Modelling of quantum information processing with Ehrenfest guided tra jectories: a case study

1.6 MB·English

by  Sai-Yun Ye

Modelling of quantum information processing with Ehrenfest guided trajectories:
a case study
Sai-Yun Ye,1 Dmitrii Shalashilin,2 and Alessio Serafini1
1 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK and
2 School of Chemistry, University of Leeds, Leeds LS2 9JT, UK
(Dated: January 31, 2012)
Weapplyanumericalmethod based onmulti-configurational Ehrenfest trajectories, anddemon-
strate converged results for the Choi fidelity of an entangling quantum gate between two two-level
systemsinteractingthroughasetofbosonicmodes. Weconsiderbothspin-bosonandrotatingwave
Hamiltonians, for various numbers of mediating modes (from 1 to 100), and extend our treatment
to include finite temperatures. Our results apply to two-level impurities interacting with the same
2 bandofaphotoniccrystal, ortotwodistant ionsinteracting with thesameset of motional degrees
1 of freedom.
0
2 PACSnumbers: 03.67.Lx,42.50.Ex,42.50.Ct,42.81.Qb

n
a
J I. INTRODUCTION to program, and yet of allowing one to follow coherent
quantum dynamics in detail, as it will be shown.
0
3 The ability of trackingthe evolutionof complex quan- In our study, we will focus on a specific, but very rele-
vant, aspectofthe quantum dynamics ofthe two qubits:
tum systems will be a crucial support to the design
] we shall consider the realisation of an entangling quan-
h and development of future quantum technologies. A
tum gate between them, namely of a controlled Z (CZ)
p paradigmofparticularinterestforthelatterisonewhere
- finite dimensional quantum systems, typically two-level gate. To estimate the quality of such a realisation we
nt systems (qubits), interact through a ‘bus’, made up of willconsiderthequantumfidelitybetweenthepurestate
a a set of bosonic field modes [1–33]. In this paper, we corresponding to the CZ gate by the standard channel-
u state duality (Choi isomorphism [44–46]) and the quan-
will consider the case of two qubits interacting with a
q tum state corresponding to the channel acting on the
common discrete set of modes in the non-perturbative
[
regime. While the possibility of creating entanglement, twoqubitsuponpartialtracingoverthefield’sdegreesof

1 even at steady state, by the interaction with a common freedom.
v bosonic bath has been highlighted repeatedly in the lit- Our main aim is demonstrating the capability of
1
erature, the case of non-perturbative interactions with Ehrenfest guided trajectories in phase space to produce
7
a bunch of 10 or 20 modes (as will be the case in our reliable and converged results for complex figures of
1
6 study) presents several major technical difficulties, es- merit,abletorevealdetailedinformationaboutthequan-
. sentially due to the impossibility of an analytic master tum dynamics of the constituents. The ‘Choi fidelity’ of
1
equationapproach–onlypossibleinthecontinuumlimit a two-qubit quantum gate is a property of the dynam-
0
2 undertheBorn-Markovapproximation–andtothehuge ics itself, and not of the initial state, and its evaluation
1 sizeofthedynamicallyrelevantpartoftheHilbertspace. requires, at any time, the evolution of ten initial states:
v: Various numerical approaches have been developed to it is, therefore, a rather cumbersome figure of merit to
compute, let alone to optimise over a rather large range
i emulate these dynamics on classical computers, such
X of values of the dynamical parameters, as we did. The
as the so called multiconfigurational time dependent
r Hartree method [34–36] and its “Gaussian” variation advantagesoftheEhrenfestguidedtrajectoriesover–ar-
a guably more precise but heavier – approaches based on
[37], various schemes based on path integral techniques
full variational principles are manifest in such circum-
[38,39],andeventheadaptationoftime-adaptivedensity
stances.
matrix renormalisation group techniques [40] borrowed
from condensed matter theory. Here, we will tackle such As for direct impact, let us remark that our study
difficulties by borrowing a method co-pioneered and de- wouldapplyonsystemsliketwo-levelimpuritiestrapped
veloped by one of the authors in the arena of chemical inaphotonicscrystalandinteractingwiththesameband
physics [41, 42]. The method is based on the adoption of allowed modes [32, 47, 48], or to the internal levels of
of a set of tensor products of time-dependent coherent two ions interacting with all the vibrationalmodes of an
states as a discrete ‘basis grid’onwhich to representthe arrayofionsinalineartrap[49]. Itisimportanttopoint
fielddegreesoffreedom(referredtoas“coupledcoherent out that our treatment can account for finite, although
states” in the literature [43]), and on letting the states relatively small, temperatures as well.
oftheco-movinggridevolveaccordingtotheirEhrenfest Ourpaperis organisedasfollows. InsectionII wewill
dynamics (whose applicationto a gridof coherentstates reviewthebasictheorybehindmethodsofsolutionofthe
goesunderthenameof“multi-configurationalEhrenfest” Schr¨odinger equation based on a set of time-dependent,
method). This approach has the advantage of being rel- Ehrenfestguidedbasisstates. We willnotdwellsomuch
atively light in terms of computational resources, easy on the technical details, which are covered elsewhere, as
2
on the basic concepts, and will try to present them in position). The quantity is hence a function of the co-
terms which will be friendly to an audience with no pre- efficientsc ,thecomplexLparametersα andtheirtime-
l,j j
viousfamiliaritywiththeterminologyofchemicalphysics derivatives c˙ and α˙ . In fact, it can be shown that
l,j j
L
or molecular dynamics. In section III we will introduce serves as a Lagrangian for the quantum system, in the
the physicalHamiltoniananddefine preciselyourchosen sense that the Euler-Langrangeequations
figure of merit. Section IV will contain the main results
∂ d ∂
ofournumericalstudy. Finally,wewilldrawconclusions, L = L (3)
and discuss advantages and shortcomings of our method ∂cl,j dt∂c˙l,j
of choice, in section VI.
are equivalent to Schr¨odinger equation [68]. See Ap-
pendix B for more details.
