Guaranteed recovery of quantum processes from few measurements

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by  Martin Kliesch

Guaranteedrecoveryofquantumprocessesfromfewmeasurements
M. Kliesch,1 R. Kueng,2 J. Eisert,3 and D. Gross2,4
1Institute of Theoretical Physics and Astrophysics, University of Gdan´sk, Poland
2Institute for Theoretical Physics, University of Cologne, Germany
3Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany
4CentreforEngineeredQuantumSystems, SchoolofPhysics, TheUniversityofSydney, Australia
Quantumprocesstomographyisthetaskofreconstructingunknownquantumchannelsfrommeasureddata.
In this work, we introduce compressed sensing-based methods that facilitate the reconstruction of quantum
channelsoflowKrausrank.Ourmaincontributionistheanalysisofanaturalmeasurementmodelforthistask:
Weassumethatdataisobtainedbysendingpurestatesintothechannelandmeasuringexpectationvalueson
theoutput. Neitherancillasystemsnorcoherentoperationsacrossmultiplechannelusesarerequired. Most
previousresultsoncompressedprocessreconstructionreducetheproblemtoquantumstatetomographyonthe
channel’s Choi matrix. While this ansatz yields recovery guarantees from an essentially minimal number of
7
measurements,physicalimplementationsofsuchschemeswouldtypicallyinvolveancillasystems. Apriori,it
1
0 isunclearwhetherameasurementmodeltailoreddirectlytoquantumprocesstomographymightrequiremore
2 measurements.Weestablishthatthisisnotthecase.
Technically,weproverecoveryguaranteesforthreedifferentreconstructionalgorithms. Thereconstructions
n
arebasedonatrace,diamond,and(cid:96) -normminimization,respectively.Ourrecoveryguaranteesareuniformin
2
a
thesensethatwithonerandomchoiceofmeasurementsettingsallquantumchannelscanberecoveredequally
J
well.Moreover,stabilityagainstarbitrarymeasurementnoiseandrobustnessagainstviolationsofthelow-rank
1
assumptionisguaranteed.Numericalstudiesdemonstratethefeasibilityoftheapproach.
1

]
h
CONTENTS 4. Nullspaceproperty 15
p
- 5. Toolsfromrepresentationtheory 16
t B. Ourboundontheminimumconicsingularvalue 16
n I. Introduction 2
a A. Motivationofourmeasurementmodel 2 1. Upperboundonthemeanempiricalwidth
u W 17
B. Ourcontribution 2 m
q 2. LowerboundonthemarginaltailfunctionQ 18
C. Relatedwork 3 ξ
[
C. Generalboundontensornetworks 21
D. Experimentalconsiderations 4

1 E. Notationandterminology 4 D. Proofsofthereconstructiontheorems 23
v 1. Constrainttracenormminimization 23
1. Mapsonoperators 4
5 2. CPT-fit 24
2. Norms 5
3 3. Constrainedtraceanddiamondnorm
1 3. Sphericalandunitarydesigns 5
minimization 24
3 4. Measurementterminology 5
E. Bosonicandfermioniclinearopticalcircuits 25
0
1. II. Results 5 IV. Conclusion 26
0 A. Measurementmodel 5
A. Outlook 26
7 B. Reconstructions 6
1 C. Noiselesscase 7 V. Acknowledgements 27
:
v 1. RecoveryguaranteesI 7
i 2. Numericaldemonstration 7 Appendix 28
X
D. Stabilityandrobustness 8
ar 1. RecoveryguaranteesII 8 A. Semidefiniteprogramsfortraceanddiamondnorm
reconstruction 28
2. Numericalexample: ReconstructingtheToffoli
gate 10
B. Anon-commutativeKhintchineinequality 28
E. Paulimeasurements 12
F. Samplecomplexity 13
C. LinearrepresentationofthesymmetricgroupS 28
4
G. Applicationstofaulttolerantquantum
computation 13 References 29
III. Analyticaldetailsandproofs 13
A. Preliminaries 14
1. Notation 14
2. Normalizationandcentralizationofthe
observables 14
3. Minimumconicsingularvalue 15
2
I. INTRODUCTION usedtopreparetheentangledinputstateinthefirstplace.
Avoidingthisdrawback,weadoptanexperimentallyfeasi-
Recent years have seen significant advances in the precise ble“sequential”measurementmodel.Whatismore,weresort
controlofquantumsystems. Complexquantumstatesofsys- tothe situationwherethe localinput statesρloc aretransmit-
temswithanincreasingnumberofdegreesoffreedomcanbe ted via individual copies of the unknown channel T in their
prepared and manipulated with high accuracy. In this devel- entirety. This setting is often referred to as “ancilla-free” or
opment,itisimportanttohavetoolsathandthatallowoneto “direct” process tomography. Data acquisition is then con-
completelycharacterizewhichstateorprocessisactuallyreal- cludedbyperformingmeasurementsontheindividualoutput
izedinagivenexperimentalsetup. Thetaskofreconstructing statesT(ρloc).
quantumstatesorprocess[1,2]fromexperimentaldataisvar- We employ the following natural measurement setup: the
iouslycalledquantumstateorquantumprocesstomography, unknown quantum channel T is applied to pure input states
estimation,orrecovery. |ψi(cid:105)(cid:104)ψi|. Subsequently, the expectation value of an observ-
able A on the output is measured. Those expectation val-
Theprecisecharacterizationofprocessesishighlyrelevant, i
ues are estimated by means of a suitably frequent repetition
e.g., inthequestforscalablequantumcomputers. Thestrin-
of the prescription choosing the same measurement setting.
gent requirements of the error correction threshold – and the
Repeating this procedure m times for different input states
adverse scaling of the error correction overhead in terms of
|ψ (cid:105)(cid:104)ψ |andobservablesA resultsinameasurementvector
the noise – make it necessary that implementations of quan- i i i
oftheform
tumgatesmatchtheirspecificationextremelyclosely.Wenote
thatfullquantumprocesstomographyisdistinctfromcertifi- (cid:2) (cid:3)
y ≈Tr A T(|ψ (cid:105)(cid:104)ψ |) , 1≤i≤m. (1)
cationprotocolsorcoarsercharacterizationschemeslikeran- i i i i
domizedbenchmarking(whichreportsonlyasinglenumber: For our theoretical recovery guarantees, the input states and
acertainerrorrate). Whilethismakesthelattertypeofpro- observables are assumed to be sampled independently from
tocolsmuchcheapertoimplement,onlyprocesstomography certainensembles–seebelowfordetails.Werestrictattention
allows one to understand in precisely which way a quantum to pure input states, as they are expected to yield the most
gatedeviatesfromitsspecification. informationaboutT,andalsobecausethepreparationofpure
Thetaskofquantumprocesstomography–importantasit inputstatesiseasilypossibleinmostexperimentalsetups.We
is – comes at a high price: This is an unfavourable scaling expectittobestraight-forwardtogeneralizethisworktotake
of the necessary effort with the system size. To learn an un- mixedinputstatesintoaccount.
