Table Of ContentDynamical Invariants for Quantum Control of Four–Level Systems
Utkan Gu¨ng¨ordu¨,1,∗ Yidun Wan,1,† Mohammad Ali Fasihi,1,2,‡ and Mikio Nakahara1,3,§
1Research Center for Quantum Computing, Interdisciplinary Graduate School of Science and Engineering,
Kinki University, 3-4-1 Kowakae, Higashi-Osaka, Osaka 577-8502, Japan
2Department of Physics, Azarbaijan University of Tarbiat Moallem, 53714-161, Tabriz, Iran
3Department of Physics, Kinki University, 3-4-1 Kowakae, Higashi-Osaka, Osaka 577-8502, Japan
We present a Lie–algebraic classification and detailed construction of the dynamical invariants,
also known as Lewis–Riesenfeld invariants, of the four–level systems including two–qubit systems
which are most relevant and sufficiently general for quantum control and computation. These
invariantsnotonlysolvethetime–dependentSchr¨odingerequationoffour–levelsystemsexactlybut
alsoenablethecontrol,andhencequantumcomputationbasedonwhich,offour–levelsystemsfast
and beyond adiabatic regimes.
3
1
0
I. INTRODUCTION RF pulse and it is typically on the order of 10 µs for a
2
π–pulse fora heteronuclearmolecule. It maytakelonger
n
Quantum computation is an emerging discipline in for a homonuclear molecule. By considering this large
a
J which quantum physics is used as a computational re- difference between the execution times, two–qubit gates
8 source [1, 2]. By making use of states and operations, often become a bottleneck in shortening the execution
which have no classical counterparts, a quantum com- time of a whole quantum circuit. This is the motivation
] puter is expected to execute some hard tasks for a clas- of considering non–adiabatic implementation of nontriv-
h
sical computer in a reasonable time. Here operations ial SU(4) gates.
p
- aremostly elements ofSU(2n), where n is the numberof Furthermore,therearemanynontrivialquantumalgo-
t qubits,onwhichtheoperationsact. Itisknownthatany rithms, such as the Deutsch Algorithm, the Grover al-
n
a classical logic operation may be realized by a collection gorithm and the Bernstein–Vazirani algorithm, just to
u ofthe NANDgate. Thecorresponding“universalitythe- name a few, which can be demonstrated with a two–
q orem” is due to Barenco et al. [3]. The theorem claims qubit system and an SU(4) gate. Execution of these al-
[ that“anyunitarygatecanbedecomposedintoone–qubit gorithms with a speed beyond the adiabatic limit shows
3 (i.e., SU(2)) gates and the CNOT gates. In other words, thepromisingfutureofarealizationofquantumcomput-
v thesetofone–qubitgatesandtheCNOTgateareuniver- ing.
4 sal in gate implementations. In many physical systems,
3 implementation of a one–qubit gate is often not hard. It
0
may be realized by the Rabi oscillation or the Raman A. Dynamical Invariants
3
transition, for example. In contrast, implementation of
.
5 the CNOT gate can be challenging and its realization is An alternative way of constructing solutions of the
0
sometimes regardedas a milestone for a physical system time–dependent Schr¨odinger equation and obtaining the
2
to be a true candidate of a working quantum computer time–evolution operator is by means of the eigenstates
1
: [4]. Later, it turned out that any SU(4) gate, which en- of an operator I = I† 1, which is a dynamical invariant
v tangles a tensor product state, may serve as an element ofthesystemwithatime–independentexpectationvalue
i
X of a universal set of quantum gates with the set of one– I [7]. Adynamicalinvariant(DI)oraLewis–Riesenfeld
qubit gates [5, 6]. Important exceptions of two–qubit Ihnviariant (LRI) obeys
r
a gates that are excluded are the SWAP gate and the “lo-
cal gates” SU(2) SU(2). I = ψ(t)I ψ(t) = ψ(0)U†IU ψ(0) =const., (1)
⊗ h i h | | i h | | i
The above observations make the importance of im-
where U =U(t;0) is the time evolution operator. Using
plementation of an SU(4) gate obvious. Adiabatic two–
Heisenberg’s equation (in units such that ~ = 1) we can
qubit gate implementation is limited in time by the cou-
restate this condition as
pling strength between the two qubits on which the gate
acts. InNMR,forexample,thecouplingstrengthJ ison (H)
d ∂I
theorderof1 100Hz,leadingtotheexecutiontimeon 0= I(H) = +i[H(H),I(H)]. (2)
∼ dt (cid:18)∂t(cid:19)
the order of millisecond to second, while the execution
time ofaone–qubitgateislimitedbythe strengthofthe
where the superscript (H) denotes a Heisenberg picture
operatorO(H) =U†OU,whereOisaSchr¨odingerpicture
∗ utkan@alice.math.kindai.ac.jp
† ywan@alice.math.kindai.ac.jp
‡ fasihi@alice.math.kindai.ac.jp 1 We willuse I to denote a dynamical invariant and 11 to denote
§ nakahara@math.kindai.ac.jp 2×2identitymatrixthroughoutthepaper.