Besides determining the state evolution, the varia-
II. EHRENFEST GUIDED TRAJECTORIES
tionalformalismalsoprovidesonewitharecipetoupdate
the basis such that the expression of Eq. (2), which
The main difficulty in dealing with a systemincluding L
clearly always equals 0 in the exact dynamics, is min-
few two-level systems and a bunch of M bosonic modes
imised during the time-evolution. Such a minimisation,
is clearly how to handle the infinite dimensional bosonic
which in essence keeps the basis in the ‘most relevant’
Hilbert space. The method we apply here, referred to
region of the Hilbert space within the constraints of the
in the literature as ‘multi-configurationalEhrenfest’ and
adopted approximation, would be obtained by consid-
firstintroducedin[41],tacklesthisdifficultyontheshoul-
ering the full Euler-Lagrange equations for the M N
ders of two major assumptions: complex parameters α and their time-derivatives. ×This
j
would be a large nonlinear system of coupled equations,
i) thestatespaceofthefieldmodesisrepresentedasa
requiring a substantial numerical effort to be solved. In-
superpositionofN timedependentcoherentstates;
stead, we introduce here assumption ii), and replace the
ii) thetimedependenceofthecoherentstatesisdeter- full variational equations for αj with a simplified ver-
mined by a simplified variational principle (which, sion thereof. In particular, we will neglect all terms cou-
in other words, dictates how the finite dimensional plingthe differentαj’s,onthegroundsthatthe overlaps
subspace spanned by the set of coherent states αj αk are typically very small if the number of modes
h | i
changes with time, trying to keep it in the dynam- M is large enough. For each j, let us then define the
ically relevant region). vector ψ˜j as
| i
Thefinitedimensionalsystemsinvolvedare,ontheother d
hand, treated by representing their entire Hilbert space, ψ˜j = cl,j l αj , (4)
| i | i⊗| i
spanned by d orthonormalbasis states l , for l 1,d . l=1
X
| i ∈{ }
In agreement with i), the ansatz wave-function of the
and the corresponding ‘approximated’ Lagrangian ˜ as
whole system reads Lj
˜ = ψ˜ H i∂ ψ˜ . (5)
d N Lj h j| − t| ji
ψ = c (t)( l α (t) ) , (1)
| i l,j | i⊗| j i Theequationofmotionfortheparameterα(m) (them-th
Xl=1Xj=1 component of the vector α ) is j
j
where α M j andeach α is a tensor productof
coherentjs∈tatCes: ∀α = M | αji(m) ,suchthat,ifa is ∂L˜j = d ∂L˜j , (6)
the annihilation|opejirator omf=m1o|dejm,ione has am αjm= ∂α(jm) dt∂α˙(jm)
N | i
αj(m)|αji. Since we will be dealing with two qubits, it where we also neglect the time-dependence of the coef-
will be d=4 for us. ficients c , such that each Lagrangian ˜ only depends
l,j j
The evolution of the dynamical parameters is more on the four complex parameters c andLon the M com-
l,j
conveniently described by adopting a Lagrangianformu- plex parameters represented by the entries of α (and
j
lation. For a Hamiltonian operator Hˆ, let us define on their time-derivatives α˙ , see Appendix A for fur-
L j
as ther details). Eq. (6) defines the “Multi-Configurational
Ehrenfest” (MCE) method we are using.
= ψ Hˆ i∂ ψ . (2)
t Notice that the assumption ii) is not, per se, an ap-
L h | − | i
proximation,but rather just a way of choosing the time-
Thetime-derivativeoperatorisdefinedasthedifferentia-
dependence of the adopted basis. However, it should be
tionofthetimedependentscoefficientsc andbythere-
l,j stressedthat, ingeneral,the exactEuler Lagrangeequa-
lationships ∂ l =0 (the finite dimensionalsystem’s ba-
t| i tionforthe full variationoftheparametersα(m) is likely
sis is time-independent) and ∂ α = M [α˙(m)(a j
t| ji m=1 j †m− to provideone with a moreaccurateresultin thatit will
α(jm)∗/2) − α˙(jm)∗α(jm)/2]|αji (derivedPfrom the time- yield a smaller Lagrangian L (which is zero in the exact
dependence of a coherentstate with varying phase space dynamics).
