.85
Ω Ω
F F 0.6
is positive, it may only vanish if x xc 0.5437, θ 0.8
≤ ≈ 2
which is the only real root of x4 2x + 1 = 0 for n
− si 0.75
x [0,1). For x x , the minimum fidelity occurs 0.4
c
if f∈(θ,φ) = (x4 2≤x+1)/(3 4x+x4). In contrast, 0.7
− −
for x > xc, ∂ Ω/∂f = 0. The minimum simply occurs 0.2 0.65
F 6
at f(θ,φ) = 0 that follows from Eq. (8). This corre-
0.6
sponds to the initialstates satisfying α/β =tan(θ/2)for
0 0.55
φ+η =2kπ, or α/β = tan(θ/2) for φ+η =(2k+1)π, 0 0.2 0.4 0.6 0.8 1
− x(T)
with k an integer.
FIG. 3. (Color online) Landscape of average transformation
B. Fidelity under noisy control phases θ and φ fidelity F¯φ in the parameter space of x(T) ≡ e−C¯φ(T) and
sin2θ.
We now consider a noise-free envelop function Ω(t).
The holonomic quantum gate then possesses an exact Afteratediousbutstraightforwardderivation,thetrans-
cyclic time T. Fluctuations in θ or φ leave the system formation fidelity is obtained as
in the computational subspace spanned by the ground
states 0 and 1 without invoking the excited state e . Fφ =cos4θ+sin4θ 1−2α2β2(1−e−4C¯φ)
| i | i | i
In the presence of random fluctuations associated with sin2(2θ) h i
dθ/dt, we have θ θ′ = θ + ∆ (t), where ∆ (t) = + e−C¯φ +α2β2sin2(2θ)
t → θ θ 2
dsδ (s). Consequently,
0 θ sin2φ˜(1 2e−C¯φ)+ 1−cos(2φ˜)e−4C¯φ
RΨ(T)ψ (T) =cos[∆ (t)]+2αβcos(φ+η)sin[∆ (t)]. ×" − 2 #
θ θ θ
h | i
+αβ(α2 β2)sin2θsin(2θ)cosφ˜(1 e−4C¯φ),(12)
Inserting it into Eq. (7), it is straightforwardto show − −
where C¯ (T) T dt t1dt δ (t )δ (t ) and φ˜=φ+
1+e−4C¯θ(T) 1 e−4C¯θ(T) η. φ ≡ 0 1 0 2h φ 1 φ 2 i
Fθ = 2 +4α2β2cos2(φ+η) − 2 , Rather than lRocatingR the “sweet spot” directly from
(10) Eq. (12), it is more instructive to first average over α, β
where C¯ (T) T dt t1dt δ (t )δ (t ) . The “sweet and η and obtain
θ ≡ 0 1 0 2h θ 1 θ 2 i
spot” for this situation, i.e., = 1, regardless of the
existence of the Rnoise, tRhen emFeθrges when α2 =1/2 and ¯φ =1 3sin2(2θ)(1 e−C¯φ) sin4θ(1 e−4C¯φ). (13)
φ+η = kπ, with k an integer. In addition, the mini- F − 8 − − 4 −
mumfidelity occurswhenα2β2cos2(φ+η)=0,inwhich Aninterestingobservationhereis thatinthe presenceof
θ = 1/2+exp[ 4C¯θ(T)]/2. Therefore, for a specific a noisy φ, the average of the transformation fidelity de-
F −
initial phase η satisfying φ+η = (k+1/2)π, the trans- pendsontheparticularvalueofθinadditiontothetime-
formation fidelity is purely dependent on the correlation integrated noise correlation C¯ . The lower-bound of the
φ
function of the noise δθ, but independent of the popu- averagefidelity is foundto be 1 3sin2(2θ)/8 sin4θ/4.
lation distribution α2 and β2 of the initial state. On − −
Therefore, the transformation fidelity depends on the
average over α, β, and η, the fidelity turns out to be
particulartype ofthe quantumgatedeterminedby θ. In
Fig.3,weshowagenerallandscapeoftheaveragefidelity.