knownunstructuredprocessactingonann-dimensionalquan- Note that while this is the common way ancilla-free pro-
tumsystems,m ∼ n4 expectationvaluesarerequired. How- cesstomographyisconsidered,thefinalmeasurementprocess
ever, common processes exhibit additional structure. Most couldpotentiallybemodeledinmoredetail. Inatypicalex-
importantly, quantum gates correspond to unitary processes, periment,onedoesnotonlylearntheexpectationvalueofan
which are quantum channels with Kraus rank equal to one. observable A, but rather acquires statistics about the POVM
Compressed sensing techniques allow one to estimate chan- associatedwithitsspectraldecomposition.Here,wedisregard
nels with (approximately) low Kraus rank from significantly thisfiner-grainedinformation. Wehavemadethisdecisionto
fewermeasurementsthannaivelyrequired. Itisimperativeto avoid technical complications: The various POVM elements
makeuseofthisstructure. associatedwithanygivensettingareclearlynotindependent
– however our proof techniques work best for independently
drawn measurements. There are now related theoretical re-
A. Motivationofourmeasurementmodel covery guarantees that can handle some form of dependency
– e.g. Refs. [3–8]. Applying such techniques to process to-
Themostgeneralquantum processtomographysettingal- mographyremainsaninterestingopenproblem.
lows “coherent measurements” of an unknown channel T.
Here,midenticalcopiesofT areavailablesimultaneously.At
thesametime,onlypartoftheinputstatemaybesentthrough B. Ourcontribution
copiesofT,whileanotherpartpotentiallyentangledwiththe
former is left unchanged. This in turn allows for perform- Weinvestigateseveralchannelreconstructionprotocolsthat
ing measurements on T⊗m ⊗id — where id denotes the exploitthemeasuredchanneltohavean(approximately)low
n n
identity map on a n-dimensional additional ancillary Hilbert Krausrank. Inspiredbycompressedsensing,theseprotocols
space that may be available – rather than “sequential” mea- requireconsiderablyfewermeasurementsoftheform(1)than
surementsonsinglecopiesofT. Thisincludesthepossibility traditional process tomography schemes and contain compu-
ofchoosingglobal,entangledinputstatesρ ,andlikewise tationallytractablereconstructionalgorithms.Weproviderig-
global
performingcorrelatedmeasurementsontherespectiveoutput orousrecoveryguaranteesfeaturingverydesirableproperties
states(T⊗m⊗id )(ρ ). (uniform,stable,androbust). Thisanalyticalworkiscomple-
n global
Whilepotentiallypowerful,coherentmeasurementsarear- mentedbyacomprehensivenumericalanalysisoftherecon-
guablyimpractical. Thisisinparticulartruewhenattempting structionprotocols. Incontrasttopreviouscompressedsens-
todiagnoseerrorsinimplementationsofaquantumgates,as ingresultsthatareapplicabletoprocesstomography, ourre-
theseareexactlythebuildingblocksofcircuitsthatwouldbe constructionprotocolsspecificallyaddressthenaturalprocess
3
tomographysetupdescribedabove. Ourfocusisonaparticu- Regarding test processes, we consider generic channels
larlysimplereconstruction: with varying Kraus rank, as well as the Toffoli gate. This
i) Aleastsquaresfitoftheobserveddatasubjecttoanaddi- is a three-qubit unitary quantum gate that is highly relevant
tionalpositivityconstraint.WecallthisapproachCPT-fit. inquantumprocessing. Moreover,ithasbeenexperimentally
A distinct advantage of the CPT-fit is that the algorithm implementedinvariousarchitectures[12–15].
doesnotdependonanyparameters. Inparticular,noes-
timate of the true rank or the noise strength is required.
Thisstandsincontrasttomostcompressedsensing-based C. Relatedwork
basedreconstructionschemes. Also,theCPT-fitappears
tobeparticularlyrobustagainstviolationsoftheassump- Recent years have seen considerable advance of com-
tionoflowKrausrank. pressed sensing tools for the tomographic reconstruction of
In order to link and compare the CPT-fit to more established low-rankquantumstates[16–20]. Tosomeextent,thesefind-
low-rank matrix reconstruction methods we also investigate ings have also been adapted to cover process tomography.
other reconstruction protocols. Their analytical analysis is Suchcompressedsensingapproachesallowforreconstructing
alsothebasisfortherecoveryguaranteesoftheCPT-fit. processeswithlowKrausrankfrommuchfewerexpectation
valuesthananaivedimensioncountwouldsuggest.
ii) The first reconstruction method corresponds resembles
Ourtracenormminimization(algorithmsii)andiii))builds
a typical low-rank matrix reconstruction protocol: min-
onabynowwell-establishedmethodsforlow-rankmatrixre-
imize the trace norm subject to convex constraints that
construction[16]. Withthediamondnormminimization(al-
takeintoaccountacquireddataoftheform(1).
gorithms iv) and v)) we continue a line of research that has
iii) Thesecondrecoveryalgorithmcloselyresemblesthefirst
been started in Ref. [9] and support it with rigorous perfor-
one,butcontainstracepreservationasanadditional(lin-
manceguarantees.
ear)constraint.
TheCPT-fithasbeensuggestedinRef.[18].Thisworkalso
iv) Otherwise similar to algorithm iii), this reconstruction
containsnumericalstudiesthatdemonstratethecapabilityof
protocol replaces the constrained trace norm minimiza-
such a reconstruction procedure. We expand on these ideas
tion by a diamond norm minimization. The constraints
by conducting additional numerical experiments. But, more
remain unchanged. The diamond norm is a well-
importantly, we also prove rigorous recovery guarantees for
motivateddistancemeasureforquantumchannels. Apri-
theCPT-fit.
ori,thisdoesnotimplyinanywaythatthediamondnorm
isalsoagoodchoiceasaregularizingfunctionforchan- Ref. [16] presents a process tomography protocol that is
nel estimation. However, Ref. [9] does provide strong based on low-rank matrix reconstruction with random Pauli
analytical and numerical arguments for why one should measurements [21–23]. These low rank recovery guarantees
expectthediamondnormtooutperformthetracenormas can be applied to the Choi-Jamiołkowski representation of a
a regularizer in this setting. The numerical studies con- quantumchannel,sincetherankofthismatrixrepresentation
ductedinthispaperlendfurthercredencetothisclaim. equalstheKrausrankoftheoriginalchannel. Onfirstsight,
v) Similar to algorithm iv), but with an additional trace suchanapproachrequirestheuseofanancillainordertoim-
preservationconstraint. plementtheChoi-Jamiołkowskirepresentationphysicallyina
concrete application [24]. However, Ref. [16] also provides
We provide rigorous performance guarantees for the ap-
a more direct implementation of their protocol that does not
proachesi),iii)andv). Theguaranteesarevalidformeasure-
requireanyancillas. Validformulti-qubitprocessesthistrick
ment setups where the pure input states |ψ (cid:105) and the eigen-
i exploitsthetensor-productstructureof(multi-qubit)Pauliop-
bases of the observables A are drawn at random from cer-
i erators. It allows for effectively measuring the Pauli expec-
tainensembles. Technically,werequirethestatevectorsand
tation value of a Choi matrix by performing several natural
theeigenbasestoformcomplex4-designsinaprecisesense.
channel measurements of the form (1) and evaluating partic-
Haar-randomvectorshavethisproperty,butrecentresultson
ularlinearcombinationsthereofinaclassicalpost-processing
the fourth moments of the Clifford group [10, 11] suggest
step. The demerit of this approach is that the number of in-
thattherearemorestructuredandexplicitensembleswiththe
dividual channel measurements required to evaluate a single
sameproperty.