3
However, Eq. (6) is much easier to treat numerically, where σ+(j) = σ(j)† = σx(j) +iσy(j). As well known, the
hence the advantage of our method, which can be eas- Hamiltonian Hˆ− is a good approximation of Hˆ in the
rw
ily programmedandapplied with modest computational almostresonant,highfrequencycasethatis,inournota-
resources and often provides results in very good agree- tion,for 2ε ω 2ε+ω , m. Inthefollowing,we
m m
ment with complete variational methods, like MCTDH will cons|ider−syst|e≪ms|with diff|e∀rent numbers of bosonic
or G-MCTDH [34–37]. modes M, various values of frequencies ω and spin-
m
Multi-configurational Ehrenfest guided trajectories bosoncouplings g(j) , and different para{met}ers ∆ and
havebeenthoroughlytestedforspin-bosondynamicsun- ∆ . Also, we wil{l smet}~=1 throughout the paper. 1
derdifferentspectraldensities,establishingthereliability 2
Inreproducingthedynamicsofthetwoqubitsbytreat-
of their converged results in several, diverse situations
ingthefieldthroughmulti-configurationalEhrenfesttra-
[41, 50]. Here, we will instead apply them to study a
jectories,wewillaimatobtainingconvergedresultsfora
composite system including discrete sets of bosonic field
figure of merit of interest in the study of quantum infor-
modes, where coherent quantum information processing
mationprocessing,namelythe fidelity withwhichanen-
can be carried out.
tanglingcontrolledZ(CZ)gatecanberealisedforthetwo
qubitsthroughthemediatingbosonicmodes. Intermsof
the basis states of Eqs. (7-10), a CZ gate is represented
III. MODEL AND FIGURE OF MERIT
as a unitary U leaving all the basis states invariant
CZ
except for 4 , which becomes 4 , that is
We set out to study coherent quantum information | i −| i
processing for a system comprising two two-level sys-
U j =f(j) j for 1 j 4, (13)
CZ
tems (“qubits”) connectedby M bosonic modes through | i | i ≤ ≤
a spin-boson like coupling. where f(j)=1 for j 1,2,3 and f(j)= 1 for j =4.
In principle, this represents the archetype of a quan- The relevance of a CZ∈g{ate to}quantuminfo−rmationpro-
tum system where complex dynamics and information cessingstemsfromitsbeingamaximallyentanglinggate
processingtaskscanbecarriedout,andwhosedynamics which,combinedwithsingle-qubitunitaries,formsauni-
is impervious to non-approximated methods. In prac- versalquantumsetforgatebasedquantumcomputation
tice, our case study may be thought of as representing [51].
two two-level atoms (or impurities) interacting with the At zero temperature, the quantum operation Γ we
t
same photonic band of a photonic crystal [47], or a sim- want to compare with the CZ unitary gate is defined as
ulation of the same setting in a linear array of trapped follows, in terms of a notional initial density matrix of
ions (where the qubits are embodied by internal levels the qubits ̺:
of the ions interacting with the same set of vibrational
normal modes [49]). Γ (̺)=Tr e iHˆt(̺ 0 0 )eiHˆt , (14)
t B −
For future convenience, let us re-label the four states ⊗| ih |
h i
of the computational basis of the two two-level systems
where Tr stands for partial tracing over the Hilbert
B
as follows:
space of the bosonic modes and 0 is the vacuum state
| i
1 = , (7) of the modes. We will also extend our treatment to in-
| i | ↓↓i
clude a finite temperature 1/β of the bosonic modes (in
2 = , (8)
| i | ↓↑i natural units where k =1), in which case the quantum
3 = , (9) B
| i | ↑↓i operation Γt,β will be given by
4 = . (10)
| i | ↑↑i
The operators σˆx(1), σˆx(2), σˆz(1) and σˆz(2), will stand for Γt,β(̺)=TrB e−iHˆt ̺ Pβ(α) α α d2Mα eiHˆt ,
⊗ | ih |
the customary Pauli operators in the Hilbert spaces of (cid:20) (cid:18) Z 2M (cid:19) (cid:21)
C (15)
qubit 1 and 2. For instance, in the adopted basis, σˆ(1)
x with
(1) (1) (1)
is defined by σˆ 1 = 3 , σˆ 3 = 1 , σˆ 2 = 4 ,
x x x
| i | i | i | i | i | i
σˆx(1I)n|4oiu=r s|t2uid.y, we shall consider both an actual ‘spin- Pβ(α)= M eβωmπ−1e−(eβωm−1)|αm|2 . (16)
boson like’ Hamiltonian: mY=1(cid:18) (cid:19)
Hˆ = 2j=1 Mm=1 εσˆz(j)+∆jσˆx(j)+ωma†mam The function Pβ(α) is just the Glauber-Sudarshan P-
(11) representation of a thermal state of the bosonic modes
P P h +gm(j)σx(j) am+a†m , (given by the product of individual P-representation for
and its rotating wave counterpart(cid:0): (cid:1)i ecaocmhpoofntehnetmofoαdesis).dIennootuerdnboytaαtion. αCle∈arCly2,Mo,nwehhialesetahcaht
m
Hˆrw = 2j=1 Mm=1 εσˆz(j)+∆jσˆx(j)+ωma†mam limInβ→o∞urΓnt,uβm=erΓict.al study, we will reproduce the oper-
P P h +gm(j) σ+(j)am+σ−(j)a†m , ations Γt by adopting the method detailed in the pre-
(cid:16) (cid:17)i(12) vious section, which is defined for pure states, and also
4
1
g =2.7
1
0.8 g =2.1
2
g =1.8
2
0.6
F
(a) (b)
0.4
FIG. 1: Choi fidelity F versus rescaled time, for Hˆrw with
ε = ∆ = 0, g1 = 1, g2 = 1.9, obtained at zero temperature 0.2
byMCE method(dot-dashed)andexactanalyticintegration
(dotted) for M =1 and ω1 =0.1 (a), and M =3 and ωm = 0
0.1m for 1≤m≤3 (b). The lines F =0.25 are reported for 0 1 2 3 4 5
reference. g1t
FIG. 2: MCE results for the Choi fidelity F versus rescaled
time, for Hˆrw with ε =∆ = 0, g1 =1, M = 10, ωm =0.1m
reconstructthe operationsΓ bysamplingdifferentini-
tial pure coherent states αt,βfor the field according to for 1≤ m≤10, zero temperature and different values of g2.
