Quantum state reconstruction with imperfect rotations on an inhomogeneously broadened ensemble of qubits

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by  Karl Tordrup

Quantum state reconstruction with imperfect rotations on an inhomogeneously
broadened ensemble of qubits
Karl Tordrup and Klaus Mølmer
Lundbeck Foundation Theoretical Center for Quantum System Research,
Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark
(Dated: February 1, 2008)
Wepresentamethodforperformingquantumstatereconstructiononqubitsandqubitregistersin
thepresenceofdecoherenceandinhomogeneousbroadening. Themethodassumesonlyrudimentary
single qubit rotations as well as knowledge of decoherence and loss mechanisms. Weshow that full
statereconstructionispossibleeveninthecasewheresinglequbitrotationsmayonlybeperformed
imperfectly. Furthermoreweshowthatforensemblequantumcomputingproposals, quantumstate
reconstruction is possible even if the ensemble experiences inhomogeneous broadening and if only
7 imperfect qubit manipulations are available during state preparation and reconstruction.
0
0 PACSnumbers: 03.65.Wj,03.67.Lx
2

n
a I. INTRODUCTION ing quantum state reconstruction.
J In quantum computing a lot of effort is put into per-

9 Thetaskofreconstructingaphysicalstatefromasetof formingqubitrotations(orgateoperations)withaslittle
1 measurement data is central to all branches of physics. error as possible. Naturally the same sources of errors
Since full knowledge of the quantum state allows only arepresentduringthereconstructiveprocedure,however
1 the prediction of a statistical distribution of outcomes it is inappropriate that these errors should detract from
v
onemustperformnumerousmeasurementsonidentically the gate fidelity. We therefore propose that decoherence
0
4 prepared quantum systems in order to reconstruct the and loss mechanisms be included in the theoretical ac-
1 state. The abstract notion of the quantum mechanical count for the reconstruction. A further complication oc-
1 ensemble thus plays a very real part in the business of curs in ensemble quantum computing systems where a
0 quantum state reconstruction. qubit consists of an entire ensemble of two–levelsystems
7 In this paper we shall consider the case of quantum withalmostidenticalphysicalcharacteristics. Qubitma-
0 state reconstruction within the framework of quantum nipulations in such schemes must be robust to the nat-
/
h computing. Since the ultimate goal of quantum com- ural variation of system parameters [3, 4, 5], but will
p puting is to coherently steer the evolution of a quantum inevitably lead to small differences in the evolution felt
t- system towards a specific state wich encodes the answer by each member of the ensemble. During readout of
n tosomecomputationalproblem,thefinalstateshouldbe such systems one obtains in fact the ensemble average
a the single state that encodes the solution with unit (or of the measured quantity and the reconstructive proce-
u
q very nearly unit) probability. Presently quantum state dure should take this averaging into account.
: reconstruction is an invaluable diagnostic tool for deter- Thepaperisarrangedasfollows. InSec.IIweanalyze
v mining the fidelity with which basic quantum gates may the case of single qubit state reconstruction. In Sec. III
i
X be performed. weextendtheresultsofSec.IItoincludetwo-qubitregis-
r Amajorchallengeinallquantumcomputingsystemsis ters. InSec. IV we presentanapplicationof our method
a that of controlling the interactions between qubits. This to a specific physical quantum computing implementa-
problem is often tackled by introducing an auxiliary de- tion. Sec.V discussesthe case withcorrelationsbetween
gree of freedom. Some examples are ion trap quantum the unknown quantum state and reconstructive process
computing where qubits arecoupled via collective vibra- and Sec. VI concludes the paper.