2
operator. With the LHS being 0, Eq. (2) simplifies to In principle, a Hamiltonian of a system is a time–
dependent operator on the Hilbert space of the system,
∂I
0= +i[H,I], (3) so is a DI associated with the Hamiltonian; they satisfy
∂t the DI equation. The first two difficulties regardthe op-
which is anequation in the Schr¨odinger picture. We will erator form of the DIs of a Hamiltonian. A Hamiltonian
refertothisLiouville–vonNeumanntypeequationasthe can spawn multiple DIs, as aforementioned; hence, one
DI equation. In terms of the eigenstates of I, φ (t) , must not only know all possible DIs of a Hamiltonian
n
| i
the general solution of the time–dependent Schr¨odinger but also select from among them the one pertinent to
equation can be written as the endofthe control,anddetermine whetherthe choice
is optimal. These difficulties worsen dramatically with
Ψ(t) = cneiαn(t) φn(t) (4) the system’s level. Unfortunately, the current literature
| i | i
Xn hasnot beenable to cope with these difficulties; instead,
itusually fumbles a wayto certainDIs. Note thatin the
and the time–evolution operator becomes
case of two–level, however, a classification of Hamiltoni-
U(t;0)= eiαn(t) φ (t) φ (0) (5) ans and their DIs does exist [29]. Consequently, insofar
n n
Xn | ih | the quantum control and computation models based on
DIs are nighlimited to two–levelsystems [8–16]. Special
where Lewis–Riesenfeld phase α (t) is given as [7]
n cases in four–level systems have also been studied [28].
t ∂ Even if the operator form of a DI is known (it may
αn(t)= φn(s) i H(s) φn(s) ds. (6) or may not belong to a subalgebra of su(4) as we detail
Z h |(cid:18) ∂s − (cid:19)| i
0 in Section II) it is still hard to obtain the DI in closed–
We observe that eigenstates of I evolve in the following form by solving the DI equation, which is in fact a set
simple form of differential equations, the number of which depends
on the number of degrees of freedom of the system. Be-
eiαn(t) φ (t) =U(t;0)φ (0) . (7)
| n i | n i sides partially causing the first limitation, this third dif-
This passage is transitionless in the eigenbasis of I, and ficulty circumscribes how and how much one can control
a quantum system based on the DIs of the system. In
is not necessarily adiabatic. The power of DIs goes be-
yond solving the time–dependent Schr¨odinger equation fact, because of this limitation, in order to gain the con-
trol of a quantum system via a DI of the system, a spe-
however. A major problem in adiabatic quantum con-
trol is that in many cases the evolution is so slow that cial method —the inversely engineered control (IEC)—
is needed, which was invented in [13] for pure states and
thesystemmaystartdecohering. Evolutionintheeigen-
states of I, however,is not restricted by the adiabaticity extended to mixed states in [8].
condition and can be made to be fast, a feature which Although this paper is not about quantum control, so
caused a recent surge of applications [8–16]. DIs have as to understand some settings for classifying DIs in the
been used in the context of quantum field theory [17]. main text, let us briefly introduce the gist of quantum
Another attractive feature of DI is that once obtained, control,inregardofAQCandIEC.Tocontrolaquantum
the time–evolution operator of the system can be con- system one needs to expose the system in an external
structed from its eigenvectors following Eq. (5) (which electromagneticfieldthatinteractswiththesystem,such
canbeotherwiseobtainedbythedirectevaluationofthe thatthetotalHamiltonianofthesystemandtheexternal
time–ordered integral [18] or through Wei–Norman ex- fieldevolvestemporallyinthewaythattakesthesystem
pansion [19–25]). to a designated state at some point of time. A system
under anAQC remainsinaninstantaneouseigenstateof
the total Hamiltonian that must satisfy the adiabaticity
B. Quantum Control Based on DIs condition, the violation of which would render the AQC
of the system impossible.
Quantum control is vital to quantum computation. In the control based on a DI of a system, however,
For example, in one of the major quantum computation the systemdoes notnecessarilyfollowanyinstantaneous
schemes,the AQC [26,27], a quantum gateis essentially eigenstateofthetotalHamiltonianofthesystemandthe
an adiabatic quantum control passage. Nevertheless, in externalfieldbutratheralwaysstaysinaninstantaneous
general,anadiabaticcontrolofasystemisveryslowrel- eigenstate of the DI. This idea of control is realized by
ativetothe decoherenceofthe systemwhichmayrender theIECmethod. Incontrasttothelogicthatoneshould
an AQC model over the system impractical. To resolve know a Hamiltonian first to determine its DIs, in IEC,
this issue, a quantum computation scheme based on DIs one only get hold of the matrix form of a Hamiltonian,
hasalsobeenproposed[28]. Nonetheless,twolimitations while leaving its physical parameters unknown, assume
hinderthequantumcontrolmethodsandhencethequan- the matrix formofaDI andpostulate some simple func-
tum computation models based on DIs are not widely tional forms of its parameters, determine the physical
applicable, due to the three difficulties in obtaining a DI parameters of the DI by the boundary conditions that
of a Hamiltonian. comply with the wanted initial and final states of the
3
control, and then use the DI equation to nail down the i) We apply the key ideas addressed in [30] to four–
Hamiltonian that bears this DI. In doing so, one avoids levelsystemswhichareofparticularimportancein
the difficulty of solving the DI equation directly from a quantum computation.
known Hamiltonian. Aside of this mathematical advan-
ii) WeshowDIsoffour–levelsystemscanbeclassified
tage of IEC,a physicaladvantageis that the controlcan
in terms of the maximal subalgebras of su(4).
bedoneregardlessoftheadiabaticityconditionandthus
serves as a shortcut to adiabatic control. But the costs
iii) Bymeansofthisclassification,weconstructDIsfor
are: 1) the Hamiltonian obtained this way may be un-
afamily offour–levelsystem,whicharesufficiently
physicalorimpractical2)a greatdealofcontrolrangeis
general for applications.
lost due to the restrictedpostulates one can make of the
parameters of the DI. iv) Ourclassificationindicatesagreatreductionofthe
In [13] the method is demonstrated in an su(2) prob- complexity in constructing exact DIs of four–level
lem. The DI is taken to be a function of the form systems.
