Table Of ContentA Robust Compressive Quantum State
⋆
Tomography Algorithm Using ADMM
Kezhi Li Shuang Cong
∗ ∗∗
ACCESS Linnaeus Centre, Royal Institute of Technology (KTH),
∗
SE 10044 Stockholm, Sweden (e-mail: [email protected]).
Dept. of Automation, Univ. of Science and Technology of China,
∗∗
Hefei, 230027, China (Corresponding authors email:
4
[email protected])
1
0
2
Abstract: The possible state space dimension increases exponentially with respect to the
n numberofqubits.Thisfeaturemakesthequantumstatetomographyexpensiveandimpractical
a for identifying the state of merely several qubits. The recent developed approach, compressed
J
sensing, gives us an alternative to estimate the quantum state with fewer measurements. It
5 is proved that the estimation then can be converted to a convex optimization problem with
2 quantummechanicsconstraints.Inthis paperwe presentanalternatingaugmentedLagrangian
method for quantum convex optimization problem aiming for recovering pure or near pure
]
T quantum states corrupted by sparse noise given observables and the expectation values of the
measurements.The proposedalgorithmis muchfaster, robustto outlier noises (evenvery large
I
. forsomeentries)andcansolvethereconstructionproblemdistributively.Thesimulationsverify
s
c the superiority of the proposed algorithm and compare it to the conventional least square and
[ compressive quantum tomography using Dantzig method.
1
v Keywords: Quantum state tomography, ADMM, rank minimization, convex optimization and
3 regularization
3
5
1. INTRODUCTION state is a mixed state that can be represented as the
6
probabilistic combination of equal to or less than r pure
.
1 The interests of applying control theory and signal pro- states.Supposetheunknownmixedstateisaprobabilistic
0
cessing techniques to quantum mechanics have increased mixture of r O(1) pure states,then it means its density
4 ∼
dramaticallyin recentdecade.One objective is to develop matrix ρ has rank not largerthan r. This pre-information
1
: aseriesofsystematicmethodsfortheactivemanipulation enables us to reduce the number ofparametersto identify
v and control of quantum systems. The foundation of such a quantum system (Gross et al. [2010]). By using a novel
Xi theory lies in the fact that we are able to prepare and signalprocessingtechniquecalledcompressedsensing(CS)
measure a given quantum state efficiently. It is not trivial (Donoho [2006]) that has been widely investigated in last
r
a since the microscopic quantum systems have their unique a few years, people are able to obtain good estimates
features,onaccountofwhichtheysignificantlydifferfrom of nearly pure quantum states with O(rdlogd) expecta-
the classic world. In practice, people often use the mea- tions and corresponding observables (Gross et al. [2010],
surement data to estimate an unknownquantum state. In Liu [2011]). Thus the required number of identifying a
mathematics,aquantumpurestate ψ canbedescribedas quantum state can be reduced dramatically by solving
averticalvectorwithsizedinHilbe|rtispace.Thisvectoris an optimization problem when d goes to large, and its
calledastatevectorandittheoreticallycontainsstatistical effectiveness has been verified by a series of experiments
information about the quantum system. For the mixed such as in Smith et al. [2013].While in currentliteratures
state that corresponds to a probabilistic mixture of pure of quantum state tomography via CS, this problem can-
states, a state vector is not enough. It usually requires not be efficiently handled by generic optimization solvers
a d d density matrix ρ to depict the quantum state because of the large number of involving variables. For
by g×iving the probabilities in each possible state, which instanceinSmithetal.[2013]theauthorssummarizedthe
implies that O(d2) parameters are needed to describe an estimation to least squares (LS) problem or compressed
arbitrary state in a d dimensional Hilbert space. sensing(CS)problem,andsolvedthembyusingthepreva-
lent convex optimization toolbox. This paper addresses
However in fact, most states people are interested in real
the problem of state recoveryusing low rank information.