Pauli expectation value scales with the dimension of the un-
Even if a particular practical setup fails to show the 4- derlyingHilbertspace. Themeasurementmodelstudiedhere
design property, our results are still a relevant proof of con- does not require such a coarse-graining: every natural mea-
cept. Indeed, incontrasttopreviouscompressedsensingap- surement (1) itself already corresponds to a valid measure-
proaches,ourreconstructionguaranteesshowthatthespecific ment instance. This considerably reduces the numberof dif-
mathematical structure present in process tomography does ferent measurement settings that are required in order to ac-
notimplythatalargernumberofmeasurementsettingsisre- quiresufficientdata.
quired. Randomizedbenchmarkinghasalsobeenadaptedtoallow
What is more, we provide numerical simulations showing forprocesstomography[19].Again,representingchannelsby
thattherecoveryalgorithmsworkwellforanumberofmea- theirChoimatrixisakeyintermediatestep.Ref.[19]adopted
surement models not having the 4-design property. This in- randomizedbenchmarkingtechniquestoobtainaprocessto-
cludestheparadigmaticcaseofPauli-typemeasurements. mographyprotocolthatisrobusttowardsstatepreparationand
4
measurement errors. This is a distinct advantage over other W V∗ W V∗ V∗
protocols that do not share this additional feature. However, M (cid:55)→J M T(cid:55)→rW M
it comes at a prize: the analytical performance guarantees in W∗ V W∗ V V
Ref. [19] require an additional “non-spikiness condition” on
the channels that are to be re-constructed. This restriction
also features in the algorithmic reconstruction as linear con- Figure1. TheChoi-Jamiołkowskiisomorphismandpartialtracein
termsoftensornetworkdiagrams(explainedinFigure9).
straints. Thetotal numberof theseadditional constraintsde-
Left: Order-4 tensor M ∈ L(L(V) → L(W)) as a map from
pends on the concrete measurement ensemble and is lower-
L(V)∼=V⊗V∗toL(W)∼=W⊗W∗.
bounded by the 4-th power of the underlying Hilbert space Middle:ItsChoi-matrixJ(M)asanoperatoronW∗⊗V ∼=W⊗V.
dimension, which affects the computational overhead of the
Right: PartialtraceTr [J(M)]oftheChoimatrixJ(M). Thisop-
1
reconstruction. eratorcorrespondstothefunctionalρ(cid:55)→Tr[M(ρ)].
Finally, Ref. [17] also considered process tomography via
compressed sensing techniques. However, this method is
somewhatdifferentfromtheotherapproachespresentedhere: experiments with cold atoms in optical lattices [35], such as
InsteadofassuminglowKrausrank,theyconsiderprocesses opticalsuper-latticesandtime-of-flightmeasurements.
that are element-wise sparse with respect to a known basis. In integrated linear optical architectures [36, 37], Haar-
Prior knowledge of this sparsifying basis is a necessary pre- random circuits are readily conceivable [38]. It is this type
requisite for this approach. In turn, techniques from com- ofsettinginwhichthemethodspresentedherearemostappli-
pressed sensing [25, 26] are applied (rather than low-rank cable,eventhoughtheyapplytoanyphysicalsysteminwhich
matrixreconstructionprotocols)toreconstructsuchprocesses unitary4-designsarefeasible. Thisincludessettingsinwhich
from a number of measurements that is proportional to the randomcircuits[39,40]orrandomlytime-fluctuatingdynam-
sparsityanddependsonlylogarithmicallyontheambientsys- ics[41],aswillbediscussedinmoredetailintheoutlook.
tem sizes. While requiring only very few measurement set-
tings, the main disadvantage of this approach are stronger
model assumptions (sparsity and knowledge of the sparsify-
E. Notationandterminology
ing basis) and measurements that may be challenging to im-
plementinpractice.
In this section we introduce some basic preliminaries
needed to understand our main results. For any integer n ∈
Z+ we use the notation [n] := {1,2,...,n}. The space of
D. Experimentalconsiderations
linearoperatorsonavectorspaceV isdenotedbyL(V).
IfV isaHilbertspace,wedenotevectorsby |ψ(cid:105) ∈ V and
In several physically important platforms, process tomog- its adjoints (conjugate transposes) by (cid:104)ψ|. So the projector
raphy has been experimentally realized [27–29], both in the onto a normalized |ψ(cid:105) is |ψ(cid:105)(cid:104)ψ|. Moreover, the real vector
direct and hence ancilla-free [27, 29] and ancilla-based [28] spaceofself-adjointoperatorsisdenotedbyHerm(V).
readingofthetask. Theseworksmakeuseofdifferentprepa-
rations and measurements. It should be clear, however, that
in several physical architectures, random measurements of 1. Mapsonoperators
thetypediscussedherecanreadilyexperimentallybeimple-
mented.
Wedenotethespaceoflinearmapstakingoperatorstoop-
Specifically,Haarrandomunitarymapshavebeenrealized eratorsbyL(Cn) := L(L(Cn)). TheChoi-Jamiołkowskiiso-
in a 16-dimensional Hilbert space associated with the 6S1/2 morphism [42, 43] J : L(Cn) → L(Cn ⊗ Cn) takes such
groundstateof133Csatoms[30]. Thishasbeenachievedby
maps to operators on a tensor product space and is given by
suitably exploiting a time-dependent Hamiltonian evolution
(seealsoFigure1foratensornetworkdescription)
givingrisetoHaarrandomunitaries,makinguseofmethods
of quantum control. Such random unitaries have been put to dim(V)
(cid:88)
use in a tomographic protocol, in an approach of perform- J(M):= M(|i(cid:105)(cid:104)j|)⊗ |i(cid:105)(cid:104)j| . (2)
ingquantumtomographybasedonHaarrandomunitariesthat
i,j=1
buildsuponearliertheoreticalideaslaidoutinRef.[31]. The
sameideaofgeneratingHaarrandomunitariesusingsuitable MapsM ∈ L(Cn)canhavedifferentproperties: M iscalled
time-dependentHamiltoniansshouldreadilyapplytosystems hermiticity preserving if it satisfies M(A†) = M(A)† for
of trapped ions [32], where a suitable type of control can be all operators A ∈ L(Cn). This is equivalent to demand-
achievedwithpresenttechnology. Indeed,inadifferentcon- ing that J(M) is hermitian. M is called trace preserving if
text,methodsofoptimalcontrolhavereadilybeenappliedto Tr[M(A)] = Tr[A] for all A ∈ L(Cn) or, equivalently, if
optimizequantumgatesfortrappedions[33]. Tr [J(M)]=1,whereTr :L(Cn⊗Cn)→L(Cn)denotes
1 1
Random unitaries, albeit not Haar distributed, that allow the partial trace over the fist tensor factor. The affine sub-
for quantum state tomography of quantum many-body sys- spaceofhermiticityandtracepreservingmapsisdenotedby
tems have been theoretically considered [34], making use of HT(Cn)⊂L(Cn).Theidentitymapid ∈L(Ck)isapartic-
k
operations only that can be considered basically feasible in ularlysimpleelementofHT(Cn). Moreover,M ∈ L(Cn)is
5
completelypositiveifforallk ≥1onehasthat(M⊗idk)(A) U |ψ(cid:105)
is positive semidefinite for every positive semidefinite oper- A0 T
ator A ∈ L(Cn ⊗Ck). We will use the shorthand notation
U∗ (cid:104)ψ|
A (cid:23) 0 to indicate positive-semidefiniteness of A. In fact, a
mapMiscompletelypositiveifandonlyif(M⊗id )(A)(cid:23)0
n
Figure2. Thetensorialstructureofonemeasurementsetting: T is
(k =n). ThisinturnisequivalenttoJ(M)(cid:23)0. mappedtoTr(cid:2)UA U†T(|ψ(cid:105)(cid:104)ψ|)(cid:3).