Ω
I = 0 (sinγcosβσ +sinγsinβσ +cosγσ ), (8) v) Welistexplicitsetofdifferentialequationsforboth
x y z
2 types, andfurther discussthe caseswhichareboth
where γ and β are polynomials in t of order 3 and 4 exactly solvable and physically feasible.
respectively, by ansatz. The system is assumed to be
Thispaperisorganizedasfollows. Inthenextsection,
prepared in a shared eigenstate of I and H initially, and
we give a review of the two types of DIs and the adjoint
due to Eq. (7) there will not be any transitions in the
DI equation. We move on to the classification of su(4)
eigenbasis of I. This situation is similar to adiabatic
DIs with respect to the subalgebra that spans the set of
quantum control where the passage is transitionless in
generators in the Hamiltonian in the following section,
Hamiltonian’s eigenbasis, but without the adiabaticity
and give detailed analysis for each possible case. Finally
condition. The corresponding Hamiltonian is found by
wesummarizeourresults. Throughouttherestofthepa-
using the DI equation as
per, summation over repeated indices is assumed unless
1 γ˙ γ˙ stated otherwise.
H = σ + cotγcosβ β˙ σ . (9)
x z
2(cid:20)sinβ (cid:18)sinβ − (cid:19) (cid:21)
The coefficients of the polynomials are fixed by employ- II. OBTAINING DYNAMICAL INVARIANTS
ing the population inversioncondition at a final time t ,
f
and requiring [H(0),I(0)]=[H(tf),I(tf)]=0 which en- We summarize the relevant findings of [30] in a self–
sures that the initial and final states are eigenstates of contained manner from an su(4) point of view in this
the Hamiltonian as well. tf can be arbitrarily small, as section. A given Hamiltonian can have infinite num-
long as it does not violate the time–energy uncertainty ber of DIs. Since I su(4) we can expand it linearly
relation[13, 28], resultingin apassagethatcanbe made in terms of a conveni∈ent set of su(4) generators I with
fast. Unfortunately, a precise implementation of the re- time–dependent coefficients as g(λ)λ = g(λ)(t)λ . (For
quired Hamiltonian is not an easy task. i i i i
brevity, we will drop the superscript denoting the rep-
One may wonder if in IEC it is still difficult to guess
resentation for the fundamental representation and set
the matrix form of a DI of a Hamiltonian. Indeed, it is g = g(λ) in what follows). We assume H su(4), since
still difficult; however, as IEC has been applied to only i i ∈
wecanalwaysdroptheidentityelementwhichwouldoth-
two–level systems so far 2, the difficulty is negligible, as
erwise commute with everything per Eq. (3) and result
onecanalwayscasttheDIinalinearcombinationofthe
inthesameequation. LettingHbethesetthatgenerates
threePaulimatrices,whichissimpleenoughandwithout
H and ignoring the coefficients if for a moment (we
loss ofgenerality. Afour–levelsystemhas 15 generators, ijk
usetheconventioninwhichthe structureconstantsobey
rendering the operator form of DI a lot more difficult to
[λ ,λ ]=if λ ), due to Eq. (3) I will obey
guess and resulting set of differential equations are less i j ijk k
likely to be solved analytically. A companion paper ad- I=[H,I]. (10)
dresses the issue by offering a Lie algebraic classification
of DIs for Hermitian, finite–level systems [30]. To this Whenwe chooseto include anygeneratorfromthe setH
end,weassumeinthispaperfour–levelHamiltoniansand in I, as a consequence of Cartandecomposition, I spans
theirDIsliveintheLiealgebrasu(4)asdiscussedin[30]. the minimal subalgebra S that encloses H, with H I,
⊆
Below we list our main results. span(I) =S su(n). We refer this class as S(uperset)–
⊆
type DI, or for the purposes of this paper equivalently
S(ubalgebra)–type DI, which has been the focus in the
literature so far.
2 In [28], where a quantum computation model based on DIs are Another class of DIs are generated by a completely
proposed,thecontrolmethodoffour–levelsystemsisessentially disjoint set of generators from that of H. As we will see
thesameasIEC. in Section III, the number of DIs obeying H I = ,
∩ ∅
4
which we will refer as D(isjoint)-type DIs, is determined su(3) u(1) 3 4+¯34+10+80
by the embeddings of the subalgebras of su(n), once the ⊕ −
so(4) (3,1)+(1,3)+(3,3)
minimalenclosingsubalgebraofHisdetermined. Clearly
when S = su(n), there are no D–type DIs. We will see so(4)⊕u(1) (1,1)0+(3,1)0+(1,3)0+(2,2)2+(2,2)−2
that in some cases, such as S=so(5) su(4), a D–type so(5) 5+10
⊂
invariant can simplify the problem.
WhenweplugI =giλi intotheDIequation,weobtain TABLEI.Maximalsubalgebrasofsu(4)intheadjointrepre-
sentation [34].