tional modes of the trapped ions [1] and quantum com-
puting with rare earth ions doped in inorganic crystals
wherequbit–qubitinteractionismediatedbyachangein II. SINGLE QUBIT STATE RECONSTRUCTION
static dipole moment when a particular ion is promoted
toanelectronicallyexcitedstate[2]. Thusalthoughqubit Aconceptuallysimplewayofreconstructingthequan-
dynamics are strictly speaking restricted to the qubit tum state of a two–levelsystem is to apply a set of rota-
Hilbert space, many systems employ a larger space for tions D andto measure the expectation value of suit-
i
{ }
the dynamics with the restriction that at the end of a able observables in the rotated states. From the mea-
gate operationthe effective evolutionis described within sured data and knowledge of the rotations performed
the qubit space. This means that the population of the one can infer the full density operator of the unrotated
auxiliary level and all coherences between the auxiliary state. For instance, in [6] coherent emission of radiation
level and the qubit levels should be sufficiently small at from rare-earth ions doped in crystals provides informa-
theendofagateoperationanditmakessensetorestrict tion about quantum coherences and is thus applied to
oneself to the qubit two–level subsystem when perform- reconstruct various quantum states of the ions. For two
2
qubits the same procedure may be employed bearing in We cantheninferhowanyinitialstatetransformsunder
mind that the two qubits must be rotated differently in the rotation D ,
i
order to distinguish contributions from the singlet and
triplet states. The case where both qubits are subject to 3
the same rotation is explored by Home et al. in Ref. [7]. ρR =DiρDi† = Di( cjσj)Di†
In this case full reconstruction of the quantum state is Xj=1
not possible and it is necessary to resort to a numerical 3
search. = cj(DiσjDi†)
We begin by recalling that the density matrix of any Xj=1
two–level system may be expanded on the set σi 3 3
{ }
= c ( b σ )
j jk k
3
ρ= 1 c σ . (1) Xj=1 Xk=1
2 j j 3 3
Xj=0
= c b σ , (6)
j jk k
with σ1−3 the Pauli matrices and σ0 the identity. The Xj=1Xk=1
condition Tr(ρ) = 1 is enforced by setting c = 1 and
0
the further condition that ρ be positive leaves three real where the bjk follow from Eqs. (5).
parameters fulfilling Weshallfinditconvenienttorepresentthestatesimply
by
3
c2i ≤1. (2) c1
Xi=1 vρ =c2 (7)
c
3
 
A. Qubit rotations
From Eq. (6) we may construct the superoperator
In order to proceed with the reconstruction we must b b b
11 21 31
now perform a set of rotations on identically prepared B =b12 b22 b32, (8)
copies of the initial state. When dealing with a sin- b b b
13 23 33
gle qubit the generic Hamiltonian in the standard basis  
0 , 1 is given by such that the rotated state may be found as
{| i | i}
H = 21σ·nˆΩ, (3) vρR =Bvρ. (9)
where σ is the vector of Pauli matrices and nˆ is a unit
Let us consider measurements which reveal the popula-
vector. WhenactingforatimetthisHamitonianeffectsa
tion of the 0 and 1 levels of the rotated state, i.e.
rotationofthe BlochvectorthroughΩtaboutthe vector | i | i
nˆ. For a perfect rotation of the Bloch vector through P = 1 +cR (10a)
θ about an equatorial axis nˆ = (cos(ϕ ),sin(ϕ ),0), the 0 2 3
i i i
rotation operator is given by P1 =1−P0 = 21 −cR3, (10b)
cos(θ /2) ie−iϕisin(θ /2) or equivalently using Eqs. (7)–(9)
Di =D(ϕi,θi)=(cid:18) ieiϕisini(θi/2) − cos(θi/2)i (cid:19).
− (4) b b b c1 P 1
DtoukentoowthheolwinetahreitPyaouflqiumaanttruimcesmtercahnasnfoicrsmituinsdseurffitchieenset (cid:18)−b1313 −b2233 −b3333(cid:19)cc23=(cid:18)P10−− 221(cid:19). (11)
 
rotations.
For the special case of Eq. (4) we obtain the explicit
Dσ1D† = cos2(θ/2)+sin2(θ/2)cos(2ϕ) σ1 equations
{ }
+sin2(θ/2)sin(2ϕ)σ
2
c
sin(θ)sin(ϕ)σ (5a) sinθcosϕ sinθcosϕ cosθ 1 P 1
− 3 (cid:18)−sinθcosϕ −sinθcosϕ −cosθ(cid:19)cc23=(cid:18)P10−− 221(cid:19).