life are nearly pure. Here nearly pure means the quantum Our work is inspired by the an alternating augmented
Lagrangian method (ADMM) which has received much
⋆
ThisworkwaspartiallysupportedbytheSwedishResearchCoun- attentions from the optimization community (Boyd and
cil, the Linnaeus Center ACCESS at KTH, the European Research Vandenberghe [2004], Boyd et al. [2011]) though it was
Council under the advanced grant LEARN, contract 267381, and developed in the 1970s. In this paper we reformulate the
the China National Key Basic Research Program under Grant No.
quantumstatetomographytoaoptimizationproblemand
2011CBA00200
design a fast algorithm based on ADMM where in each yi =(A(ρ))i+ei =ctr(O∗iρ)+ei, i=1,··· ,m, or
step trying to optimize the density matrix and project y=Avec(ρ)+e,
it onto the constraint set of quantum states according to (1)
the Karush-Kuhn–Tucker (KKT) conditions, and the al-
where A Cm d2 is the normalized operator whose ith
gorithmfinallyreachesasolutionwithgoodaccuracy.Due ×
row is the∈concatenation of O ’s rows, e Rm represents
to the alternating propertiesof the algorithm,we are able ∗i ∈
thenoiseduetothesystemormeasuringprocess.cissome
to run it distributively, and normally obtain the quantum
normalized constant. If we set E( ) = , c would be
state with most purified result by minimizing the nuclear ∗
A A I
normofthe density matrix,a heuristic for minimizing the d . Conventionally,people use the least squareapproach
√m
rank (Candes and Plan [2011]). People have provedmany to estimate ρ
matrices bases including Pauli matrices satisfy the rank ρˆ=argmin [y ctr(O ρ)]2,
reetstarli.c[t2e0d07is]owmietthryrepgraorpdetrotylo(wRIrPa)nkinrterocodvuecreydsiunchRtehchatt ρ Xi i− ∗i (2)
s.t. ρ =ρ, ρ 0, tr(ρ)=1.
theyarecapabletorecovertheuniquedensitymatrixwith ∗ (cid:23)
sufficientmeasurementsviacompressivesensingapproach. Because the degrees of freedom of ρ is O(d2), normally
This work can be deemed as the reconstruction part of O(d2) measurements are needed to identify the unique
the compressivequantumstatetomography,whichgivesa state.
solutiontoidentifyingthedensitymatricesaccuratelyand
Yet assuming the underlying quantum system is pure or
efficiently for standard tomography as well as continuous
nearly pure, ρ becomes a probabilistic weighted combina-
tomography (Smith et al. [2013]).
tion of equal to or less than r rank-1 matrices derived
This paper is organized as follows. In Section II, we will from a series pure states (see details in Appendix). When
explain the idea of quantum state tomography via com- r is small, people have suggested that O(rdlogd) settings
pressive sensing and the framework of ADMM. In Sec- would possibly suffice instead of d2. Minimizing the rank
tion III the compressive state tomography with quantum ofa matrix belongsto NP-hardproblems,so alternatively
constraints is formulated formally, and the proposed al- people pursuit the solution by minimize the nuclear norm
gveorriiftyhmtheiseffinetcrtoivdeunceesdsaonfdthaenparloypzeodseidnadpeptraoila.cShiminuSlaectitoionns wofhidcehnsisitya mcoantvriexx |f|uρn||c∗tio=n ttrh(a√tρc∗aρn) b=e oPptimi=min1i{zmed,n}eσffii-,
IV, and finally the conclusion is summarized in Section ciently. The nuclear norm has been proved as the best
V. Moreover,some necessary supplementary knowledge is convex approximation of the rank function over the unit
explained in the Appendix. ball(Rechtetal.[2007]),sominimizing ρ isaheuristic
for minimizing the rank (Gross et al.||[2||0∗10]). Thus ρ
Notation: Bold letters are used to denote a vector or a
with low rank can be estimated by compressed sensing
matrix. For vectors, , represent the l ,l norm,
1 2 1 2
respectively.Format|r|ix·|,|A|T|·a|n|dA denotethe transpose approaches such as in Liu [2011]
∗
and Hermitian transpose of A, respectively. denotes Dantzig with quantum constraints:
p
||·||
the Schatten p-normwith A = σ (Ap)1/p , where ρˆ=argmin ρ
σi(A) are the singular va||lue|s|pof (cid:0)AP. iSpiecifically(cid:1), ρ || ||∗
is the nuclear norm and ||·||F represents the Frob||en·i|u|∗s s.t. [yi−ctr(O∗iρ)]2 ≤ǫ,ρ∗ =ρ, ρ(cid:23)0, or
norm. tr() is the operator to calculate the trace. vec(A) Xi (3)
·
represents the vertical vector concatenates A’s columns, LASSO with quantum constraints:
and “mat” is its inverse operator to convert a vector to a 1
matrix.Bra-ketnotations|ψiiareusedtodenotequantum ρˆ=argmρin2||yi−ctr(O∗iρ)||22+µ||ρ||∗
states. A 0 means A is a positive semi-definite matrix.