Theconvexsubsetofcompletelypositiveandtracepreserv- 0
ing (CPT) maps is denoted by CPT(Cn) ⊂ L(Cn). Impor-
tantly, these maps take density matrices to density matrices,
3. Sphericalandunitarydesigns
even when applied to subsystems. Therefore, they are also
calledquantumchannelsandareaverygeneraldescriptionof
The probability measures on the unitary group, respec-
quantumprocesses,e.g.dynamics.
tively, the unit complex sphere that is invariant under multi-
plication with any unitary is called Haar measure and is the
naturaluniformmeasureonthesesets. Acomplexprojective
2. Norms t-design [48–50] is a probability distribution µ on the unit
sphere that reproduces the moments of the Haar measure up
Forq ∈[1,∞[,the(cid:96) -normofavectorv ∈Cnis toordert+t,
q
E (cid:2)(|ψ(cid:105)(cid:104)ψ|)⊗t(cid:3)=E (cid:2)(|ψ(cid:105)(cid:104)ψ|)⊗t(cid:3). (9)
 1/q |ψ(cid:105)∼µ |ψ(cid:105)∼Haar
n
(cid:107)v(cid:107)(cid:96)q :=(cid:88)|vj|q (3) Sνiomniltahrelyu,naiutanriytagryroduepsisgantis[5fy1i,n5g2]isaprobabilitydistribution
j=1
E (cid:2)U⊗tXU†⊗t(cid:3)=E (cid:2)U⊗tXU†⊗t(cid:3) (10)
U∼ν U∼Haar
and the (cid:96) -norm is (cid:107)v(cid:107) := max |v |. The Schatten
q-norm (cid:107)∞A(cid:107) of an oper(cid:96)a∞tor A corrje∈sp[no]ndjs to the (cid:96) -norm for all operators X ∈ L(Cn)⊗t. Typically, designs are con-
q q
of A’s singular values arranged in an n-dimensional vector. sideredtobeuniformdistributionsoverfinitesets.
Here, we will specifically encounter the following Schatten
norms,
4. Measurementterminology
√
(cid:2) (cid:3)
(cid:107)A(cid:107) =Tr AA† (tracenorm), (4)
1
(cid:113) Inthecompressedsensingliterature,asinglemeasurement
(cid:107)A(cid:107) = Tr[AA†] (Frobeniusnorm), (5) is usually given by an inner product, which is a Hilbert-
2
SchmidtinnerproductTr[AT(|ψ(cid:105)(cid:104)ψ|)]inourcase. Inquan-
(cid:107)A|ψ(cid:105)(cid:107)
(cid:107)A(cid:107) =max (cid:96)2 (spectralnorm). (6) tum physics terminology, a (ψ,A) would correspond to a
∞ |ψ(cid:105) (cid:107)|ψ(cid:105)(cid:107)(cid:96)2 measurement setting, whereas a corresponding measurement
would be the von-Neumann measurement of the observable
Notethatrank-oneprojectors |ψ(cid:105)(cid:104)ψ|areunitnormalizedwith A in the state T(|ψ(cid:105)(cid:104)ψ|) giving rise to an expectation value
respecttoanySchatten-pnorm. Tr[AT(|ψ(cid:105)(cid:104)ψ|)]inthelimitofinfinitelymanyvon-Neumann
Thespectralnormisalsoanexampleofaninducednorm. measurements. In order to also account for finite sample er-
This concept can be generalized to norms on maps M ∈ rorswealwaysallowforadditivenoiseontheobtainedexpec-
L(Cn),andtheinducedtracenormamountsto tationvaluesTr[AT(|ψ(cid:105)(cid:104)ψ|)].
(cid:107)M(A)(cid:107)
(cid:107)M(cid:107) :=max 1 . (7)
1→1 A (cid:107)A(cid:107) II. RESULTS
1
Finally, the diamond norm is a stabilized version of the in- In this section, we explicitly specify our measurement
ducedtracenorm: model of natural measurements, explain how we computa-
tionallyreconstructthequantumchannelsfromthemeasure-
(cid:107)M(cid:107) :=(cid:107)M ⊗id (cid:107) . (8) ments, state our recovery guarantees, discuss their stability
(cid:5) n 1→1
androbustnessproperties,discussournumericsonPauli-type
Thisnormisaparticularlymeaningfulandwidelyappreciated measurements, and derive a sample complexity upper bound
distance measure for quantum processes [44]. Perhaps sur- fromourrecoveryguarantees.
prisingly,itcanbecalculatedefficiently[45–47]asasemidef-
inite program. Regarding the reconstruction of structured
maps, thediamondnormcanserveasaconvexsurrogatefor A. Measurementmodel
lowKrausrank[9].Thisisthecaseforchannelreconstruction
andtwoofthefivereconstructionprotocolspresentedinthis We consider the task of reconstructing a quantum channel
workbuildonthisidea. T ∈ CPT(Cn)frommeasurementdataoftheform(1): The
6
unknown channel receives pure states |ψ (cid:105)(cid:104)ψ | as input and approach, wealsoinvestigatethereconstructionmethodsT∗
i i η
subsequently expectation values of observables A are mea- andT∗casversionsoflow-rankmatrixreconstructionaswell
i η
suredfortheoutputstate;seeFigure2forapictorialdescrip- as T(cid:5) and T(cid:5)c as reconstruction methods based on the dia-
η η
tion. Thisprocedureisrepeatedmtimeswithdifferentmea- mondnorm.
surement settings (input states and observables). Hence, the Tostartwiththeformer,weminimizethesquarelossunder
entiremeasurementprocessleadstoameasurementvector themodelconstrainedthatT isaquantumprocesstoobtain
y =A(T)+e ∈Rm, (11) T(cid:96)2 :=argmin{(cid:107)A(T)−y(cid:107) : T ∈CPT(Cn)}. (13)
(cid:96)2
withsingleexpectationvalues This minimization is essentially a fit under a CPT-constraint
andwewillcallitCPT-fit. Ithasbeensuggestedandnumer-
A(T) =Tr[A T(|ψ (cid:105)(cid:104)ψ |)]+e . (12)
i i i i i ically investigated in Ref. [18]. This approach makes use of
Thevectore∈Rmdenotesadditivenoisepresentinthemea- themeasurementdatayalone.