0=g˙ +iAg, (11)
hereafter called the adjoint DI equation, where g =
(g ,...,g )T is a vector made of the expansion coeffi- (second) qubit, and is sometimes referred to as a con-
1 15
cientsofI. Atfirstsight,iAisa15 15anti–symmetric, trol parameter. There could be an extra term 11 11 in
× ⊗
real matrix whose entries are linear combinations of the theHamiltonianwhichwedeliberatelydropped;itwould
coefficients in H. By casting Eq. (3) into commute with everything else and would not appear in
theadjointDIequation. Wewillchoosetodropanygen-
0=g˙iλi+gj[H,λj] (12) erator that commutes with all the others for the same
reason. For purposes of quantum control we prefer to
and comparing with Eq. (11), we see that the A matrix
work with a respresentation involving elements of the
canbeobtainedfromtheHamiltoniandirectlyasad(H).
This means A matrix lives in the adjoint representation form {σi ⊗11,11⊗σi,σi ⊗σj} which allows us to read
off the physical content directly. Moreover, this repre-
of su(4), that is A ad[su(4)] in general. A obtained
∈ sentationis compatible with the spinor representationof
this way will be a block diagonalized 15 15 matrix,
× so(4) and so(5), and will be referred as such from here
withdifferentblocksrepresentingS–typeandD–types(if
on. Another advantage of this spinor basis is that the
there is any). Once the adjoint representation is known,
commutatoroftwospinorelementsisneveralinearcom-
this serves as a very practical way of obtaining A.
bination of more than one generators, which lets us use
DuetotheDIequation,ifIisavalidDIandifaHermi-
Eq. (10) smoothly. We will use the shorthand notation
tiantime–independentoperatorΛissuchthat[H,Λ]=0,
XY to denote σ σ interchangeably throughout the
thenI′ =c1I+c211+c3Λwhere{ci}areconstantsisalso paper. x ⊗ y
avalidDI.Thisfreedomallowsonetoworkwitha“min-
The A matrix corresponding to the subalgebra (S–
imal” DI in which the time–independent trace part and
type)anddisjoint(D–type,mayormaynotexists)cases
terms that commute with H are dropped. We will ex-
can be read off directly as separate blocks after block–
ploit this freedom in what follows, and work with this
diagonalization,inwhichcasegbelongstooneofthetwo
kind of minimal DIs. We remark that since I obeys the
subspaces that correspond to two distinct DIs. There
same Liouville–vonNeumann equation just like the den-
can be more than one S–type or D–type DIs which will
sity matrix ρ, it is possible to obtain a density matrix
cause further block diagonalizationin the respective sec-
fromaDIviceandversa,someexamplesgivenin[31,32].
tor. The overall block structure and the algebraic struc-
Tracelesspart of ρ is alwaysa valid DI, but the converse
turearedeterminedthroughtheembeddingofthecorre-
isnottrue. Adensitymatrixhastraceoneandispositive
sponding subalgebra of the adjoint representation which
semi–definitive whereas a DI is not necessarily, however
is the 15 dimensionalrepresentationofsu(4)in our case.
this situation may be remedied using a “non–minimal”
Possible group embeddings are listed in Table I.
DI when necessary.
In the spinor basis, span(I) can be so(5), so(4) u(1)
⊕
and so(4) subalgebras of su(4). It can not, however, be
su(3) u(1)inthisrepresentation;onecanconstraintthe
III. FOUR–LEVEL SYSTEMS
⊕
coefficients of the generators to effectively transform the
representationintoasuitableonebutthe constraintsare
In this paper, we will be working with a form of four–
found to be too tight when we force the “cross–terms”
level Hamiltonian which is general enough to cover al-
such as XY and ZX to be zero in order to adapt Eq.
most all practical applications 3
(13): the number of generators reduces to four, leaving
Jiσi⊗σi+h(i1)σi⊗11+h(i2)11⊗σi (i=x,y,z)(13) udsetianilss)u.(2s)o⊕(4)u(1)u⊂(1)sus(u3b)a⊕lgeub(r1a) a(sreiseesAfprpoemndHixamAiltfoor-
⊕
with at least one non–zero J . Each J is a coupling co- nians that can have —but not necessarily present— a
i i
efficient,andeachh(1) (h(2)),isafieldactingonthe first u(1)generatorrepresentinganon–interactingsubsystem,
i i
howeverbecausewechooseignoresuchtrivialityfromthe
beginning we will not have the u(1) generator in the in-
variant.
3 By doing that we leave out some systems such as ENDOR in- While we can obtain A directly by taking the adjoint
volvinganisotropy withelectron spin and nuclear spin coupling ofH whichis practical,we maydesire toswitchto a dif-
[33]. ferent choice of generator basis afterwards, such as the
5
spinor basis 4 with a particular ordering or a new set
su(3) u(1)
of their linear combinations. In this case, we may sim- ⊕ −
ply perform a time–independent similarity transforma- so(4) {σiσj,σjσj,σk11,σkσi,σkσk,11σj}.
tion on the adjoint DI equation: let Σ be the new so(4) u(1) 11σ ,σiσ σi11 ,
basis for DI such that I = giλi = gi(Σ{)Σii}. The trans- ⊕ {σi11,11σj,σ{jσk∗,σjσi∗,}σk⊔σ{i,σkσ}k}⊔{σiσj},
formationmatrixrelating two sets ofgeneratorsthrough σi11,11σi,σjσk,σjσj,σkσj,σkσk σiσi .
{ }⊔{ }
Σi = [S]ijλj can be obtained using tr(λiλj) = 2δij as so(5) {σi11,11σ∗,σjσ∗,σkσ∗}.