The line F =0.25 is reported for reference.
the distribution given by|Pi(α). We will describe the
field in terms of coupled coherent states during the time
evolution and then trace it out to achieve the quantum
operation acting on the two qubits. The quantity F captures, in one real number, a rele-
To define the gate fidelity F, we will make use of the vant facet of the quantum dynamics governing the two
classic channel-state duality (Choi isomorphism) map- qubits. Itsrelationshiptoquantumcoherenceismanifest
ping linear quantum operations over a Hilbert space inthatiftheoff-diagonalelementsbetweenthebasisvec-
H
into quantum states on the Hilbert space [45]. torsofEqs.(7-10)aresetto zero,then onehas F 1/4.
H ⊗ H ≤
Turning to the two qubits Hilbert space spanned by AnyvalueofF largerthan1/4isthusinasenseasigna-
H
the basis states (7-10), let us define the maximally en- ture of quantum coherence. More importantly, F is also
tangled fiducial state ψ (belonging to 2) as: a measure of how well a coherent quantum task can be
| i H
performedandisalsostrictlyrelatedtotheentanglement
1
ψ = ( 1 1 + 2 2 + 3 3 + 4 4 ). (17) generated between the two qubits (in that entanglement
| i 2 | i⊗| i | i⊗| i | i⊗| i | i⊗| i
a perfect CZ gate would get entanglement equal to 1
For a generic CP-map Ω, the corresponding quantum ebit for a properly chosen initial state). Moreover, F,
state ̺ may be defined as althoughpartial to the chosenreference gate (CZ in this
Ω
case) is completely independent of the initial state, and
̺ =(Ω )( ψ ψ ), (18)
Ω ⊗1 | ih | represents a property of the dynamics alone.
where is the identity map acting on . Of course, we could have chosen more generic quanti-
Since1the CZ gate is unitary, the quHantum state ̺ fierslike,forinstance,thelargesteigenvalueoftheopera-
CZ
is bound to be pure: ̺CZ =|ϕCZihϕCZ|, with tor̺Γt,β,whichwouldequal1intheidealcasewherethe
qubits undergo a unitary evolution and would quantify,
1
ϕ = ( 1 1 + 2 2 + 3 3 4 4 ) in a sense,the overallcoherenceof the qubits’ evolution.
CZ
| i 2 | i⊗| i | i⊗| i | i⊗| i−| i⊗| i However, we deem such choices to be less informative
1 4 withregardtothe applicativepotentialofacomplex dy-
= f(j)( j j ) . (19)
2 | i⊗| i namics.
Xj=1 Given a potentially useful quantum dynamics, the
The state ̺ corresponding to Γ is instead given by knowledge of F is instead very desirable to possess.
Γt,β t,β
Demonstrating the use of a numerical technique capable
1 4 of providing one with reliable estimates of F in relevant
̺ = Γ ( j k ) j k . (20)
Γt,β 4 t,β | ih | ⊗| ih | situations is, in a nutshell, the aim of the current analy-
jX,k=1 sis.
WecanthennaturallydefinetheCZoperationfidelityF
(which we shall informally refer to as ‘Choi fidelity’) as
the overlap IV. CHOI FIDELITY OF THE CZ GATE
4
1
F = ϕ ̺ ϕ = j Γ ( j k ) k . Here, we will slightly deviate from the previously
h CZ| Γt,β| CZi 16jX,k=1h | t,β | ih | | i adopted notation by setting gl(j) = gj for all j and l.
(21) It is important to remark that assuming equal couplings
5
1 0.8
gg1==22..71 0.7 g2=2.7
0.8 g2=1.8 0.6 g2=2.1
2 g =1.8
2
0.5
0.6
F F0.4
0.4
0.3
0.2
0.2
0.1
0 0
0 1 2 3 4 5 0 1 2 3 4 5
g1t g1t
FIG. 3: MCE results for the Choi fidelity F versus rescaled FIG. 5: MCE results for the Choi fidelity F versus rescaled
time, for Hˆrw with ε =∆ = 0, g1 = 1, M =10, ωm = 0.1m time, for Hˆrw with ε =∆ = 0, g1 =1, M = 10, ωm =0.1m
for 1 ≤ m ≤ 10, β = 10 and different values of g2. The line for 1 ≤ m ≤ 10, β = 5 and different values of g2. The line
F =0.25 is reported for reference. F =0.25 is reported for reference.