(cid:23) s.t. ρ∗ =ρ, ρ 0,
(cid:23)
where ǫ,µ are parameters. In this paper we develop a
convex optimization algorithm based on ADMM to solve
2. COMPRESSIVE QUANTUM STATE aboveproblemscorruptedbysparseoutlierswithquantum
TOMOGRAPHY AND ADMM constraints.
2.2 Alternating Direction Method of Multipliers (ADMM)
2.1 QuantumStateTomography ViaCompressive Sensing
ADMM is an optimization method with good robustness
The task of quantum state tomography is to reconstruct
andcansupportdecomposition.Considertheoptimization
the quantum states processed and produced by physical
problem such as
systems.Duetothespecialcharacteristicsofthequantum
mechanics, a d d,d = 2q density matrix ρ, a quantum- minimize f(x)+g(z) s.t. Ax+Bz=c (4)
mechanicalanal×oguetoaphase-spaceprobabilitymeasure, for some variable x,z Rn, where f,g : Rn R are
∈ →
is used to describe a quantum system. Since the degree of convex functions. The augmented Lagrangian of (4) is
thefreedomofρisd d,usuallypeopleneedthenumberof defined
×
measurementsincreasewithexponentialgrowthregarding λ
L (x,z,y)=f(x)+g(z)+yT(Ax+Bz c)+ Ax+Bz c 2.
the state space dimension d in order to identify ρ. If we λ − 2|| − ||2
make the measurements discretely and denote the ob- (5)
servable matrix O , the the expectation of measurements whereλ>0isatunableparameter.Thenthekthiteration
i
y Rm, and measuring operator : d d m, then of ADMM algorithm consists of three steps as follows
i × →
∈ A C
1) xk+1 =argminL (x,zk,yk) // x-minimization andWright[2011],Zhouetal.[2010],KyrillidisandCevher
x λ
[2012] and has many applications in face recognition,
2) zk+1 =argminL (xk+1,zk,yk) // z-minimization
z λ etc. While in most previous papers the authors aimed
3) yk+1 =yk+λ(Axk+1+Bzk+1 c) // dual-update for solving a matrix completion problem however here
− we want to recover the density matrix from observable
(6)
measurements with special constraints on ρ. To involve
the quantum constraints in ADMM, we re-write (9) as
From above steps one can see that if we minimize over x
and z jointly, the approach reduces to the classic method minimize ρ +I (ρ)+ S 1
|| ||∗ C || || (10)
of multipliers. Instead, people split the augmented La- s.t. y Avec(ρ+S) 2 ǫ,
grangianandminimize overxwith z fixed andvice versa. || − ||2 ≤
whereI (ρ)istheindictorfunctiononaconvexset with
The three steps are repeated until convergence. Certain I (ρ) =C 0 for ρ , and I (ρ) = for ρ / C, (ρ)
stopping criteria is made to decide when the algorithm hCere is the Hermit∈ianCp.s.d. sCet satisfy∞ing ρ =∈ρ,Cρ C 0.