Dσ2D† = sin2(θ/2)sin(2ϕ)σ1   (12)
+ cos2(θ/2) sin2(θ/2)cos(2ϕ) σ It is evident that the two equations are linearly depen-
2
{ − }
dentandthusonemustchooseatleastthreesetsofrota-
+sin(θ)cos(ϕ)σ (5b)
3
tionangles(θ ,ϕ )inordertoobtainacoefficientmatrix
i i
of rank 3. The coefficients c are then found by sim-
i
Dσ3D† = sin(θ)sin(ϕ)σ1 sin(θ)cos(ϕ)σ2 ply invertingthe coefficientmatrixofthe resultingsetof
−
cos(θ)σ . (5c) equations.
3
3
B. Relaxation effects
Wenowconsiderthecomplicationsofdecoherenceand
couplingtoauxiliarydegreesoffreedom. Theproblemof P
quantum state reconstruction in damped systems is well
studied, for instance a very generaltreatmentis givenin
[8]. Here we shall assume that the decoherence mecha- δ
nisms and the coupling to auxiliary degrees of freedom
FIG. 1: The probability that a member of the ensemble
arewellunderstoodandmaythusbemodelledaccurately
experiencesthe errorparametervalue δ is givenby P(δ).
whendescribingthefullsystemdynamicswhichmaytake
place in a larger space than the qubit space. The proce-
durenowfollowsthesamelinesaspreviously. Wechoose
member of the ensemble reacts slightly differently to the
a set of rotations (θ,φ) and identify a set of parame-
{ i} externalcontrolsandthatthesedifferencesmaybequan-
ters describing decoherence effects. Taking these effects
tified by some set of parameters δ. For example δ could
into account we simulate the effect of each rotation on
signifyavariationinthelevelstructurebroughtonbythe
each of the basis states
microscopic environment around each ensemble member
3 oravariationintheexternalcontrolfieldoverthespatial
σR = b σ , (13)
j jk k extentoftheensemble. Whenperformingthequbitrota-
kX=0 tionseachmemberoftheensembleexperiencesaslightly
bynumericalintegrationoftheLindbladmasterequation different evolution from that of its neighbours so that
with probability P(δ) an ensemble member undergoes
ρ˙ =i[ρ,H]+ (ρ). (14)
relax the evolution
L
The function Lrelax given by [9] ρ ρR(δ)=D(θ,ϕ,δ)ρD†(θ,ϕ,δ). (18)
→
(ρ)= 1 C† C ρ+ρC† C + C ρC† ,
Lrelax −2 { m m m m} m m The final state is given by the ensemble average
Xm Xm
(15)
ρR = ρR(δ)P(δ)dδ. (19)
takes the relevant relaxation mechanisms into account. Z
For instance spontaneous transitions from an auxiliary
Itisavalidpointthattheeffectsofδ wouldalsobefelt
state aux to the qubit state 0 with decay rate Γ is
| i | i during the preparationof the state ρ. Means exist, how-
modelled by a term with
ever, to eliminate the effects of error Hamiltonians that
C =√Γ0 aux. (16) are constant in time and thus to compensate for varia-
m
| ih |
tions in δ [4]. In this section we shall thus assume that
Notice in Eq. (13) that the sum now runs from j = 0
the ensemble represented by ρ is uncorrelated with the
since some population may be lost to the auxiliary state
valueofδ. Inordertoproceedwiththestatereconstruc-
forcing us to relax the normalisation constraint and de-
tion we choose a set of N sample values of the variable
scribe the state as a four component vector. From the
δ as illustrated in Fig. 1 and assign to each a weight p
i
calculatedb andthemeasuredpopulationsP forasin-
jk i proportional to the probability P(δ) such that
gle set of rotation angles (θ,ϕ) the equation for the c is
i
given by N
p =1. (20)
c i
0 Xi=1
b +b b +b b +b b +b c P
00 03 10 13 20 23 30 33  1= 0 .
(cid:18)b00 b03 b10 b13 b20 b23 b30 b33(cid:19) c2 (cid:18)P1(cid:19) We then simulate the effects of a rotation D(θ,ϕ,δ) on
− − − − c  each basis state for each sample point
 3
(17)
N 3
SincewearenowforcedtorelaxtheconstraintTr(ρ)=1
σR = p b σ . (21)
because population may be lost to other levels the two j i jk k
Xi=1kX=0
equationsofEq.(17)areingenerallinearlyindependent.