More common low-rank matrix reconstruction methods in
surementprocess.Incontrasttopreviousapproaches[16,23],
compressedsensingusethetracenormasaso-calledregular-
nopriorassumptionsonthenatureofthisnoisecorruptionare
izer to favour low-rank solutions [22, 53–55]. These recon-
required.
structions do notonly make use of the data y, but requirean
Throughout this work, we consider instances of random
measurements {A ,|ψ (cid:105)}m that are independent instances a-prioriboundη >0onthenoisee,
i i i=1
ofameasurementensemble(A,|ψ(cid:105))thatmeetsthefollowing
(cid:107)e(cid:107) ≤η. (14)
requirements: (cid:96)2
Definition 1 (4-generic measurement ensemble). We call AppliedtotheChoimatrixJ(T)ofaquantumchannelT the
a measurement ensemble (A,|ψ(cid:105)) with observable A and usualtracenormregularizationleadstothereconstructions
state |ψ(cid:105)4-genericifitfulfilsthefollowingcriteria:
i) Thedistributionof |ψ(cid:105)inCn isaspherical4-design(9), Tη∗ :=argmin{(cid:107)J(T)(cid:107)1 :T ∈L(Cn), (cid:107)A(T)−y(cid:107)(cid:96)2 ≤η},
i.e., it reproduces the first four moments of the unitarily (15)
invariant(Haar)measureonthecomplexunitsphere. T∗c :=argmin{(cid:107)J(T)(cid:107) :T ∈HT(Cn), (cid:107)A(T)−y(cid:107) ≤η}.
ii) A = UA U†, whereA ∈ Herm(Cn)isfixedandU in η 1 (cid:96)2
0 0 (16)
U(n) is chosen from a unitary 4-design (10), i.e., repro-
ducesthefirstfourmomentsoftheHaarmeasure.
Thesetwoapproachesareverysimilar. However,thesecond
Themeasurementensembleiscalled normalized4-generic
one contains an additional constraint that enforces hermicity
if the observables are traceless and normalized in spectral
andtracepreservation.
norm,i.e. Tr[A ]=0and(cid:107)A (cid:107) =1.
0 0 ∞ Thediamondnormiswell-knowninquantuminformation
Corresponding expectation values Tr[AT(|ψ(cid:105)(cid:104)ψ|)] of a
theoryasameasureofdistinguishabilityofquantumchannels
quantumchannelT ∈ CPT(Cn)arereferredtoas4-generic
by expectation values [56, Chapter 11], [57] and can practi-
measurementandnormalized4-genericmeasurement,respec-
cally be calculated and minimized using a semidefinite pro-
tively.
gram[47]. Fornon-obviousreasons, itcanalsobeusedasa
reguralizerforthereconstructionofcertainmapsonoperators
This definition encapsulates a variety of process measure-
[9],leadingtothereconstructions
ment ensembles. In particular, the generic measurement en-
semble(|ψ(cid:105)andU areHaarrandomstatesandunitaries,re-
T(cid:5) :=argmin{(cid:107)T(cid:107) :T ∈L(Cn), (cid:107)A(T)−y(cid:107) ≤η},
spectively) meets the requirements by definition. However, η (cid:5) (cid:96)2
(17)
ourrecoveryguaranteesdonotrequiresuchastrongnotionof
randomness in the measurement design: 4-designs are suffi- T(cid:5)c :=argmin{(cid:107)T(cid:107) :T ∈HT(Cn), (cid:107)A(T)−y(cid:107) ≤η},
cient. η (cid:5) (cid:96)2 (18)
Wealsoexpectthattheactualchannelrecoveryworkssim-
ilarly well if |ψ(cid:105) and U are chosen from other distributions where η ≥ 0 is again an anticipated bound on the measure-
aswell,providedthattheycoverthecomplexunitsphereand mentnoise.
U(n) “evenly” enough (see Section IV for a further discus- Superior reconstruction properties over simple inversion
sion). We confirm this expectation with numerical simula- methodsareexpectedforchannelswithsmallKrausrankwith
tions,seeSectionIIE. unitarychannelsbeingtheoptimalcase. Perhapssurprisingly
at first sight, the CPT-fit performs equally well as the other
reconstruction methods based on the trace (15), (16) or di-
B. Reconstructions amond norm (17), (18) minimization. However, a quantum
channel T has a Choi matrix J(T) with constant trace norm
In this section, we lay out how a quantum channel T can (cid:107)J(T)(cid:107) =n. Hence,onecanomitthetracenormminimiza-
1
bereconstructedfromdataoftheform(11)formeasurement tioninthetheprogram(16)whentheconstraintJ(T) (cid:23) 0is
settings given by A. We put an emphasis on describing the addedtothatonehasT ∈ CPT(Cn). Minimizingthesquare
fittingmethodT(cid:96)2 referredtoasCPT-fit. Tocomplementthis loss(cid:107)A(T)−y(cid:107) insteadyieldstheCPT-fit(13).
(cid:96)2
7
n=4,Krausrankr=2 n=4,Krausrankr=2
100 140
s 120
la 80 .k
irt 001 / 50 60 || . ||1 es / emit U18000
rorr-ee 1# 5 2400 |||C||| P... |||T|||1}} f,,i t TTPP--ccoonnssttrr.. PC egarevA 246000
0 0
50 100 150 50 100 150
# measurement settings m # measurement settings m
(a)Successrateofthereconstructionoverthenumberofmeasure- (b)AverageCPUtimerequiredforthereconstructionin(a)overthe
ments m for Kraus rank r = rank(J(T )) = 2. The rate is the numberofmeasurementsm.
0
numberoftrailsoutof100withsmallerrors(cid:107)Trec−T (cid:107) ≤10−5,
0 F
forreconstructedchannelsTrec.
Figure3. RetrievalofrandomquantumchannelsT ∈CPT(C4)forvanishingnoisee=0. ThemeasurementsettingsAaredrawni.i.d.in
0
eachtrial.Theparameterηinthereconstructions(15),(16),(18)and(18)ischosentobetentimesmachineprecision.
We mention once more that existing recovery guarantees Remarks:
for low-rank matrix reconstruction [4, 22, 23, 54] do not ap- i) Thenumberofrealparametersrequiredtospecifyaher-
plytothemeasurementmodelconsideredhere. Ourmeasure- miticity preserving map T ∈ L(Cn) is n4. If said map
0
mentsetting(12)carriesacertainproductstructure(seeFig- is also trace preserving, then n4 −n2 many parameters
ure 2) that is more restrictive than the structure covered in arerequired. ThesameholdsforT ∈ CPT(Cn). Ifthe
0
priorworks. Kraus rank of T is rank(J(T )) = r then T is com-
0 0 0
pressible in the sense that only a number of parameter
scaling as rn2 is required and also sufficient to describe
T . Therefore, up to the factor ln(n) the scaling (19) of
C. Noiselesscase 0
thenumberofmeasurementsisoptimal.
ii) Thescaling(19)alsocontainsaconstantC thatwehave
Forsakeofclarity,wefirstdiscussthenoiselesscase(e =
notboundedexplicitly. Numerically,wefindforn=4,8
0)andwilldiscussstabilityandrobustnessinSectionIID.
that m ≥ 5rn2 generic (Haar-random) measurements
aresufficientforrecovery,seeSectionsIIC2andIID2.