achieves a convergence. For instance, the algorithm is ∗ (cid:23)
So we have obtained two sets of variables with separable
iterated until the primal and dual residuals are bounded
objective. The augmented Lagrangiancan be derived as
Axk+Bzk c ε ,
2 pri L (ρ,S,u)=( ρ +I (ρ))+ S
xk xk−1||2+ zk zk−−1||2 ≤εdual, (7) λ1 ′ +u||T(||A∗vec(Cρ)+Av||ec|(|S1) y)
|| − || || − || ≤ ′ − (11)
mwhoerreedεeptariils>an0d,εaducaolm>pl0etaerceontovleerrgaenncceepaanraalmyseist,erpse.oFpoler + λ21||Avec(ρ)+Avec(S)−y||22,
who have interests may refer to Boyd et al. [2011].
where λ is a parameter that can effect the rate of
1
convergenceand the number of iterations required. Or we
3. PROBLEM FORMULATION AND METHOD may combine the linear and quadratic terms in (11) and
it becomes
In this section, we formulate the the problem of robust L (ρ,S,u)=( ρ +I (ρ))+ S
quantum state tomography and derive an efficient opti- λ1 λ|| ||∗ C || ||1 (12)
mization algorithm using ADMM. Here “Robust” means + 1 Avec(ρ)+Avec(S) y+u 2,
2 || − ||2
thealgorithmfitsforthecircumstancetheexistenceofnot
with u=(1/λ )u.
only small random noises, but also sparse outlier noises 1 ′
involved in the density matrix.
3.2 ADMM Steps
3.1 Robust Compressive Quantum State Tomography We carry out the following steps in eachiteration to solve
(10).
During the measuring process of quantum state tomog-
raphy, noises are involved due to the system or mea- Step 1 In the ρ minimization step, we update low rank
surement errors. Normally we assume e satisfying certain ρ with fixed S,u.
distribution (like Gaussian) and it can be minimized with ρk+1 :=argmin ρ +I (ρ)
least square techniques (2), similar in Danzig or LASSO ρ{|| ||∗ C
λ (13)
(3). However there exist abnormal circumstances in the + 1 Avec(ρ)+Avec(Sk) y+uk 2 .
measuringprocessthatmay causethe perturbationinthe 2 || − ||2(cid:27)
density matrix, and it can be reflected by sparse outlier First,weminimizetheunconstrainedquadraticfunctionin
entriesinρandofcoursenotsatisfyingGaussiandistribu- termsofρ.Theanalyticsolutiontoleastsquareestimation
tion.We formulatethese outlierentriesas asparsematrix can be written as
S∈yCid=×d(,Ath(ρen+(S1)))bie+coemi =esc·tr(O∗i(ρ+S))+ei, (8) Secρok1n+d1,=prmoajetc(cid:16)t(ρAk∗+A1)t−o1ρAk+∗1(cid:0)yon−tuokt−heAcvoencs(tSra)(cid:1)in(cid:17)t.s se(1t4)
i=1, ,m. 1 2 C
··· at the same time with low rank, i.e.
In this case the result of least square method in (2) will
ρk+1 =Π (ρk+1), (15)
changesignificantlysometimesduetotheexistenceofout- 2 1
C
where Π denotes the Euclidean projection onto and at
liers.Inaddition,giventheinformationthatρisrelatively
thesameCtimewithlowrank.FortheparticularcoCnstraint
pure which implies it has low rank, the Dantzig/LASSO
set of quantum state, is a proper cone of the Hermitian
solver in (3) with low rank constraints based on trun-
C
p.s.d. matrices. We will show the projection process in
cated Singular Value Decomposition (SVD) might also
Section 3.3 with efficient approach.
fail because the sparse outliers effect the classic principle
component analysis (PCA) dramatically in the process
Step 2 In the S minimization step, we update sparse
of dimensionality reduction. To reduce the influence of
matrix S with fixed ρk+1 =ρk+1,u.