[S] =tr(Σ λ )/tr(λ λ ). Notethatthismixestheorig-
ij i j i i
inalgeneratorsratherthantransformingthemaltogether
TABLE II. List of maximal (semi–)simple subalgebras S
in the same way. The g vector will transform the same su(4) when H σ 11,11σ ,σxσx,σyσy,σzσz with at leas⊂t
way as λ does, since giλi as a whole forms as a single one σiσi typeg⊆ene{ra∗tor, in∗accordance with E}q. (13). There
“component” of I, from which we deduce that the ad- is no summation over repeated indices. The indices i,j,k
∈
joint DI equation will become x,y,z are distinct. σ serves as a wildcard in such a way
{ } ∗
that σiσ denotes three elements σiσx,σiσy,σiσz. The com-
0=g˙(Σ)+iA(Σ)g(Σ); plementa∗rytableisobtainedbyswappingtheroleoffirstand
(14)
second terms in all generators. Alternative tables can be ob-
A(Σ) =SAS−1, g(Σ) =Sg.
tained by similarity transformations. Tensor product symbol
is omitted for brevity. The table is referred and used in Sec-
Clearly, for the differential equation
tion IIIB1.
∂
0= (Sg)+i(SAS−1)(Sg) (15)
∂t B. Exact Dynamical Invariants
to keep its form after the similarity transformation, we
must have (∂S/∂t)g = 0, which is always the case when TheadjointDI equationforthe su(2)=so(3)casehas
∼
the new frame is stationary as in our case. been studied in the literature extensively under different
We observe that the choice of axes is not of any im- contexts, with many known exactly solvable cases, and
portance, as any similarity transformation on the set of is known as the Bloch equation with infinite relaxation
generatorsasawholeleavesthesubalgebraspannedbyI [35–50]. Thus the Bloch equation is a special case of the
and the structure constants intact, resulting in the same adjoint DI equation.
equation as above. Nevertheless, the choice of generator Inbothso(4)=su(2) su(2)andso(4) u(1)caseswe
∼ ⊕ ⊕
basis does make a difference by mixing the components essentially have two decoupled Bloch equations, mean-
of g and determining the block structure of A. ing that the mathematical problem reduces to solving
single–qubitproblems. Notethatthetwophysicalqubits
are actually coupled in so(4) u(1) case but not in
⊕
A. Classification of Hamiltonians and Invariants so(4) case. The explicit block diagonalization to bring
(3,1) + (1,3) into 3 + 3 can be achieved by a time–
LetusdenoteanS–typeIwithI andaD–typeIwith independent similarity transformation. To this purpose,
S
I . GivenaHamiltonian,thesubalgebraspannedbyI we first notice that so(4) is readily split into two dis-
D S
can be determined directly by means of Table II. When tinct su(2), 11σ∗ σ∗11 (note that some generators
{ }⊔{ }
H 3, due to the simplicity, one can manually check mayrequireacorrectionofsigndepending i,j,k being
w| h|e≤ther these generators belong to an su(2). In other an odd permutation of x,y,z or not). {Clearl}y, such
{ }
cases, finding the smallest set in the table such that H a Hamiltonian represents composite systems with two
fits into gives the answer. However,we can alternatively non–interacting su(2) subsystems. The other so(4) case
decide on the desired subalgebra as our starting point 11σj,σkσi,σkσk σk11,σiσj,σjσj is equivalentto the
{ }⊔{ }
and pick a subset of corresponding generators from the formercaseuptoa similaritytransformation. Moreover,
table to forma Hamiltonian, which will be ourapproach theD–typeDIforso(4)resultsina9 9Amatrix,which
×
inthis paper to exhaustallpossibilities ina concise way. does not help simplifying the problem. It is because of
Depending on the list of generators appearing in I , thesefacts thatwewillnotanalyzeso(4)subalgebraany
S
further embeddings are possible through the embedding further 5. In so(4) u(1) cases however, we respectively
⊕
of the subgroups listed in Table II. Note that so(4) take the followingsimple linear combinationsof the gen-
su(4) and so(4) so(4) u(1) su(4) will result in di⊂f- erators to form two separate su(2) algebras
⊂ ⊕ ⊂
ferent D–types and different block structures for A. The (11 σi)σ∗ /2,
block structure is dictated by the maximal embedding. { ± }
σj11 11σk,σiσj σkσi,σkσj σiσi /2, (16)
{ ± ∓ ± }
σi11 11σi,σjσj σkσk,σjσk σkσj /2,
{ ± ∓ ± }
4 Because there is no standard ordering for the spinor basis and
wewillneedtousedifferentorderingslateron,wedonotlistthe 5 Although the physical context is different, [51] studies an so(4)
structureconstants forthem. HamiltonianinaLie–algebraicframework.
6
inthenotationandorderofTableII.Notethatthesum- form A(Σ) =A(Σ) (0) A(Σ) A(Σ) where
(3,1)+(1,3)⊕ ⊕ (2,2)−2 ⊕ (2,2)2
mandsineachnewgeneratorarerelatedtoeachotherby
the u(1) generator of the subalgebra; we have exploited
0 h h 0 0 0
the following to obtain these pairs: let T ,T ,T q z y
1 2 3 −
(where {T1,T2,T3}⊂IS and q2 =1) be a{n su(2)⊕}u(⊔1), hz 0 −hx 0 0 −J
Dathnieffdner{{eTTn11t,T±c2h,qoTTic31e,,sqTT2s1u±,cqhqTT2a,2sq,TTth33e}±⊔oqnT{eq3}}g/ivi2seniasniansp[o2a(2i4r,)o2⊕f3]suu(c(1a2)n). A((Σ3,)1)+(1,3) =2i−0hy h0x 00 00 −Jhz h0y ,
be obtained through time–independent similarity trans- 0 0 −J hz 0 −hx
0 J 0 h h 0
formations. Furthermore, the sector corresponding to − y x
the D–type DI further splits into two 4 4 blocks in 0 h h J
so(4) u(1), resulting in four smaller, deco×upled adjoint h −0z hy 0
DI eq⊕uations. These two sets of 4 generators are conju- A(Σ) =2i z − x . (20)
gate of each other under the respective u(1) generator. (2,2)±2 −hy hx 0 0
Therearethreetypesofso(4) u(1)(seeTableII)which −J 0 0 0
⊕
we will discuss in detail below.