The minimum set of anglesneeded to reconstructsuch a Again choosing two sets of angles (θ ,ϕ ) gives a linear
i i
state is two. systemofequationsanalogoustoEq.(17)forthec which
i
may be inverted to obtain the initial state v .
ρ
C. Ensemble quantum systems
III. TWO QUBIT STATE RECONSTRUCTION
As a final level of complication we now consider the
problem of state reconstruction in the context of ensem- In the previous section we assertedthat a single qubit
ble quantumcomputing. Explicitly we assume that each state can be expanded on the Pauli matrices. The state
4
of a two qubit system may therefore be expandedon the |e(cid:1)
∆
tensor product space n
o
3 3 Ω1 Ω0 |0(cid:1) Absorpti
ρ= cijσi σj. (22) δ |1(cid:1)
Xi=0Xj=0 ⊗ Frequency
FIG. 2: Left: simplified level diagram of a single rare
Thematrixrepresentationofσ σ isconstructedasthe
i⊗ j earth ion. Right: qubits are prepared as peaks in a hole
Kronecker product of the two matrices σ and σ . The
i j burnt structure.
state vector v now becomes a 16 component vector of
ρ
the c . By applying rotations to both qubits we obtain
ij
the rotated two–qubit state
IV. APPLICATION TO QUANTUM
COMPUTING WITH RARE EARTH IONS
ρ = (D D )ρ(D† D†)
R 1⊗ 2 1⊗ 2 DOPED IN CRYSTALS
3 3
= (D D ) c σ σ (D† D†)
1⊗ 2 ij i⊗ j 1⊗ 2 As an example we shall now apply the methods de-
Xi=0Xj=0 veloped in the previous sections to the case of quantum
3 3 state reconstruction in the context of quantum comput-
= c (D σ D†) (D σ D†)
ij 1 i 1 ⊗ 2 j 2 ing with rare earth ions doped in inorganic crystals. We
Xi=0Xj=0 shallbase ourstudy ona rareearthquantumcomputing
3 3 3 3 proposalfirstdevelopedbyOhlssonetal. [2]. Inthispro-
= cij( bikσk) ( bjk′σk′) posal qubit states are encoded in hyperfine levels of the
⊗
Xi=0Xj=0 Xk=0 kX′=0 rareearthionsembeddedinthecrystal. Thequbitlevels
3 3 3 3 are coupled via an electronically excited state accessi-
= cijbikbjk′σk σk′. (23) ble via optical fields. Due to the large inhomogeneous
⊗
Xi=0Xj=0kX=0kX′=0 broadeningofthe excitedstatequbits maybe definedby
isolatedpeaks in frequency space as seen in Fig. 2. Each
As mentioned previouslyitis importantthat D andD qubit thus consists of an ensemble of ions centered at
1 2
implement different rotations in order to allow full re- a given transition frequency. Qubit–qubit interaction is
construction of the state. As in the single qubit case we mediatedbythedipole–dipolecouplingbetweendifferent
obtainanexpressionforthepopulationofthebasisstate ions. In rare earth quantum computing arbitrary single
nm qubit rotations as described by Eq. (4) may be imple-
| i
mented using two fields Ω and Ω simultaneously [10].
0 1
3 3 Theimplementationofsinglequbitrotationsis madero-
Pnm = sgn(n,m,k,k′)cijbikbjk′, bust to variations in the excited state shift ∆, within
Xi=0Xj=0k=X{0,3}k′=X{0,3} the width of the qubit channel (see Fig. 2), by sweeping
(24) the frequency of the laser across the width of the qubit
where sgn(n,m,k,k′) represents the sign of the entry in frequency space. Typical parameters for the system
(kn′,=m)0,i3n.tEheq.te(n24so)rispraondaulocgtoσuks⊗toσEk′q.w(it1h7)ka=nd0t,h3earned- RareeceΩn0tly∼wΩe1h∼av1e sMhoHwz,n∆tha∼t1e0rr0orksHdzuaentdoδth∼e i1n0hokmHoz-.
sults of the previous section now extend naturally to the geneous broadening δ of the hyperfine splitting between
two qubit case. We include the effects of decoherence the qubit levels are in fact dynamically suppressed [11]
and when working with an ensemble quantum comput- during single qubit rotations, but they cannot be made
ing system we include effects of inhomogeneity over the vanishingly small.