2.7 2.7
0.8 0.7
2.6 2.6
2.5 0.7 2.5 0.6
2.4 0.6 2.4
0.5
2.3 0.5 2.3
g2 g2 0.4
2.2 0.4 2.2
0.3
2.1 0.3 2.1
2 0.2 2 0.2
1.9 0.1 1.9 0.1
1.8 1.8
0 1 2 g1t 3 4 5 0 1 2 g1t 3 4 5
FIG. 4: MCE results for the Choi fidelity F versus rescaled FIG. 6: MCE results for the Choi fidelity F versus rescaled
time, for Hˆrw with ε =∆ = 0, g1 = 1, M =10, ωm = 0.1m time, for Hˆrw with ε =∆ = 0, g1 =1, M = 10, ωm =0.1m
for 1≤m≤10, β =10 and different valuesof g2 (red stands for 1≤m≤10, β = 5 and different values of g2 (red stands
for higher values, blue for lower values). for higher values, blue for lower values).
betweeneachqubitandallthe modesisinnowayessen- ∆ = 0 can be easily solved analytically. The agreement
tial to our numerical approach. Such an assumption can between such analytical solutions and the MCE results
be – and will be, in the following – relaxed if needs be. has been tested for up to ten modes and is excellent.
Let us then, to begin with, set the coupling between Figs. 1(a) and 1(b) show such an agreement in terms of
thefirstqubitandthefieldmodesg to1,andessentially CZ Choi fidelity F for g = 1.9 and, respectively, one
1 2
choose it as the unit of time. Let us also, until further mode with ω = 0.1 and three modes with ω = 0.1,
1 1
notice, set ε = ∆ = 0, β (zero temperature) and ω = 0.2 and ω = 0.3. An initial peak with fidelity
2 3
→ ∞
consider the rotating wave Hamiltonian Hˆ . Note that larger than 0.9 is immediately apparent: this peak will
rw
the Hamiltonian Hˆ with ε = ∆ = 0 can be derived be the main object of our investigation, for larger num-
rw
fromthe full HamiltonianHˆ with ∆=0 by switching to bers of modes too. For three modes, the peak appears
interaction picture and applying the rotating wave ap- atatime whichis approximatelyreducedby a√3factor
proximation: ε can be thus be set to zero and each field with respect to the single mode case. This cooperative
frequency ω is shifted as per ω ω 2ε. effectis confirmedforallnumber ofmodesup to 20,and
m m m
Byexploitingtheconservationof→thenu−mberofexcita- is simply due to the fact that the qubits are coupled to
tions,thedynamicsgovernedbytheHamiltonianHˆrwfor the field through the mode √1M Mm=1am, with an ef-
P
6
0.8 2.7
g=2.7
2 0.7
0.7 g2=2.2 2.6
g=1.8
2 2.5 0.6
0.6
2.4 0.5
0.5
2.3
0.4
F0.4 g2
2.2
0.3
0.3
2.1
0.2 0.2
2
0.1 1.9 0.1
0 1.8
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 4 5
∆t g1t
FIG. 7: MCE results for the Choi fidelity F versus rescaled FIG. 8: MCE results for the Choi fidelity F versus rescaled
time at zero temperature, for Hˆrw with ε = ∆ = 1, g1 = 1, time at zero temperature, for Hˆrw with ε = ∆ = 1, g1 = 1,
M = 10, ωm = 0.1m for 1 ≤ m ≤ 10 and different values of M = 10, ωm = 0.1m for 1 ≤ m ≤ 10 and different values of
g2. The line F =0.25 is reported for reference. g2 (red stands for higher values, bluefor lower values).
fectivecouplingwhichscaleslike√M (clearly,thisisthe 0.7
g=2.7
2
consequenceofassumingequalcouplingswithallmodes). g=2.2
0.6 g2=1.8
Figs. 2-6 depict a detailed analysis of the Choi fidelity 2
for M = 10 bosonic modes with frequencies ωm = 0.1m 0.5
for 1 m 10, three different temperatures (β ,
β = 1≤0 and≤β = 5), and different values of the cou→pli∞ng 0.4
F
g , scanned over the range 1.8 2.7. The zero tempera-
2 − 0.3
ture case (Fig. 2) shows how the dispersion of quantum
coherence among the field’s degrees of freedom affects 0.2
the gate’s fidelity (whose maximum is smaller than in
the one- and three-modes cases),althoughin the consid- 0.1
ered region of dynamical parameters the effect is not as
0
pronouncedasonecouldimagine. Theplotsclearlyshow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
∆t
thedetrimentaleffectofthermalfluctuationsoncoherent
quantum effects, even at such relatively small tempera-
FIG. 9: MCE results for the Choi fidelity F versus rescaled
tures. Inpractice,this suggestssevereconstraintsonthe
temperature for the observationof coherentquantumef- time, for Hˆrw with ε =∆ = 1, g1 =1, M = 10, ωm =0.1m
for 1 ≤ m ≤ 10, β = 10 and different values of g2. The line
fects mediated by discrete vibrational modes (typically
F =0.25 is reported for reference.
much more susceptible to thermal excitations than opti-
calmodesduetotheirlowerfrequency),consideringthat
the highesttemperatureaccountedfor isaround0.2g in
1
natural units. Most importantly, we were able to deter- displayed in Figs. 7-10. The initial peak in F is still
minetheoptimalvalueofthecouplingg2 withrespectto apparent, but the breaking of the phase invariance by
a vast range of values (much wider than what reported the term∆clearlydegradesthe qualityofthe gate,with
in the plots), in terms of the maximal converged Choi a maximum Choi fidelity which is now around 0.7, even
fidelity F, and to establish that g2 2.2 yields the clos- at zero temperature.