and(0)isa1 1matrixcontaining0. Thecorresponding
×
linearcombinationsofso(4)generatorslistedinEq. (16)
are the two independent sets of su(2), Σ± = (11
{ i } { ±
X)X,(11 X)Y,(11 X)Z /2. Transformingto the basis
1. so(4)⊕u(1) {trΣ(Σ′i}′Σ=){/±Σ4+,}⊔{Σ−±}byem}ployingEq. (14)with[S]ij =
i j
1. This firstcaseis also knownas single–qubitcontrol
Ising model, and the corresponding Hamiltonian can be
1 0 0 1 0 0
written as
0 1 0 0 1 0
1 0 0 1 0 0 1
S = (21)
H =JXX+hx11X +hy11Y +hz11Z. (17) 2 1 0 0 1 0 0
−
0 1 0 0 1 0
−
A quick comparison of the set of generators H = 0 0 1 0 0 1
−
11X,11Y,11Z,XX of this Hamiltonian with Table II re-
{ }
vealsspan(H) so(4) u(1)(the firstentryinthe table,
⊂ ⊕ will block diagonalize A(Σ) , yielding
duetothedistinctivecontrolterms11σ∗,alongwithσiσi), (3,1)+(1,3)
with the algebraic groupings given as
0=g˙±+iA±g± (22)
Σi = 11X,11Y,11Z,XX,XY,XZ X11
{ } { }⊔{ }⊔ where
ZX,ZY,ZZ,Y11 YX,YY,YZ, Z11 (18)
{ }⊔{ − }
=IS ⊔Q⊔ID⊔ID¯. 0 hz hy
−
A± = 2i h 0 (h J),
The sets ID and ID¯ representing two D–type invariants − hzy hx J − x0± (23)
obey Eq. (10), and are related to each other by an in- − ±
finitesimal u(1) charge conjugation g± =(g11X ±gXX,g11Y ±gXY,g11Z ±gXZ)T/2.
2. The secondtype ofHamiltonianinso(4) u(1)can
[Q,ID]=−2iID¯, [Q,ID¯]=2iID. (19) be written as ⊕
Exploitingtheu(1)factorofthesubalgebra,itispossible H =JXX+h(1)Y11+h(2)11Z. (24)
toconsideranadditionaltime–dependentXI terminthe
Hamiltonian as well, but it will not affect the discussion
GeneratorsoftheS–type,itsrespectiveu(1)generatoras
that follows. Although the (3,1)+(1,3) so(4) sector is
well as the generators of the D–types are grouped as
nottworeadilydecoupledsu(2)systems,suchasituation
canalwaysbe remediedbymeans ofa time–independent
similarity transformation which we will employ below. {Σi}={XX,Y11,ZX,ZY,11Z,XY}⊔{YZ}⊔ (25)
This is of course not the case for the (2,2) sector. X11,Z11,YY,YX ZZ, XZ,11X, 11Y .
±2 { }⊔{ − − }
In the generator basis Σ the adjoint DI equation is
i
0 = g˙(Σ) +iA(Σ)g(Σ) an{d A}(Σ) has the block diagonal TakingtheadjointA(Σ) =ad(H),weobtainA(Σ)
(3,1)+(1,3)⊕
7
(0) A(Σ) A(Σ) , The generators are grouped as
⊕ (2,2)−2 ⊕ (2,2)2
Σi = XX,YX,Z11,YY,XY,11Z ZZ
{ } { }⊔{ }⊔ (30)
0 0 h(2) 0 0 h(1) YZ,11Y,XZ,11X X11,ZX, Y11, ZY ,
y − z { }⊔{ − − }
0 0 J 0 0 0
− x and A(Σ) is A(Σ) (0) A(Σ) A(Σ) where
A((Σ3,)1)+(1,3) =2i−h0(y2) J0x h(z01) −h0(z1) 00 h0(y2) (3,1)+(10,3)⊕h(1)⊕ 0(2,2)−20⊕ (2h,2()22) 0
−
0 0 0 0 0 Jx h(1) −0 Jx h(2) −0 Jy
h(1) 0 0 h(2) J 0 − −
0z h(y2) 0 y 0 − x A((Σ3,)1)+(1,3) =2i 00 hJ(x2) 00 00 −h(J1y) 00
A(Σ) =2i−h(y2) 0 0 Jx (26) h(2) 0 Jy −h(1) 0 −Jx
(2,2)±2 0 0 0 h(z1) 0 −Jy 0 0 Jx 0
0 −Jx −h(z1) 0 0 0 h(1) −Jy
A(Σ) =2i 0 0 −Jx h(2) . (31)
Similar to the previous case, we note that XX (2,2)±2 h(1) Jx 0 0
{ ± −
ZY,Y11 11Z,ZX XY /2 generates an su(2) algebra Jy h(2) 0 0
whichen±ables us to∓furthe}r block diagonalizeA(3,1)+(1,3) XX YY,YX XY,Z−11 11Z /2 is an su(2) and
{ ∓ ± ± }
into A decomposes into
(3,1)+(1,3)
0 (h(2) h(1)) 0
(2) (1)
0 0 h h − ∓
A± =2i 0 0 y ±J z , (27) A± =2ih(2)∓h(1) 0 Jy±Jx
x
(h(y2) h(z1)) Jx −0 0 −(Jy ±Jx) 0
− ± g± =(gXX ∓gYY,gYX ±gXY,gZ11±g11Z)/2. (32)
with
2. so(5)
g± =(gXX ±gZY,gY11±g11Z,gZX ∓gXY)T/2.(28) We now turn to the most general so(5) case of the
HamiltonianEq. (13),whichalsoisthelastcaseinTable