ensemble in δ choosingagaina setof sample points from Inthefollowingwesimulatereconstructionofthequbit
the ensemble probability distribution as shown in Fig. state ψ = 1/√2(0 + 1 ) using three sets of rota-
| i | i | i
1. For each set of angles (ϕ1,θ1,ϕ2,θ2) we obtain four tion angles. The inhomogeneous profile is modelled as
equationsin thecij. Inorderto obtainaninvertiblesys- a Lorentzian with variable width and we evaluate the
tem of rank 16 we must thus choose at least four sets fidelity of the reconstruction as
of angles (ϕ ,θ ,ϕ ,θ ). This approach assumes perfect
1 1 2 2
measurement statistics which requires an infinite num- F = ψ ρ˜ψ , (25)
h | | i
ber of measurements in the laboratory and numerical
precision on the order of the machine precision during where ρ˜is the reconstructed state. In order to evaluate
the integration of the master equation. In practice we the usefulness of our method we first perform the recon-
choose at least five and preferably more sets of angles structiontakingtheensembleaveragingintoaccountand
and perform a least squares fit to the obtained data. A compare this with the reconstructionassuming δ =0 for
thoroughdescriptionofquantumstatereconstructionby allensemblemembers. Inthe lattercasethe linearequa-
least squares inversion is given in [8]. tion system of Eq. (17) is not exactly soluble so a least
5
1 1
0.98
0.99
F F
0.96
0.98
0.94
0 5 10 0 5 10
Width [kHz] Width [kHz]
FIG. 3: (Color online) Fidelity of the quantum state FIG. 4: (Color online) Fidelity of the quantum state
reconstruction of a single qubit state as a function of reconstruction of a two qubit state as a function of the
the width of the inhomogeneous profile. The dashed width of the inhomogeneous profile. The dashed line is
line is found by averaging correctly over the ensemble found by averagingcorrectly over the ensemble while
while the full line is found by ignoring variations over the full line is found by ignoring variations over the
the ensemble. ensemble.
squares fit is performed. In Fig. 3 we plot the fidelity
as a function of the width of the inhomogeneous profile.
As expected the fidelity drops when the width of the
Lorentzian increases if we do not properly include the δ
dependent contributions to the final state. We have also
simulated the reconstruction of the fully entangled two
qubit state ψ =1/2(00 + 01 + 10 11 ) using six
| i | i | i | i−| i
sets ofangles. In Fig.4 we plot the fidelity as a function (a) (b)
of the width of the inhomogeneous profile. In this case
the drop in fidelity as the width of the Lorentzian in- FIG. 5: (Color online) (a): Applying a gate operation
creasesis significantif ensemble averagingis disregarded to a pure state qubit causes ensemble members to
and failure to take this into account during the recon- diverge on the Bloch sphere. Only a few representative
structive procedure may lead to overly pessimistic esti- paths corresponding to the dashed lines in Fig. 1 are
mates of the fidelity of the operation producing the en- shown. (b): In the subsequent reconstructive rotations
tangled state. the paths followed by different ensemble members are
correlated with the paths followed during the gate
rotation.
V. CORRELATED PERTURBATIONS IN
PREPARATION AND RECONSTRUCTION
STAGES tification of the state v . Intuitively it should be clear
ρ
fromFig.5thatasimilarproblemoccursnomatterwhich
Considerasinglequbitensembleinthepurespindown aspectoftheHamiltonianofEq.(3)isaffectedbyδ. Any
state v(i) = (0,0, 1). Performing a gate operation cor- aspectoftheexternalcontrolthatvarieswithδwillcause
ρ
− different ensemble members to follow different paths.