the noise to the rank estimation, we may reformulate the 2
robust Dantzig solver with sparse outliers and quantum Sk+1 :=argmin S
S 1
{|| ||
constraints to
λ
minimize ||ρ||∗+||S||1 (9) + 21||Avec(ρk+1)+Avec(S)−y+uk||22(cid:27).
s.t. ||y−Avec(ρ+S)||22 ≤ǫ,ρ∗ =ρ, ρ(cid:23)0, (16)
where A is with the same definition in (1). The idea of It is a conventionalLASSO problemand can be solvedby
minimizingsparsenoisescanalsobefoundinE.J.Cand´es iterations.Howeverhere we avoidsolvingitby a sequence
ofconvexprogramsandadoptthe shrinkoperatordefined by decomposing ρ into the difference of two orthogonal
previously to calculate a solution efficiently. In detail, the elements: one with nonnegative eigenvalues and one with
least square estimate S can be approximated by negative part. After that the shrink operator leads to a
solution satisfying low rank constraints. In addition, if
Sk1+1 =mat(cid:16)(A∗A)−1A∗(cid:0)y−uk−Avec(ρk+1)(cid:1)(cid:17)(,17) gthiveenprtohbeabinilfiosrtmicaltiinoenarthcaotmtbhienaotbiojenctoivfeleqsusatnhtaunmosrtaeqteuaisl
andthenshrinkthemagnitudetoachieveasparsesolution to r pure states, we may project ρ to the set of r-rank
Sk2+1 =Sτ′(s)=sgn[s]max(|s|−τ′1,0) (18) minaΣtr+iceins b(2y2s)e.lFecotrinthgetdheetmailasxoimftuhmedrepriovsaittiiovne ethigeenrevaadlueerss
where is the shrink operator also explained in Section
3.3, s =S vec(Sk+1), τ is a shrink parameter depends on may refer to (Boyd and Vandenberghe [2004]).
1 ′
the sparsity level of S. Remark:
1)RegardingtheconvergenceofADMManderrorbounds
Step 3 At last we proceed the dual update step of recovering low rank matrix from its measurements the
uk+1 =uk+(y Avec(ρk+1) Avec(Sk+1)). (19) readersmayrefertoBoydandVandenberghe[2004],Boyd
− − et al. [2011],J. Wright and Ma [2013],Lin et al. [2011].If
Thisstepistorecordthe alternativeupdatedirectionand
there is no analytical solution to (6), we may also use the
contribute to the next step.
semidefinite programs. Details and softwere can be found
in Sturm [1999].
Stop Criteria and Parameter Settings The algorithm
followsthesteps1-3tocarryouttheupdatinginformation 2) In practice, the observable O is not necessary the
i
iteratively. In practice, relatively small numbers of itera- tensor product of Pauli matrices. For instance, in Smith
tions, like 30-40,are sufficient to achievea goodaccuracy. et al. [2013]the author developed a device to proceed the
Thereareseveralstoppingcriterions,e.g.adoptingbounds quantum state tomography by continuous measurements
in (7) we have where O is affected by outer radio frequency magnetic
i
y Avec(ρk+Sk) 2 ε y , fields. In this case we can still use the proposed algorithm
|| − ||2 ≤ 1|| ||2 (20) to recover the quantum state, as long as that O satisfy
ρk ρk 1 ε , Sk Sk 1 ε . i
|| − − ||2 ≤ 2 || − − ||2 ≤ 3 the rank RIP and number of measurements are sufficient
where ε1,ε2,ε3 are parameters need to be tuned. Some large. Regarding the details of rank RIP and number of
methods of tuning parameters of alternating direction measurementsofcompressivequantumtomography,please
methodsareindicatedinYuanandYang[2009],E.J.Cand´es refer to the Appendix.
and Wright [2011].