≃
est results to an idealCZ gate. Clearly,in practice, such We now turn to the full spin-boson like Hamiltonian
couplings will not always be tunable at will, or possibly Hˆ (including the counter-rotating terms in the qubits-
only within a given windows of values: it is anyway re- field coupling), and consider the case ε = ∆ = 1 and
markable to be able to identify optimal values given a ω = 0.1m for 1 m 10. The inclusion of the
m
specific dynamical figure of merit. counter-rotatingterm≤s mak≤es the simulation much more
Letusnowmoveontoacasewhichisnotanalytically challenging to run and converge. Roughly speaking, the
treatable, with ε = ∆ = 1 and ω = 0.1m for 1 m main difficulty one encounters comes down to the fact
m
≤ ≤
10. As usual, we set g =1 and consider different values that the time-derivatives of the phase space positions of
1
of g and different temperatures (zero temperature and the basis grid, determined by the Ehrenfest dynamics as
2
β = 10): the values of F for such a configuration are perEq.(6),aremuchlargerifthecounter-rotatingterms
7
2.7
2.6
0.6
2.5
0.5
2.4
2.3 0.4
g2
2.2 0.3
2.1
0.2
2
0.1
1.9
1.8
0 1 2 3 4 5
g1t
FIG. 12: MCE results for the Choi fidelity F versus rescaled
FIG. 10: MCE results for the Choi fidelity F versus rescaled time, for Hˆ with ε = ∆ = 1, g1 = 0.5, M = 10, ωm = 0.1m
time, for Hˆrw with ε =∆ = 1, g1 = 1, M =10, ωm = 0.1m fFor=10≤.25mis≤re1p0o,rtβed=fo1r0raenfedrednicffee.rent values of g2. The line
for 1≤m≤10, β =10 and different valuesof g2 (red stands
for higher values, blue for lower values).
0.9
g=2.7
2
0.8 g2=2.2
g=1.8
2
0.7
0.6
0.5
F
0.4
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
∆t
FIG. 13: MCE results for the Choi fidelity F versus rescaled
tFMiImG=e.a11t10:,zeMωrmoCtE=emr0ep.s1eumrlatstfuoforrer1,tfh≤oermCHˆh≤owii1fit0hdeaεlnitd=ydF∆iffve=erer1snu,tsgv1rael=suceas0l.eo5df, Mt1i1m≤=e am2t0,≤zeωr2mo0t=aenm0dp.1demriffaetfuroerrne1,tfv≤oarlmuHˆesr≤wo1fw0g,i2t.ωhmTεh==e l0∆in.1e=(mF1−,=g1010.2)=5fo1irs,
g2. The line F =0.25 is reported for reference.
reported for reference.
areincluded. Thetime-dependentgridthusevolvesmuch the field,whicharebarelynoticeableforβ =10(andare
more rapidly in phase space and is likely to leave the insteadmanifestintherotatingwaveregimeatthe same
dynamically relevant region and accumulate substantial temperature).
errorsearlier. Wewerehoweverabletoobtainedwellcon- In order to simulate the effect of a band of a 1-
verged results by reducing the coupling g to 0.5, which dimensional photonic band-gap medium, where modes
1
is still very far from the perturbative regime. Likewise, are usually doubly degenerate in frequency (since they
we scanned values of g up to 0.5. Quite significantly can propagate in either spatial direction), we have also
2
– as confirmed by Figs. 11 and 12, respectively for zero considered a case with M = 20 modes, two for each
temperature and β = 10 – we could not find any value equally spaced frequency. All the other parameters have
of g such that the Choi fidelity of the CZ gate reaches been kept as above, with ε = ∆ = 1 and g = 1 in the
2 1
0.25. In fact, strong enough couplings are necessary to rotating wave case (Fig. 13), and g = 0.5 in the full
1
entanglethe two qubits onshortenoughtime-scalesbut, Hamiltonian (Fig. 14). Comparing Fig. 13 with Fig. 7
with such strong couplings, the counter-rotating terms showsthattheinitialpeakinChoifidelityisstillpresent:
heat the qubits up too quickly for coherent effects to moreover,notonlydoesitoccurearlierbyafactor√2(as
take place, at least in this region of parameters. This expected because of the cooperation between the modes
heating overshadowsthe effect ofthermalfluctuations in duetothebalancedcoupling),butitisalsohigher. Con-
8
FIG. 14: MCE results for the Choi fidelity F versus rescaled FIG. 16: MCE results for the Choi fidelity F versus rescaled
time at zero temperature, for Hˆ with ε = ∆ = 1, g1 = 0.5, timeatzerotemperature,forHˆrwwithε=∆=1andacom-
M = 20, ωm = 0.1m for 1 ≤ m ≤ 10, ωm = 0.1(m−10) for mon Ohmic bath with α = 0.09, different cutoff frequencies
11≤m≤20 and different values of g2. The line F =0.25 is anddifferentnumbersoftotalbathmodes. ThelineF =0.25
reported for reference. is reported for reference.
communities, and has been explored under a number of
– either more specific and applied or more general and
abstract – viewpoints [52–66]. However, the problem of
studying the non-perturbative interaction of two qubits
with a common bath is still in general a difficult one.