Such an approach is very common in compressed sens-
1. RecoveryguaranteesI ing: Recoveryguarantees,oftenwithunknownconstants
and logarithmic factors in the scaling of the number of
We have recovery guarantees for the minimizations (16), measurements, ensure the functioning of the reconstruc-
(18),and(13),whichhaveatracepreservationasaconstraint. tionprocedurealsointhelimitoflargedimensions. The
precisenumberofmeasurementsandtheexpectederrors
Theorem 2 (Uniform recovery guarantees (noiseless case)). inagivensettingareoftendeterminednumerically.Here,
Fixr ≤nandsupposethatA:CPT(Cn)→Rmcontains alsomorespecialmeasurementsettingscanpracticallybe
considered. In Section IIE, we numerically investigate
m≥Crn2ln(n), (19) Pauli-type measurements, which are not covered by our
recoveryguarantees.
4-genericmeasurementswhicharechosenindependentlyfrom
the measurement ensemble introduced in Definition 1. Then,
withprobabilityatleast1−e−λm, allT ∈ CPT(Cn)with 2. Numericaldemonstration
0
Krausrankatmostrmaybereconstructedexactlyfromnoise-
less measurements y = A(T0). This exact reconstruction Our variational reconstructions can be recast as standard
is achieved by each of the minimizations (16), (18), or (13). convex optimization problems (see also Appendix A). These
Here,C andλareuniversalconstants(whichcan,inprinci- canbesolvedcomputationallyefficientlyandalsopractically
ple,beextractedexplicitlyfromtheproof). usingstandardsoftwaresuchasCVX[58,59]. Fortherecon-
structions(16), (17), and(13)thathavetracepreservationas
This statement follows from more general results, namely aconstraintwehaveproventhatanumberof
Theorems20, 21, and25below. Stabilityandrobustnessare
discussedwithTheorems3and4. m≥m :=Crn2ln(n) (20)
0
8
measurement settings is sufficient to recover quantum chan-
CPT-t:n=4
nelsofKrausrankr. Weexpectsimilarreconstructionprop- 250 100
erties for the unconstrained trace and diamond norm recon-
m
structions (15) and (17), although our proves do not directly sg200 80 sla
cmasincaagnarDrtluryii(minf2cfgoeb0evse)[re2ercrno0aotft,noft2reemb2nthce,eoah2artna3esscvcu,taeorr5suev4ecmae,.tri6eedIon0ndinft]dfs.vepeiwerraeodinct,trhtefadoccsureuormcnenhssoatarnawmnynittshemCistitsu.nheaientmitFooiiaonpzlrstaliytmliionooawsnplt-atfsirnrmcaocanmaelk-l, nittes tnemerusaem #11505000 246000 irt 001 / r5o0rr-ee1 # 5
Ref. [61] shows that a constant of size C (cid:39) 6 provably suf-
0
fices for Gaussian measurement matrices (even without the 2 4 6 8 10 12 14 16
ln(n)factor).Ontheotherhand,theresultspresentedinRefs. Kraus rank r
[22,23]arevalidformorestructuredPaulimeasurementsand
require a much larger constant. Numerical studies typically Figure4. RetrievalofrandomquantumchannelsT ∈CPT(C4)of
0
highlightasimilarbehaviour. differentKrausranksr = rank(J(T ))fromtheCPT-fit(13). The
0
Exploiting additional prior information – such as the fact whiteregioncorrespondsto100%observedrecoveryandtheblack
that quantum channels preserve both hermicity and traces in oneto0%.ParametersandmeasurementsareasinFigure3.
the algorithmic reconstructions (16), (18) – can only lead to
an improvement. In fact, these constraints also facilitate the
linearly on the Kraus rank r of the channels to be recon-
mathematicalanalysis. Besidesbeingabletoproverecovery
structed. This dependence is confirmed for small r in our
guarantees,wealsoobservethebenefitofsuchadditionalcon-
straintsnumerically: Figure3(a)showstherecoveryratesof numerics for Hilbert space dimension n = 4, see Figure 4.
Fordimensionn = 4theKrausrankisr ≤ n2 = 16andfor
differentapproaches.
Thisrecoveryrateisdeterminedasfollows. Wesaythata r =16thenumberm=dimR(HT(Cn))=n4−n2 =252of
measurementvaluesarerequiredforthereconstruction.Here,
channel T is successfully reconstructed if the reconstructed
channel T0rec is close to the original one: (cid:107)Trec−T (cid:107) ≤ dimR(HT(Cn))denotestherealdimensionoftheaffinespace
10−5. We have chosen 10−5 as threshold because w0e Fhave HT(Cn). Thisobservationexplainsthenon-linearbehaviour
forlargerr.
observed the reconstruction errors to be typically well sepa-
rated from this value. Varying it changes the curves in the
plot only slightly. The precision of the convex optimization
software CVX [58, 59] in terms of the machine precision D. Stabilityandrobustness
√
eps = 2−52 ≈ 2.2·10−16 is eps ≈ 1.4·10−8. Thissome-
what limits the choice of thresholds. In order to obtain the We consider two types of errors: i) the measurement er-
rateshownintheplots,werun100independentinstancesand rors e already indicated in the measurement model (12) and
obtain the rate as the percentage of trials with a successful ii) model mismatches capturing violations of the assumption
reconstruction. rank(J(T ))≤rontheKrausrank.
0
Our numerical studies are based on generic measurement Forthispurpose,wewriteT ∈CPT(Cn)asT =T +T
|r ¬r
ensembles(Haarrandominputstates,Haarrandomunitaries) where T is defined to be a CPT map of Kraus rank r en-
|r
andwehavechosenA tobeadiagonaln×nmatrixwhose compassingther largesteigenvalues[63]oftheChoi-matrix
0
spectrumevenlycoverstheinterval[−1,1]. J(T) and T = T − T is a trace-annihilating map (i.e.,
¬r |r
Interestingly, the unconstrained diamond norm minimiza- Tr[T¬r(X)] = 0forallX ∈ L(Cn))thatiscalledthemodel
tion(17)performsalmostasgoodasitsconstrainedcounter- mismatch. In turn we relax our model assumption of low
part(18).Theheuristicreasonforthisbehaviouristhatthedi- Kraus rank. In the interesting cases, the model mismatch is
amondnorm“favours”mapsthatsatisfyacertaintracepreser- small.
vation condition [9, 62], which is fulfilled for CPT maps. Areconstructioniscalledstableifittoleratesmeasurement
Moreover,theso-calleddescentconeatCPT-mapsofthedia- errorsanditiscalledrobustifittoleratesmodelmismatches.