3) If the dataset is large, our algorithm equipped with
ADMM technique can be extended to a distributed man-
3.3 Projection onto Constraint Set with Low Rank
ner as a consensus optimization problem. Assume the N
agentscancommunicatewitheachother,denoteeachcost
We utilize a positive eigenvalue thresholding operator
τ
to calculate ρk2+1. Let Sτ : Rd → Rd denote the shriDnk tfuonction fi(·), i=1,2,··· as in (1), in this case (6) turns
operator such that
Sτ(x)=sgn[x]max(|x|−τ1,0) (21) xki+1 =argminxi(cid:16)fi(xi)+yikT(xi−x¯ki)+λ/2||xi−x¯ki||22(cid:17),
here 1 is a vector with all elements 1. The definition also yk+1 =yk+λ(xk+1 x¯k+1),
can be extended to the matrix form. Then the positive i i i − i (26)
eigenvalue thresholding operator is defined as
Dτ where x¯k = 1/n ni xk represents the average of n
ρk2+1 =Dτ(ρk1+1)=VSτ(Σ+)V∗ (22) neighbouirs of ageniPt ii.=G1enierally speaking, we gather xk
i
where Σ,V are obtained from the eigenvalue decomposi- from outside and scatter x¯k to processors, then update
tion of a symmetrized matrix 1/2(ρk1+1+ρk1+1∗), xi,yi in each processor locally in parallel. See details of
VΣV∗ =1/2(ρk1+1+ρk1+1∗), (23) consensus optimization via ADMM in Boyd et al. [2011].
Σ+ only keep the positive part of the eigenvalues where
Σ+ =max(Σ,0), (Σ+)isashrinkoperatorondiagonal 4. NUMERICAL EXAMPLES
τ
matrix Σ+ whichShas eigenvalues as entries, τ = 1/λ .
1
This approach can be derived from its Karush-Kuhn- In the following we demonstrate the reconstruction per-
Tucker (KKT) conditions of the optimal projection from formance of the proposed algorithm for quantum state
ρk+1 toset withleastsquareerrors.Takingtheindicator tomography.Twoexperimentsarecarriedouttoshowthe
2 C superior of the proposed algorithm. Consider a quantum
function I (ρ) for instance, under mild assumptions on a
proper conCe the KKT conditions of state consisting of q = 5 qubits, its density matrix ρ has
C size d d, d=25. Let the true quantum state as ρ , ρ is
minimize ρ¯ ρ 2 × ∗ ∗
|| − ||2 (24) generatedfromnormalizedWishartrandommatriceswith
s.t. ρ¯ I form as (Zyczkowski et al. [2011])
∈ C
are given by ΨrΨr∗
ρ¯∈IC, ρ¯−ρ=θ, (25) ρ∗ = tr(ΨrΨr∗) (27)
θ ∈IC, θ∗ρ¯=0. where Ψr is a complex d×r matrix with i.i.d. complex
The third term is because positive semidenite cone is random Gaussian entries, the denominator is due to the
self-dual. Then the Euclidean projection can be derived trace 1 constraint. We construct A as a M d2 sampling
×
matrix whose M rows are chosen randomly without re-
d=25, R=2
placement from an d2 d2 matrix whose rows are the set 1
of all vecterized tensor×product of Pauli matrices. Maltab Least Square (LS)
0.9 Dantzig
R2012b version is used to run the numerical simulations ADMM
and each value in figures is recorded after averaging 200 0.8
experiments. 2ρ) 0.7
m(
At first we consider the scenario when the system has nor 0.6
smallrandomnoises.Herewesetei in(1)satisfiesrandom *2ρ)/
Gaussian distribution N(0,0.001||ρ||2). In this case there ρm(− 0.5
are two terms in (10) without S, then the problem is or 0.4
n
simplified tominimize y Avec(ρ) +I (z), Error: 0.3
2
s.t. ρ=z||, − || C (28) 0.2
where represents the low rank Hermitian p.s.d. matrix 0.1
C
set. We may update the quadratic term and I (ρ) itera-
0
tively using ADMM. Specifically, the iteration sCteps are 0.1 0.15 0.2 0.25 0.3 0.35 0.4
η=M/(d2)
ρk+1 =mat argmin y Aρ 2+λ/2 ρ zk+uk 2 ,
ρ || − ||2 || − ||2
zk+1 =Π (ρ(cid:0)k+1+uk)(cid:8), (cid:9)(cid:1) Fig. 1. The comparison of reconstruction performances
C of different algorithms, including the least square
uk+1 =uk+(ρk+1 zk+1).