Thus, to provide the reader with further evidence of the
versatilityandpowerofourapproach,wealsoreportthe
applicationofthe MCEmethodtothe controlled-ZChoi
fidelity for the case of the spin boson Hamiltonian Hˆ,
withε=∆=1,andbothqubitsinteractingwithacom-
mon bath at zero temperature and with Ohmic spectral
density J(ω) given by
2
ω
FIG. 15: MCE results for the Choi fidelity F versus rescaled J(ω)= παωe−ωc , (22)
time at zero temperature, for Hˆ with ε=∆=1 and a com-
mon Ohmic bath with α=0.09, ωc =2.5 and different num- where α is the Kondo parameter, which we fix at 0.09,
bers of total bath modes. The line F = 0.25 is reported for and ω is a cutoff frequency.
c
reference. We use a standard approach to discretise the bath,
which has already been proven very reliable for single
spin Ohmic spin-boson systems [41]. In particular, the
frequencies and coupling strengths are chosenas follows:
trary to common intuition, this example shows that a
largernumberofmediatingmodes,infavourabledynam-
ωmax
ical configurations such as this, can actually be advan- m 1 e− ωc
−
ω = ω ln 1 , (23)
tageous for the implementation of locally coherent dy- m − c  − (cid:16) M (cid:17)
namics. Note that, for M =20 modes, we needed about
N = 400 coupled coherent states to achieve converged  
results. This is as large a basis set as we used in this
study. ωmαωc 1 e−ωmωcax
g1 =g2 =v − , (24)
m m u (cid:16)2M (cid:17)
u
t
A. Zero temperature Ohmic spin-boson bath
where ω is a free parameter of the numerics, which
max
we willconvergeourresults against(in thatwe chooseit
Thenotionofentanglingseparatedsystemsandofdis- large enough to obtain convergedresults). In particular,
tributing quantum coherence by interaction with com- we choose ω = 12.5 for ω = 2.5 and ω = 6 for
max c max
mon heat baths or other incoherent means is well estab- ω =1. Also,thecouplingstrengthsgj aredefinedasin
c m
lishedinthequantuminformationandcondensedmatter Eqs. (11) and (12).
9
(a) (b) (c)
FIG.17: MCE results for theconcurrenceversusrescaled timeat differenttemperaturesfor Hˆrw with g1 =1and g2=2.1. In
(a), ε = ∆ = 0, M = 10 (with ωm = 0.1m for 1 ≤ m ≤ 10) and the initial state is |4i = | ↑↑i; in (b), ε = ∆ = 0, M = 10
(with ωm = 0.1m for 1≤ m ≤ 10) and the initial state is |2i = | ↑↓i; in (c), ε = ∆ = 1, the initial state is |2i = | ↑↑i and,
respectively,M =10(with ωm=0.1m for1≤m≤10) for thedash dottedlineandM =20(with ωm =0.1m for1≤m≤10
and ωm =0.1(m−10) for 11≤m≤20) for the dashed line.
Fig. 15 shows the convergence of our results for ω = VI. CONCLUSIONS
c
2.5andthefullHamiltonianHˆ intermsofthetotalnum-
ber of modes in the bath (M =50 and M =100). Quite
interestingly, if the counter-rotating terms are included
Wehavepresentedanextensivenumericalstudy,based
we obtain larger gate fidelities when mimicking a bath
on Multi-Configurational Ehrenfest trajectories, on the
than for a smaller set of discrete bus frequencies. Off
dynamicsoftwoqubits interactingwithacommonsetof
resonant modes are more influential in the full Hamilto-
bosonic field modes, obtaining converged results for the
nianand seems to be capturedfaithfully by our method.
ChoifidelityofanentanglingCZgatebetweenthequbits
In Fig. 16, instead, we report results for the rotating for a rather wide range of Hamiltonian parameters and
wave Hamiltonian Hˆrw and different cutoff frequencies field temperatures, which cannot be covered by pertur-
and number of bath modes. In this case, the effect of bationtheoryorotherapproximateapproaches. Wethus
the counter-rotating terms is rather limited. Also, the demonstratedthe capabilityoftracking,analysinginde-
influence of larger cutoff frequencies at longer times is tail,andevenoptimisingwithrespectofcertainrangesof
clearly visible in the plot. some parameters, specific aspects of the coherent quan-
tum dynamics of the qubits.
V. ENTANGLEMENT GENERATION We were able to properly take into account the effect
of finite bath’s temperatures on the reduced dynamics
Typically,alargeChoifidelity for the (entangling)CZ ofthequbits,andalsotohighlightsomecounterintuitive
gate corresponds to the generation of substantial entan- featuresrelatedtothescalingofcoherentsignatureswith
glement between the two qubits. To support this state- the number of field modes (which we varied over the
ment, we report here a brief study on the entanglement range 1 100), showing that at times more mediating
−
generatedbetweentwoqubits. Asanentanglementquan- modes can actually be advantageous for the distribution
tifier, we adopt the concurrence,anentanglementmono- of quantum coherence.
tone that can be easily calculated for a system of two
qubits[67]. Figs.17(a),17(b)and17(c)showtheconcur- The main limitations of our approach lie in the dif-
rence versus rescaled time for the rotating wave Hamil- ficulty of handling counter-rotating qubit-field coupling
tonian Hˆ with different initial states, temperatures, terms in the strong coupling regime (i.e., when the cou-
rw
dynamicalparametersandnumberofmodes. Thedegra- pling strengths are comparable to the inherent dynami-
dation of quantum entanglement due to temperature is calfrequenciesofthequbits). Eveninsuchinstances,we
apparent(Figs.17(a)and17(b)),alongwiththespeedup could however reach convergence by somewhat limiting
in the entanglementgenerationinduced by a doubling of the range of the coupling strengths.
the modes (Fig. 17(c)).