3. Finally, we consider the Hamiltonian II:
H =J XX+J YY+
x y
(33)
H =JxXX+JyYY +h(1)Z11+h(2)11Z. (29) h(x2)11X +h(y2)11Y +h(z2)11Z+h(z1)Z11
This time, the generators are grouped into two sets per
which is the last instance of the so(4) u(1) listed in Table I:
⊕
Table II. This Hamiltonian can implement a two–qubit
gate because it can represent Josephson junctions when Σ = 11X,11Y,11Z,XX,XY,XZ,YX,YY,YZ,Z11
{ } { }⊔
one of the coupling terms Ji vanishes [52]; also with the ZX,ZY,ZZ,X11,Y11 (34)
addition of the u(1) generator ZZ and the assumption { }
J =J =J ,the HamiltoniancanimplementD model A(Σ) matrix is a direct sum of two blocks A(Σ) A(Σ),
x y z 3 10 ⊕ 5
[53, 54] and quantum dots [55]. where
8
0 h(2) h(2) 0 0 0 0 0 J 0
z y y
−
h(2) 0 h(2) 0 0 J 0 0 0 0
z x x
− −
h(y2) h(x2) 0 0 Jx 0 Jy 0 0 0
− −
0 0 0 0 h(2) h(2) h(1) 0 0 0
− z y − z
A(1Σ0) =i 00 J0x −0Jx hh(z2(y)2) h0(x2) −h0(x2) 00 −h0(z1) h0(z1) J0y ,
− −
0 0 J h(1) 0 0 0 h(2) h(2) J
y z − z y − x
0 0 0 0 h(z1) 0 h(z2) 0 h(x2) 0 (35)
(1) (2) (2) −
Jy 0 0 0 0 hz hy hx 0 0
− −
0 0 0 0 J 0 J 0 0 0
y x
−
0 h(2) h(2) 0 J
z y x
−
h(2) 0 h(2) J 0
z x y
A(5Σ) =i h(y2) h(x2) −0 −0 0 .
−0 Jy 0 0 h(z1)
−
J 0 0 h(1) 0
− x z
This is an example case where a D–type DI greatly normalized eigenvectors and eigenvalues of a simpler DI
simplifies the problem. We further observe that there
aretwoblocksinA(Σ) whichare“coupled”toeachother IS =IS++IS− =2JXX+(2hx−ω)11X+ (38)
5
through J and J . The first block is a Bloch equation 2Bcosωt11Y +2Bsinωt11Z
x y
similar to Eq. (23), whereas the second block is solved
follow as
by
1 0
g2 +g2 =C, g˙2 +g2 =Ch(1)2. (36) 0 1
X11 Y11 X11 X11 z φ± =i(g++ig+) +(g+ g+) ,
This feature can be exploited in construction of pertur- (cid:12)(cid:12) +(cid:11) 2 1 10 3 ± 01
bative solutions in the weak–coupling limit.
λ± = g+;
+ ± (39)
1 0
−
C. Solutions of the Bloch Equation 0 1
φ± =i(g−+ig−) +(g− g−) − ,
We have encountered A so(3) for IS so(4) u(1) (cid:12)(cid:12) −(cid:11) 2 1 10 3 ± 01
∈ ∈ ⊕
in Eqs. (23), (27) and (32) . Such a matrix lives in the
λ± = g−
adjoint representation of su(2), as a result we can find a − ±
two–levelHamiltoniansatisfyingA± =ad(H±),meaning where g± = g± . Using these eigenvectors in Eq. (5),
wehaveeffectivelyreducedfour–levelproblemswithH wecanobtain|the|analytictime–evolutionoperatorofthe
∈
so(4) and so(4) u(1) into a pair of independent two– system. It would be straightforward to design quantum
level problems, a⊕ssociated with the Hamiltonian H± = control of such systems accordingly.
(hx J)σx+hyσy +hzσz for Eq. (23). (see Table III). Finally, the type of Bloch equations that appeared in
±
For the special case of Eq. (23) with hx = const., Eq. (27) and Eq. (32) has several exact solutions [43]
hy = Bcosωt, hz = Bsinωt where B and ω are corresponding to experimentally realizable fields. When
time–independent, the Hamiltonian Eq. (17) becomes the time–dependenttermisperiodicandismuchgreater
the NMR Hamiltonian in rotating–frame, and the prob- thatthe time–independent part(weak–couplinglimit), a
lem has a simple exact solution g± = ( J + hx perturbative solution can be constructed [44].
ω/2,Bcosωt,Bsinωt)T such that we obtain± −
IS± =(±J +[hx−ω/2])([11±X]X)+ (37) IV. DISCUSSION AND OUTLOOK
Bcosωt([11 X]Y)+Bsinωt([11 X]Z).