− method in (2) using cvx toolbox, compressive quan-
(29)
tumtomographysolvingDantzigin(3)usingcvxtool-
The ρ updating step can be completed by calculating its box, and compressive quantum tomography solving
analytic solution, (28) using ADMM.
ρ=(A∗A+λI)−1 A∗y+λ(zk uk) . (30)
−
Theprojectionprocessinste(cid:0)p2followstheexp(cid:1)lanationin
Section3.3thatexploitstheshrinkortruncatedeigenvalue d=25, R=2
decomposition as in (22) (23). In addition, we set rank 1
r = 2 in the generation of true state ρ in (27), λ = 1
0.9
in(29).Thereconstructionperformancesareevaluatedby
the error defined as 0.8
error= ||ρ|∗|ρ−∗||ρˆ22||22, (31) 2ρnorm() 00..67
where ρ∗ and ρˆ denote the true state and the estimate *2ρ)/
state, respectively. The error is calculated verses the in- ρ− 0.5
creasing measurement rate η = M/d2. Fig.1 depicts the orm( 0.4
n
rceacnonosbtsreurcvteiotnhaetrrgoivrsenwtihthe lionwcreraansiknginηfo.rmFraotmionFaigs.1proionrei Error: 0.3
knowledge, the number of measurements is dramatically 0.2
Least Square (LS)
reduced. Specifically the Dantzig using cvx performs bet-
Dantzig
0.1
ter than the least square approach, and our simplified ADMM
algorithm using ADMM has smaller errors comparing to 0
0.1 0.15 0.2 0.25 0.3 0.35 0.4
Dantzig given the same number of measurements M. η=M/(d2)
In the second simulation we add the outlier noises in the
density matrix. We set the measurements y = (ρ + Fig. 2. The comparison of reconstruction performances
S) + e where S d d has (0.01d2) nonzero Avalues with sparse outlier noises, including the least square
located randomly∈with×magnitudes satisfying Gaussian using cvx toolbox, Dantzig using cvx toolbox, and
distribution (0,0.1 ρ ). If the error is larger than 1, compressive quantum tomography solving (28) using
2
we record it aNs 1. Th|e|r|e|sults are demonstrated in Fig. 2. ADMM.
Fig. 2 shows that the Dantzig approach fails under such
scenariosincethenuclearnormminimizationisinfluenced
significantly by the large outliers, however the proposed forrecoveringpureornearlypurestatewithsparseoutlier
method may overcome this shortage and still lead to a noises. The algorithm updates the density matrix and
recovery much better than that of LS method. the sparse estimate noises iteratively and finally obtain
a reconstruction result efficiently. Simulations show that
5. CONCLUSION the proposed algorithm achieves better recovery accuracy
comparing to the conventional least square and compres-
After reviewing several existing algorithms of compres- sive Dantzig method with the same number of measure-
sive quantum state tomography and ADMM method, ments. For the case of existence of sparse outlier noises,
this paper proposesanalternating augmentedLagrangian the proposed algorithm beats the Dantzig method due to
method for quantumconvexoptimizationproblemaiming the fact that the influence of outliers has been reduced.