¯ = 5+3e−4C¯θ(T), (11) We see that when sin2θ is sufficiently small, the average
θ
F 8 transformationfidelity shows a “sweetspot” regime. On
the other hand, if the decay function x(T) = e−C¯φ(T) is
which has a lower bound of 5/8, larger than that for the sufficiently large,equivalently if T is sufficiently smallor
case with a random envelop function Ω′(t). if the correlation function is in a strong non-Markovian
Similarly, in the presence of a random control phase regime,westillhaveanaveragefidelity closetounity for
φ φ′ =φ+∆ (t), where ∆ (t)= tdsδ (s), we have an arbitrary sin2θ. In addition, it is found that when
→ φ φ 0 φ sin2θ = 3/5, ¯ arrives at its minimum value 11/20,
φ
R F
hΨ(T)|ψφ(T)i=sin2θ α2ei∆φ(T)+β2e−i∆φ(T) winhgicehnvieslolaprgfuernctthiaonn Ωth′a(tt)ianntdhesmparlelseerntcheaonftahafltucintutahte-
+cos2θ+iαβsin(2θ)[sinh(φ′+η) sin(φ+η)]. i presence of a fluctuating θ′.
−
6
t
Considering averages over input states, values of the where ∆ (t) = dsδ (s). We study the decay func-
φ 0 φ
transformationfidelityminimizedwithrespecttovarious tion expressed by x(T) = e−C¯φ(T), where the definition
gates and input states follow F¯Ωmin < F¯φmin < F¯θmin. In of C¯φ(T) in Sec.RIIIB for the single-qubit case remains
summary,thereliabilityofthisnonadiabaticnon-Abelian valid. Theoverlapbetweentheoutputstatesoftheideal
quantum gate is most susceptible to fluctuations occur- and nonideal unitary transformations is found to be
ring on the envelop function of Rabi frequency but is
most resilient against that on the control phase θ. Ψ(T)ψφ(T) =1 α2sin2θ 1 ei∆φ(T)
h | i − −
h i
β2sin2θ 1 e−i∆φ(T) .
IV. TWO-QUBIT GATE − −
h i
Therefore, the transformation fidelity reads
We have considereda direct and exact constructionof
asingle-qubitnonadiabaticnon-Abelianholonomicquan- (2) =1 2sin2θ[(α2+β2)(1 x)]
Fφ − −
tumgatebymodulatingtwolaserpulsesinteractingwith
+2sin4θ[(α4+β4)(1 x)+α2β2(1+x4 2x)],
a three-level atomic system (see Sec. II). In contrast, an − −
existing design [15, 45] of a two-qubit gate is only ap- where x = x(T) = e−C¯φ(T). Averaging over θ, α and β,
proximately holonomic as a result of an adiabatic elimi-
we find
nationunderarestrictedregimeofthecouplingstrength
between the laser and atoms. In the so-called Sørensen- 93 3
¯(2) =1 (1 x)+ (1+x4 2x), (16)
Mølmer setting [45], a pair of ions constitute two inter- Fφ − 256 − 64 −
nal Λ-configuration three-level systems. The transitions
e 1 and e 0 for these two ions are coupled by whichisamonotonicincreasingfunctionofxintherange
| i↔| i | i↔| i
lasers with envelop functions of Rabi frequencies Ω (t) (0,1) and tightly lower-bounded by 175/256 0.6836.
1 ≈
andΩ (t)anddetunings ν δ and ν δ,respectively, Thus in this particular situation, the minimum value of
0
± ± ± ∓
where ν is a phononfrequency andδ is anadditionalde- theaveragetransformationfidelityforthetwo-qubitgate
tuning. The indirect interaction between the two ions is is even larger than that for the single-qubit gate.
induced by the lasers. When the Lamb-Dicke parame-
ter ζ satisfies ζ2 1, the effective Hamiltonian can be
≪
approximated as V. DISCUSSION AND CONCLUSION
ζ2
H = Ω4(t)+Ω4(t)H0, (14) In this work, for the nonadiabatic non-Abelian holo-
2 δ 0 1 2
H0 =sinqθeiφ2 ee 00 cosθe−iφ2 ee 11 +h.c., npoenmdiecnqcueaonfttuhmetsrianngslefo-qrmubaittiognatfied,ewlietyoobntatihnegiennpeurtal(idnei--
2 2 | ih |− 2 | ih |
tial)states,thequantumgateparameters,andstatistical
where θ = 2tan−1(Ω2/Ω2) and φ is the phase difference properties of classicalnoises acting on these parameters.