± ±
In this work we offered a Lie–algebraic classification
WenotethattheHamiltonianandtheinvariantareboth and construction of DIs of four–level systems. Such sys-
periodic with T = 2π/ω and thus can be used to obtain tems covertwo–qubitsystems which arerelevantandes-
cyclic non–adiabatic geometric phases [56, 57]. The un- sential for quantum computation. Applications of DIs
9
0 1 0 0 0 i 0 0 1 0 0 0 0 0 1 0 0 0 i 0
− −
1 0 0 0 i 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
λ1 = λ2 = λ3= − λ4 = λ5=
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 i 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 i
−
λ6 =00 01 10 00 λ7 =00 0i −0i 00 λ8= √1300 10 02 00 λ9 =00 00 00 00 λ10 =00 00 00 00
−
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 i 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
λ11 =00 00 00 10 λ12 =00 00 00 −0i λ13 =00 00 00 01 λ14 =00 00 00 0i λ15 = √1600 10 01 00
−
0 1 0 0 0 i 0 0 0 0 1 0 0 0 i 0 0 0 0 3
−
TABLEIV.λmatrices,obeyingtr(λiλj)=2δij. λ1 3 isansu(2)andλ1 8 isansu(3). λ8 commuteswithλ1 3,λ15 commutes
with λ1 8, giving rise to su(2) u(1) and su(3) u−(1) subalgebras. − −
− ⊕ ⊕
variouscontroloffour–levelsystems. Inanongoingwork
0 0 0 0 0 1
to be reported elsewhere, we try to devise a CNOT gate
ad(σx)=2i 0 0 1 ad(σy)=2i 0 0 0
− asacontrolpassagebasedontheDIs constructedinthis
0 1 0 1 0 0
− paper.
0 1 0
−
ad(σz)=2i 1 0 0
0 0 0
ACKNOWLEDGMENTS
TABLE III. Adjoint representation of su(2), [ad(σi)]jk =
This work is partially supported by ‘Open Research
i(fi)jk where [σi,σj]=ifijkσk.
− Center’ Project for Private Universities: matching fund
subsidyfromtheMinistryofEducation,Culture,Sports,
go beyond the exact solutions of the time–dependent Science and Technology,Japan(MEXT). MN would like
Schr¨odinger equation, such as fast control of four–level to thank partialsupports of Grants–in–Aidfor Scientific
systems beyond adiabatic regimes and non–abelian geo- Research from the JSPS (Grant No. 24320008). UG is
metric phases. We have shown that the span(I)=so(4) supportedbytheMEXTscholarshipforforeignstudents.
and span(I) = so(4) u(1) cases are reduced to the YWthanksLing–YanHungandRobertMannforhelpful
solutions of two–level⊕systems, with analytic solutions discussions.
corresponding to practically realizable four–levelparam-
eters. One such solution was explicitly given. The
span(I)=so(5)wasnotedforitsD–typeDI,whosesolu-
Appendix A: No su(3) S–Type DIs
tioncanbereducedtosolvingtwoweaklycoupledsetsof
differential equations. These DIs can be applied to gain
The fundamental representation of su(4) is 15 Hermi- 2H written in terms of λ is
j
tian set of λ matrices (Tables IV and V). The first 8
i
of these, al{ong}with λ which commutes with the rest h(2)λ +h(2)λ +(h(2)+J )λ +
15 x 1 y 2 z z 3
form the su(3) u(1) subalgebra. They are related to
⊕ 2h(1) h(2)+J
spinor generators as h(1)λ +h(1)λ +(J +J )λ + z − z zλ +
x 4 y 5 x y 6 √3 8
2λ1 =(11+Z)X, 2λ2 =(11+Z)Y, (Jx−Jy)λ9+h(x1)λ11+h(y1)λ12+h(x2)λ13+h(y2)λ14+
2λ3 =(11+Z)Z, 2λ4 =X(11+Z), 2 (h(1)+h(2) J )λ (A2)
2λ5 =Y(11+Z), 2λ6 =XX+YY, √3 z z − z 15
2λ7 =YX −XY, 2√3λ8 =2Z11−11Z +ZZ, We see that in order to get rid of λ9−14 terms, we must
√6λ15 =Z11+11Z ZZ. (A1) have h(x1) = h(y1) = h(x2) = h(y2) = 0 and Jx = Jy. This,
−
10
however,reduces the Hamiltonian to
i j k f i j k f i j k f
ijk ijk ijk
1 2 3 2 1 9 12 1 6 12 13 -1
11 45 76 −11 12 190 1111 −11 77 1112 1134 11 (h(z2)+Jz)λ3+2Jλ6+ 2h(z1)−√h3(z2)+Jzλ8+
2 4 6 1 2 10 12 1 8 9 10 1/√3
2
2 5 7 1 3 9 10 1 8 11 12 1/√3 √3(h(z1)+h(z2)−Jz)λ15, (A3)
3 4 5 1 3 11 12 1 8 13 14 2/√3
− −
3 6 7 1 4 9 14 1 9 10 15 2 2/3
− p
4 5 8 √3 4 10 13 1 11 12 15 2 2/3 where we defined J = Jx = Jy. The coefficients are
− p
6 7 8 √3 5 9 13 1 13 14 15 2 2/3 linearly independent but the set of generators belong to
p su(2) u(1) su(3) u(1),fromwhichweconcludethat
5 10 14 1 6 11 14 1 ⊕ ⊂ ⊕
one cannot emulate and gain the full SU(3) control of a
qutrit using two qubits with a Hamiltonian of the form
TABLE V.Structureconstants for λ matrices. Eq. (13).
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