6. APPENDIX Gross, D. (2011). Recovering low-rank matrices from
few coefficients in any basis. 57(3), 1548–1566. doi:
Proposition 1. When the quantum state consisting of q 10.1109/TIT.2011.2104999.
qubits is the probabilistic combination of r pure states, Gross, D., Liu, Y., Flammia, S.T., Becker, S., and Eisert,
then its density matrix ρ with size d d has rank not J. (2010). Quantum state tomography via compressed
larger than r, d=2q. × sensing. Phys. Rev. Lett., 105(15), 150401150404.
J. Wright, A. Ganesh, K.M. and Ma, Y. (2013). Com-
Proof: The proof is simple however it seldom appears pressive principal component pursuit. A Journal of the
in literatures and lays the foundation of the compressive IMA, (2), 3268.
quantum tomography, so we give a proof here. Suppose Kyrillidis, A. and Cevher, V. (2012). Matrix alps: Ac-
the quantum state under consideration celerated low rank and sparse matrix reconstruction.
r In IEEE Statistical Signal Processing Workshop (SSP)
ρˆ= p ψ ψ , (32) 2012, 185–188.
i i i
| ih |
Xi=1 Lin,Z.,Liu,R.,,andSu,Z.(2011). Linearizedalternating
which means the quantum system may be found in state direction method with adaptive penalty for low rank
ψ with probability p , i =1,2, ,r. If we concatenate representation. NIPS.
i i
|theicolumnvectors ψ asamatri·x·,·then(32)isequivalent Liu, Y. (2011). Universal low-rank matrix recovery from
i
to | i pauli measurements. NIPS, 1638–1646.
ρˆ=Ψr Ψr∗,where Recht, B., Fazel, M., and Parillo, P. (2007). Guaranteed
· (33) minimum rank solution of matrix equations via nuclear
Ψr =[√p1 ψ1 ,√p2 ψ2 , ,√pr ψr ],
| i | i ··· | i norm minimization. SIAM Rev., 52, 471501.
Ψr is of size d r. Thus the density matrix ρˆhas rank at Smith, A., Riofro, C., Anderson, B., Martinez, H.,
×
most r due to the rank property of multiplication of two
Deutsch, I., and Jessen, P. (2013). Quantum state to-
matrices.
mography by continuous measurement and compressed
Definition 1. (Rank RIP). Rechtetal.[2007],Liu[2011] sensing. Phys. Rev. A, 030102(87).
The satisfies the rank restricted isometry property Sturm, J.F. (1999). Using sedumi 1.02, a matlab toolbox
A
(RIP) if for all d d X, we have foroptimizationoversymmetriccones. Optim. Methods
×
(1 δ) X (X) (1+δ) X (34) Softw., 11-12, 625653.
F 2 F
− || || ≤||A || ≤ || || Yuan,X.M.andYang,J.(2009). Sparseandlow-rankma-
where some constant 0<δ <1.
trix decomposition via alternating direction methods.
Proposition 2. When we formulate the measurement pro- Pacific Journal of Optimization.
cess as equations in (1), and the observable Oi are the Zhou, Z., Li, X., Wright, J., Candes, E., and Ma,
tensor/Kronecker product of a series of complex and uni- Y. (2010). Stable principal component pursuit.
tary elemental 2 2 Pauli matrices Pi chosen from the In IEEE International Symposium on Information
×
four possibilities randomly, Theory Proceedings (ISIT), 2010, 1518–1522. doi:
1 0 0 1 10.1109/ISIT.2010.5513535.
I2 =(cid:18)0 1(cid:19),σx =(cid:18)1 0(cid:19), Zyczkowski,K., Penson,K.A., Nechita,I., and Collins, B.
(35) (2011). Generating random density matrices. J. Math.
0 i 1 0
σy =(cid:18)i −0 (cid:19),σz =(cid:18)0 1(cid:19). Phys, 52, 062201.
−
Then sampling operator satisfies the rank RIP and we
A
are able to recover the rank-r density matrix ρ by using
numberofmeasurementsm c rdlog6dforsomeabsolute
≤ ·
constant c with high probability.
Proof: The details of the proof can be found in Gross
[2011], Liu [2011].
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