responds to rotating the Bloch vector as illustrated in
Fig. 5a. Each member of the ensemble follows a distinct Inthe precedingsectionsweassumedthatthe stateto
path along the Bloch sphere and the final state v is the bereconstructeddidnotpossesanycorrelationsbetween
ρ
mixed state that results from taking the statistical aver- the quantum state and δ. Whether dealing with an en-
ageofallstatesalongtheupperblueline. Ifwenowwish semble quantum computing system where a single qubit
toreconstructv wemustperformanumberofrotations is encoded in an ensemble of physical two-level systems
ρ
as exemplified in Fig. 5b, but since each ensemble mem- or ”single instance” systems where a single qubit is en-
ber experiences the same error during the tomographic coded in a single physical two–level system, this case is
rotation as during the initial rotation, the path followed welldescribedby the quantumoperationsformalism[12]
on the Blochsphere by eachmember is correlatedto the
path followedduring the initial rotation. It is important ρ E ρE†. (26)
toverifythatthesecorrelationsdonotprecludetheiden- →Xk k k
6
The E are linear operators on the qubit Hilbert space obtain the ensemble average
k
fulfilling the relation
EkEk† =1. (27) vρR = Z P(δ)vρR(δ)dδ
Xk = B(0)v(0)+δ(B(0)ρ(1)+B(1)v(0))
ρ ρ
Intheensemblecasestudiedabovewhereensemblemem- +δ2(B(1)v(1)+B(2)v(0)+B(0)v(2)). (32)
ρ ρ ρ
bersundergowithprobabilityP(δ)theunitaryevolution
U we have simply
δ If the distribution P(δ) is symmetric around the origin
the first order term averages to zero. However from Eq.
E = P(δ)U . (28)
k δ (32) it is evident that v may be reconstructed to sec-
ρ
p
ond order simply by expanding the rotation operator to
This isthe casedepictedgraphicallyinFig.5a. We shall
the same order. Similarly the state can in fact be found
nowconsiderthecasedepictedinFig.5bwheretheevolu-
to any order as long as the rotation operator is known
tionofeachensemblememberiscorrelatedtothedistinct
to the same order. In order to determine the individual
evolutionexperiencedby thatmemberduringthe prepa-
terms in the expansionof the state givenin Eq.(29) one
rationofthe state. Thiscaseis not containedwithinthe
must employ successively more sets of rotation angles.
quantum operations formalism and so it is not a priori
If, however we are only interested in the measured en-
clearthatstatereconstructionispossibleinthiscase. We
semble average the discussion of Sec. III still holds. In
shall now prove that full state reconstruction is in fact
this case the expansion of the state is contained implic-
possible. We can assume that at some time in the past
itly in the averaging over δ and four sets of angles is the
the ensemble was in a state possessing no correlations
minimum to reconstruct the state. In a single instance
withδ. Thecumulativeeffectsofallsubsequentmanipu-
quantumcomputing proposalthe aboveargumentsshow
lations including the gate operation used to prepare the
thatquantumstatereconstructioncanbeperformedeven
stateisthatwithprobabilityP(δ)agivenensemblemem-
in systems with intrinsic systematic errors quantified by
ber evolves into v (δ). We proceed by expanding v (δ)
ρ ρ δ.
in powers of the inhomogeneity parameter δ
v (δ)=v(0)+δv(1)+δ2v(2)+ (δ3). (29)
ρ ρ ρ ρ O
VI. CONCLUSION
We also expand the superoperator B describing the to-
mographic rotation in powers of δ
In conclusion we have described a simple and effec-
B(δ)=B(0)+δB(1)+δ2B(2)+ (δ3). (30) tive method of performing quantumstate reconstruction
O in the context of imperfect quantum computing. The
We may think of each ensemble member as being finally methodcanbeadaptedtoincludedecoherenceeffectsin-
rotated with probability P(δ) into the state herent in a given specific quantum computing proposal.
Moreover we have shown that full quantum state recon-
vR(δ) = (B(0)+δB(1)+δ2B(2))(v(0)+δv(1)+δ2v(2)) struction is possible even in systems that interact with
ρ ρ ρ ρ
environmental degrees of freedom that were also impor-
= B(0)v(0)+δ(B(0)v(1)+B(1)v(0))
ρ ρ ρ tantinthepreparationofthestate. Thisisagoodexam-
+δ2(B(1)v(1)+B(2)v(0)+B(0)v(2)), (31) ple of a process which is not contained within the quan-
ρ ρ ρ
tum operations formalism, and we note that such a sce-
where we have kept only terms up to (δ2). When per- nario was already mentioned with a rather contrivedyet
O
formingameasurementontherotatedstatewetherefore illuminating example in [13].
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