It is also worth noticing that we did not find any Withinsuchlimitations,the MCEapproachhashence
region of parameters where entanglement between the beenestablishedasapowerfultoolforthedetailedstudy
two qubits is generated for the full Hamiltonian Hˆ and of complex quantum dynamics even with relatively lim-
M = 10, thus mirroring our failure in obtaining Cz fi- itedresources(desktopcomputers),typicallyforsystems
delities larger than 0.25. wherediscretesets ofupto 100fieldmodes areinvolved.
10
1.0001 1.2
m00..999999891 NNNN532200000000−−−−ccccoooommmmpppp22220000000 1.11.51 NNNN532200000000−−−−ccccoooommmmpppp22220000000 eρ130.000..100512 NNNN532200000000−−−−ccccoooommmmpppp22220000000
nor00..99999967 E1.05 0.005
0.9995 0
1
0.9994 −0.005
0.99930 1 2 ∆t 3 4 5 0.950 1 2 ∆t 3 4 5 −0.010 1 2 ∆t 3 4 5
(a)Normversusrescaledtime (b)Energyversusrescaledtime
(a)̺11 versusrescaledtime (b)̺13 versusrescaledtime
FIG.18: Normandexpectation valueoftheenergyforMCE
results at zero temperature, for Hˆrw with ε = ∆ = g1 = 1, FfoIrGM. 2C1E: rEensutrltiessaotfztehreo qteumbiptes’radteunrsei,tyformHaˆtrwixit̺h11εa=nd∆̺1=3
dg2iff=ere2n.t7,vaMlue=s o1f0N(waintdhcωommp=. 0.1m for 1 ≤ m ≤ 10) and g1 =g2 =1, M =10 (with ωm =0.1m for 1≤m≤10) and
different values of N and comp.
1 0.05
0.9 0.04
0.8 0.03 Appendix A: Ehrenfest dynamics of the coherent
0.7 0.02 states
0.6 0.01
eρ110.5 eρ13 0
0.4 −0.01
0.3 −0.02 Here,weelaborateonEq.(6)andderiveexplicitly the
0.2 −0.03
0.1 −0.04 equationofmotionofeachcomplexparameterα(m). The
00 1 2 ∆t 3 4 5 −0.050 1 2 ∆t 3 4 5 approximated Lagrangian j reads j
L
(a)̺11 versusrescaledtime (b)̺13 versusrescaledtime
d d
FIG.19: Entriesofthequbits’densitymatrix̺11 and̺13 for = c c α ,l Hˆ α ,n i c c˙
MCEresultsatzerotemperature,forHˆrw withε=∆=g1 = Lj l,n=1 ∗l,j n,jh j | | j i− l=1 ∗l,j l,j
1, g2 = 2.7, M = 10 (with ωm = 0.1m for 1 ≤ m ≤ 10) and X X
different values of N and comp. i d c 2 α˙j ·α∗j α˙∗j ·αj , (A1)
l,j
− | | 2 − 2
l=1 (cid:18) (cid:19)
X
where l,α = l α . Note that α ,l Hˆ α ,n is
j j j j
just af|unctioin o|ftih⊗e v|ectiorα andthehinteg|ers| l andin,
j
promptly evaluated by normal ordering Hˆ.
(m)
The Euler-Lagrangeequation for α then is:
j
4
∂ ∂
(a)Norm versusrescaledtime (b)Energyversusrescaledtime ∂αL(mj) = c∗l,jcn,j∂α(m)hαj,l|Hˆ|αj,ni
j l,n=1 j
X
FreIsGul.t2s0a:tNzeorromteamndpeerxapteucrtea,tfioornHˆvawluiethofεt=he∆en=ergg1y=forg2M=C1E, +i d |cl,j|2α˙(jm2)∗ = (A2)
M = 10 (with ωm = 0.1m for 1 ≤ m ≤ 10) and different Xl=1
values of N and comp. = id d c 2α(jm)∗ = d ∂Lj
− dt | l,j| 2 ! dt∂α˙(m)
l=1 j
X
Acknowledgments which, by neglecting the time dependence of c (setting
l,j
d d c 2 =0), can be rearrangedto obtain
WethankHannuWichterich,RuiZhangandLianHeng dt l=1| l,j|
(cid:16) (cid:17)
Tong for their help during the writing of the code. SYY P
hppauosrrtsubfrienoegmntEshuPispSprRoeCrste,eatdrhcrhbo.yugDahSgKraCacnktWnsooEwnPlgeSdRSgcCehsEofilPanr/asInh0ci1ipa4l5w0sh0ui/ple1- α˙(jm)∗ =iP4l,n=1c∗l,jcn,jdl∂=α1∂j(m|c)lh,jα|2j,l|Hˆ|αj,ni. (A3)
andNSF/EPSRCEP/J001481/1. ASthankstheCentral
P
Research Fund of the University of London for financial This equation governs the evolution of the vectors α in
j
support. our numerics.