mondnormiscontainedinanintersectionofdescentconesof
nuclearnorms[9]. Ifthiscontainmentisstrict,italsoleadsto
animprovedrecoveryrecoveryguarantee. 1. RecoveryguaranteesII
Arguably the simplest reconstruction procedure is the fit
under the CPT constraint (13). Here, we observe that this
We prove all reconstructions from Theorem 2 to be sta-
protocol achieves a rate that is very similar to the other con-
bleagainstmeasurementnoise. Moreover,weprovethetrace
strained procedures (16), (18). Moreover, the CPT-fit (13)
normminimization(16)andtheCPT-fit(13)alsotoberobust
clearlyhasthefastestcomputationtimeinoursimpleimple-
againstmodelmismatches.
mentation in CVX [58, 59] with Mosek 7.1 as a solver, Fig-
ure3(b). Theorem 3 (Stability of the diamond norm reconstruction
Therequirednumberofmeasurementsettings(20)depends (18)). Consider normalized 4-generic measurements from
9
Definition 1. Then, under the hypotheses of Theorem 2, re- iii) We emphasize that the dimensional scaling of the error
constructionviatheconstraineddiamondnormminimization boundsinEqs.(22),(23)isaconsequenceofthenormal-
(18) is stable towards additive noise in the measurements ization we adopted. To make this precise, it is useful to
y = A(T )+e: for any η ≥ (cid:107)e(cid:107) , the associated recon- viewthesestatementsasresultsaboutstablyreconstruct-
structione0rrorobeys (cid:96)2 ing (particular) Choi matrices J(T) ∈ L(Cn ⊗Cn) (cid:39)
L(Cn2). Rewritingthesingleexpectationvalues(12)as
(cid:13)(cid:13)J(T0−Tη(cid:5)c)(cid:13)(cid:13)2 ≤c˜√mn(cid:107)2A0(cid:107)2 η. (21) yi =Tr(cid:104)Ai⊗(|ψi(cid:105)(cid:104)ψi|)T J(T)(cid:105)+ei (24)
TheconstantsC,λ,andc˜onlydependoneachother. reveals that all individual measurement matrices M =
i
A ⊗ (|ψ (cid:105)(cid:104)ψ |)T ∈ L(Cn2) have constant Frobenius
Amoregeneralversionofthistheorem–whichallowsfor i i i
norm
quantifying the noise strength by any (cid:96) -norm – is also true;
q
(cid:13) (cid:13)
seeCorollary25below. (cid:107)M (cid:107) =(cid:13)A ⊗(|ψ (cid:105)(cid:104)ψ |)T(cid:13) =(cid:107)A (cid:107) . (25)
i F (cid:13) i i i (cid:13) 0 F
F
Theorem 4 (Stability and robustness, trace norm minimiza-
Choosingaparticularnormalizationinfluencesthesignal-
tion (16) and CPT-fit (13)). Under the hypotheses of Theo-
to-noiseratio(SNR)in(24)and,inturn,thestabilityas-
rem 2, reconstruction via constrained trace norm minimiza-
sertions. In order to avoid these ambiguities, it is useful
tion(16)isbothstabletowardsadditivenoisecorruptionand
torewriteEq.(22)forvanishingmodelmismatchas
robustwithrespecttorelaxingthemodelassumptionofKraus
√
ra(cid:13)(cid:13)nJk(rT:0−Tη∗c)(cid:13)(cid:13)2 ≤4(cid:107)J(T0¬r)(cid:107)1+c˜√mn(cid:13)(cid:13)2A0(cid:13)(cid:13)2 η. (22) with d :(cid:13)(cid:13)=J(dTim0−(cid:0)CTnη∗2c(cid:1)).(cid:13)(cid:13)2N≤otec˜t(cid:80)hamit=t1hm(cid:107)eMsduim(cid:107)Fsηcales like(2m6).
Up to our knowledge, all stable low-rank matrix recon-
Thisboundisvalidforany(cid:107)e(cid:107) ≤ η. Similarly,theCPT-fit
(cid:96)2 structionguaranteesareessentially[65]ofthisform!
(13)hasanerrorboundedas
Inparticular,thisincludesGaussianmeasurementensem-
bles [55, Theorem 2.3], random Pauli matrix measure-
(cid:13)(cid:13)J(T0−T(cid:96)2)(cid:13)(cid:13)2 ≤4(cid:107)J(T0)¬r(cid:107)1+c˜√mn(cid:13)(cid:13)2A0(cid:13)(cid:13)2 (cid:107)e(cid:107)(cid:96)2 . mdaerndtsGa[2u3ss,iPanrovpeocstiotirosn[220.,3T]haenodreomute2r].products of stan-
(23) iv) Regarding sample complexity, an order of rn5ln(n)/(cid:15)2
Once more, the constants C, λ, and c˜again only depend on independentchannelevaluations(“samples”)arerequired
eachother. toachieveareconstructionerrorofatmost(cid:15)inFrobenius
norm;seeSectionIIFbelow.Weexpectthisscalingtobe
This statement summarizes two results – Theorems 20
closetooptimaluptoln(n)-factors(atleastforchannels
and 21 below – that are slightly more general: they allow
withverylowKrausrankr).Evidenceforthisisprovided
forquantifyingthereconstructionerrors(22)and(23)byany
bythediscussionabove:thestabilityguaranteesinTheo-
Schatten-pnormwithp∈[1,2]. Moreover,thenoisestrength
rem3andTheorem4essentiallymatchthebestexisting
maybecharacterizedbyany(cid:96) -norm: η ≥ (cid:107)e(cid:107) .
q (cid:96)q resultsonstablelow-rankmatrixreconstruction. Among
Corollary 5 (Normalization of the measurements). Similar thesearetwothatareapplicabletotherelatedproblemof
statementsasinTheorems3and4holdalsointhecaseof4- quantumstatetomography: randomPaulimeasurements
genericmeasurements. Inthiscase,inthebounds(21),(22), [23]andouterproductsofstandardGaussianvectors[20].
and(23),ηneedstobereplacedbyη/(cid:107)A (cid:107) and(cid:107)A (cid:107) by Thesamplecomplexityoftheformerapproachhasbeen
0 ∞ 0 2
the2-normofthetracelesspartofA . determinedinRef.[16]. Moreover,ithasbeenshownto
0
beclosetooptimalinthesensethatitreproducesafun-
Remarks: damentallowerbound–validforanytomographicproce-
i) Theorem 4 guarantees approximate reconstructions for durebasedonPaulimeasurements–uptoasingleln(n)-
any quantum channel T ∈ CPT(Cn) without a prior factor.