0 1
oftwocontrolpulses. Notenowθisnolongeranindepen- Thegateparametersconsideredincludetheenvelopfunc-
dent physicalparameterin contrastto its counterpartin tionofRabifrequencyΩ(t),andtwocontrolphasesθand
the single-qubit protocols. In the protocol for two-qubit φoftwodrivinglaserpulses. The formeronedetermines
gateprovidedbyRef.[15],theparameterθ isdetermined theruntimeofthegatewhilethelattertwodeterminethe
by the ratio of the amplitudes Ω and Ω of the pulse typeofthegate. Valuesofthetransformationfidelityav-
0 1
pairbut not aphase under controlas thatfor the single- eragedorminimizedoverallinputstatesarealsostudied.
qubitgate. Toachieveadesiredtwo-qubitgate,theratio The location of the high-fidelity regimes, i.e., the “sweet
Ω2/Ω2 and the phase φ should be kept constant during spots” implies that the nonadiabatic non-Abelian holo-
0 1
each pulse pair. Meanwhile, Ω and Ω are constrained nomic quantum gate is often robust against fluctuations
0 1
by the π pulse criterion T dtζ2 Ω4(t)+Ω4(t) = π to in the control parameters. In the presence of the noise,
0 δ 0 1 wefindthatperfect“sweetspot”doesexistundercertain
attain an effective evolution operator on the computa-
R p conditions, in particular in the case of nonideal parame-
tionalsubspacespannedby 00 , 01 , 10 , 11 forming
{| i | i | i | i} terΩ(t)orθ. Wealsofindanonvanishingmemoryofthe
a holonomic two-qubit gate
classical noise can relieve the requirement on the speed
U˜ =cosθ 00 00 cosθ 11 11 +sinθe−iφ 00 11 of a cyclic evolution of the logic subspace.
h
| ih |− | ih | | ih |
+sinθeiφ 11 00 + 01 10 + 10 01. (15) We then extend the analysis into a special two-qubit
| ih | | ih | | ih | quantumgateleadingtoauniversalsetofquantumgates.
We parameterized the input state as ψ(0) = α00 + Itisinterestingtofindthatthenonadiabaticnon-Abelian
| i | i
ǫ01 +η 10 +β 11 , where the four coefficients are real holonomictwo-qubitgateismorerobustthanthesingle-
| i | i | i
and follow the normalization condition. Then, we con- qubit gate against classical noise for the setups we have
sider the classicalnoise perturbing the controlphase dif- considered.
ference φ, which is a physically relevant parameter and In conclusion, we have investigated nonadiabatic non-
independent from Ω and Ω . Thus φ φ′ + ∆ (t), Abelian holonomic quantum gates for both single-qubit
0 1 φ
→
7
and two-qubit operations. By studying a unitary trans- ACKNOWLEDGMENTS
formation fidelity, we clarify generic properties concern-
ing the gate reliability, which are independent of the de-
tails of the classical noise correlation function. We com- We acknowledge grant support from the Basque Gov-
pare the effect of classical noise from different sources. ernment (Grant No. IT472-10), the Spanish MICINN
The analyses on the “sweet spot” and minimum values (GrantNo. FIS2012-36673-C03-03),theNationalScience
of the transformation fidelity are general and apply to Foundation of China Grant No. 11575071, Science and
Gaussiannoisewith arbitrarycorrelationfunctions. Our Technology Development Program of Jilin Province of
investigation provides a systematic estimation over the China (Grant No. 20150519021JH),and the HongKong
error of HQC caused by classical noise. It is expected GRFGrantNo. 501213. Wealsoacknowledgethehelpful
to be useful for optimizing the performance of quantum discussions with Dr. Junxiang Zhang and Dr. Fengdong
gates for nonadiabatic non-Abelian holonomic quantum Jia on the exclusive modulation over the parameters of
computing. Rabi frequency in experiments.
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