0
rank constraint. The deviation T from a channel of Fundamental lower bounds on the sample complexity
0¬r
low Kraus rank, i.e., the model mismatch enters the er- achievablebyanytomographicprocedurehavebeende-
ror bound linearly. Such a robustness is desirable for rived in Ref. [66]. Said work also determines the sam-
practicalapplicationswherethemodelassumptionoflow plecomplexityassociatedwithmeasuringouterproducts
Krausrankistypicallyonlyapproximatelytrue. of standard Gaussian vectors. Interestingly, this sample
ii) For the constrained norm minimizations (18) and (16) a complexitymatchesthefundamentallowerbounduptoa
prior error threshold is required. This additional model factorofrln(n),whererdenotestherankofthedensity
selectiontaskisimportantinactualapplicationsandcan operator in question. Thus, state tomography via low-
beanon-trivialtask;seec.f.,Ref.[64]. TheCPT-fit(13) rank matrix recovery from outer products of Gaussian
has the distinct advantage that the reconstruction can be vectors is close to optimal, at least for states that have
donewithoutsuchapriornoiseestimation. verylowrank.
10
We expect that similar results are true for quantum pro- (cid:107)e(cid:107) and λ, we first run a uniform reconstruction with dif-
(cid:96)2
cess tomography via low-rank matrix reconstruction of ferent m in the ideal case with λ = 0 and e = 0. Again,
Choimatrices. However,whileconceptuallysimilar,the wesimulate100trialswithHaarrandomunitarychannels. It
results regarding sample complexities of state tomogra- turnsoutthatalreadyform=320allreconstructionsaresuc-
phyprocedures[16,66]arenotdirectlyapplicabletopro- cessful with extremely high probability, see Figure 5(a). To
cesstomography. Weintendtoaddresssuchanextension putthisnumberintoperspective,notethatforn=8aunitary
infuturework. channelisgivenby63realparametersandthefulldimension
isdim(CPT(C8))=4032.
We reconstruct T from m = 320 noisy measurements
2. Numericalexample:ReconstructingtheToffoligate Toff
withdifferentvaluesof(cid:107)e(cid:107) withoutmodelmismatch(λ =
(cid:96)2
0), see Figure 5(b). For the trace and diamond norm re-
The Toffoli gate is a three qubit gate that has drawn a lot
constructions (16), (17), and (18) we set the error parame-
of attention from theorists as well as experimentalists. It ter η = (cid:107)e(cid:107) +10eps, where eps = 2−52 ≈ 2·10−16 is
is universal for classical computation and, together with the (cid:96)2
the machine precision. As predicted by Theorems 3 and 4,
Hadamard gate, also for quantum computation [67]. More- the reconstruction error (cid:107)Trec−T (cid:107) scales linearly in the
over, it has played an important role in the theory of gate 0 F
noisestrength(cid:107)e(cid:107) ,seeFigure5(b).TheCPT-fit(13)hasthe
sets [68] and can help to significantly reduce the number of (cid:96)2
smallest reconstruction error, see Figure 5(b). If one further
gates in quantum algorithms, such as in quantum error cor-
increases the number of measurements, then the reconstruc-
rection. It also has been implemented experimentally in nu- tionerror(cid:107)Trec−T (cid:107) decreasesfurther. Thisisguaranteed
clearmagneticresonance[12],linearoptics[13],trappedions 0 F
byTheorems3and4,butnotshownhere.
[14],andinsuperconductingcircuits[15]. Hence,theToffoli
gate is a good candidate to benchmark our process tomogra- Inordertoproberobustnessagainstmodelmismatches,we
physchemes. testthereconstructionofT definedin(27)fordifferentvalues
0
We demonstrate uniform recovery, i.e., we draw a fixed ofλ ∈ [0,1]. Forthesakeofclarity,wecompletelysuppress
number of measurement settings at random and keep them additivenoise(e = 0)andsettheerrorthresholdη = 10eps.
fixedthroughoutthesimulationinFigure5. Then,wealways The results are presented in Figure 5(c) and confirm the ro-
usethefirstmofthemforreconstructionswithmsettings. bustnessguaranteedbyTheorem4. Thereconstructionerror
Sincereconstructionviatheunconstrainedtracenormmin- (cid:107)Trec−T0(cid:107)F dependsroughlylinearlyonthenoisestrength
imization(15)isnumericallyinferiortotheotherreconstruc- (cid:107)e(cid:107) . Here,thediamondnormminimizations(17)and(18),
(cid:96)2
tions(Figure3(a)and(b))wewillnotinvestigateitinallsim- performworsethantheconstrainedtracenormminimization
ulations. (16)andtheCPT-fit(13).
There are two types of error sources: (i) imperfect mea-
surements give rise to measurement noise e ∈ Rm in our Inthecaseofmeasurementnoise, thereconstructionerror
model(12)and(ii)theimplementedchannelT0 ∈CPT(Cn) (cid:107)Trec−T0(cid:107)Fapproachestheoptimalvalue(cid:107)e(cid:107)(cid:96)2 inthelimit
of large m, see Figure 5(d) left plot. In the case of a model
could have violated the model assumption of a low Kraus
mismatch, the reconstruction error decreases below the mis-
rank. Here, we confirm our analytic stability result towards
matchparameterλandvanishesifmapproachesthefulldi-
botherrorsourcesnumerically.
mensionofCPT(Cn),seeFigure6ontheright.
Our theorems put minimal assumptions on the potential
noise corruption e ∈ Rm. In particular, it does not need to
Wefinditworthwhiletoshareoneinterestingnumericalob-
followaspecificstatisticalmodel. Forsakeofournumerical
servationontheunconstraineddiamondnormreconstruction
analysis, however, we draw e i.i.d. from a zero-mean Gaus- (17): Theoptimalvalue(cid:107)Trec(cid:107) ofthereconstructionseems
(cid:5)
siandistributionandscaleittoadesiredvalue. Suchanerror to decrease with the reconstruction error (cid:107)Trec−T (cid:107) , see
0 F
modelfrequentlyoccursinpractice: Whenestimatingexpec-
therightcolumnofFigure5. Hence,thisreconstructionpro-
tation values from observed frequencies such an error model
cedure does not only yield good approximations to the mea-
arisesnaturallyinthelimitofmanymeasurementspersetting. sured channel T , but its optimal value also provides some
0
In order to demonstrate robustness of our reconstructions, indicationofthereconstructionerror’ssize.
we set T to be a convex combination of the Toffoli gate
0
T and a completely depolarizing channel T (ρ) = One can exploit this observation by using both the CPT-
Toff dep
n−1Tr(ρ)1, fit (13) and the unconstrained diamond norm reconstruction
(17). The CPT-fit is the fastest reconstruction procedure and
T =(1−λ)T +λT , (27) yields the smallest error. Complementing this, the diamond
0 Toff dep
norm minimization provides an indication of what the error
whereλ ∈ [0,1]. Thedepolarizingchannelcorrespondstoa mightbe.IntheCPT-fit,thereconstructedmapTrecisalways
physically relevant error model that maximally violates our a quantum channel, i.e., Trec ∈ CPT(Cn). In contrast, the
model assumption of low Kraus rank; see, e.g., Ref. [69, solutionofthediamondnormminimizationcan,inprinciple,
Chapter8.3.4]. be any map in L(Cn). The latter also holds true for the un-
Inordertodetermineareasonablenumberofmeasurement constrainedtracenormreconstruction(15), butwecouldnot
settings m for the investigation of the noise dependencies observeasimilarfeatureofitsminimumvalue.