DTIC AD1033721: An Optimal Dissipative Encoder for the Toric Code

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An optimal dissipative encoder for the toric code
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2014 New J. Phys. 16 013023
(http://iopscience.iop.org/1367-2630/16/1/013023)
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An optimal dissipative encoder for the toric code
John Dengis1, Robert König1,3 and Fernando Pastawski2
1InstituteforQuantumComputingandDepartmentofAppliedMathematics,Universityof
Waterloo,200UniversityAvenueW,Waterloo,ONN2L3G1,Canada
2InstituteforQuantumInformationandMatter,CaliforniaInstituteofTechnology,
1200ECaliforniaBoulevard,Pasadena,CA91125,USA
E-mail:[email protected]
Received31October2013,revised3November2013
Acceptedforpublication9December2013
Published16January2014
NewJournalofPhysics16(2014)013023
doi:10.1088/1367-2630/16/1/013023
Abstract
We consider the problem of preparing specific encoded resource states for
the toric code by local, time-independent interactions with a memoryless
environment. We propose the construction of such a dissipative encoder which
converts product states to topologically ordered ones while preserving logical
information. The corresponding Liouvillian is made up of four local Lindblad
operators. For a qubit lattice of size L×L, we show that this process prepares
encodedstatesintime O(L),whichisoptimal.Thisscalingcomparesfavorably
with known local unitary encoders for the toric code which take time of order
(cid:127)(L2)andrequireactivetime-dependentcontrol.
1. Introduction and main result
Dissipation, while generally seen as detrimental for quantum computers, can nevertheless be
a useful resource if suitably engineered. Appropriately chosen system–bath couplings can
resultinnon-equilibriumdynamicswhereinitialstatesconvergetowardsomedynamicalsteady
state. This kind of ‘quantum reservoir engineering’ has been proposed as a viable approach to
the experimental preparation of interesting many-body states [11, 18]. Remarkable examples
include the preparation of pure states with long-range order in Bose–Einstein condensates [6],
photonic arrays [17], as well as topologically ordered states [2, 18]. More generally, Verstraete
etal[18]arguedthat,atleastinprinciple,anarbitraryquantumcomputationcanberealizedby
3Authortowhomanycorrespondenceshouldbeaddressed.
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NewJournalofPhysics16(2014)013023
1367-2630/14/013023+11$33.00 ©2014IOPPublishingLtdandDeutschePhysikalischeGesellschaft
NewJ.Phys.16(2014)013023 JDengisetal
dissipation. The corresponding process is similar to Feynman’s clock construction and has the
final state of the computation as its steady state. Subsequent work [8] following this program
proposed dissipative gadgets allowing the realization of different dissipative dynamics during
subsequenttimeintervals.
Hereweexaminethedissipativepreparationofspecifictopologicallyorderedstates.While
not realizing a fully dissipative computation, this basic primitive could act as a building block
inahybridschemewhereinitialstatesforquantumcomputationarepreparedbythermalization
and subsequent computations are performed in the usual framework of topological quantum
computation. We ask whether dissipative processes can be used to realize an encoder, i.e.
a map which turns states on individual physical qubits into encoded (many-qubit) states. In
contrast,previousworkonlyconsideredthedissipativepreparationofsomegroundstatewithout
guaranteesonthelogicalinformation.
It is worth mentioning that various unitary encoders are known for topologically ordered
systems. For the toric code [9] on an L×L lattice, Dennis et al [5] gave a unitary circuit with
two-local controlled-not (CNOT) gates of depth 2(L2) acting as an encoder. Bravyi et al [3]
showed for any evolution under a local time-dependent Hamiltonian acting as an encoder
requires time at least (cid:127)(L). In turn, Brown et al [4] present a duality transformation from a
two-dimensional (2D) cluster state to a topologically ordered state, which can be interpreted as
a geometrically local quantum circuit of matching depth. Dropping the requirement of locality,
Aguado and Vidal [1] gave an encoder with depth O(logL) with geometrically non-local two-
qubitgates.
The dissipative encoder considered here may be realized by designing suitable
system–environment interactions. This is to be contrasted with schemes involving error
correction, which generally consist of syndrome extraction by measurement and associated
correction operations. For the toric code, an encoding procedure of this form was given [7].
It involves active error correction operations similar to the minimal matching technique used
in[5].
Todefinethenotionofanencoderinmoredetail,consideraquantumerror-correctingcode
Q∼=(C2)⊗k ⊂(C2)⊗n encoding k logical qubits into n physical qubits. Informally, an encoder
isamaptakinganystate|9i∈(C2)⊗k intoitsencodedversion|9i∈Q⊂(C2)⊗n.(Thisnotion
implicitly assumes a choice of basis of Q.) Since we are interested in a physical system of n
qubits,wewillrequiretheencodertoconverta‘simple’unencodedinitialstateintoanencoded
logical state. That is, we ask that for a fixed subset A ,..., A of qubits and a fixed product
1 k
state|ϕi ⊗···⊗|ϕi ontheremainingqubits,theencodermaps
Ak+1 An
|9i ⊗|ϕi ⊗···⊗|ϕi 7→ |9i∈Q forall|9i∈(C2)⊗k. (1)
A1···Ak Ak+1 An
WeareinterestedinencodersrealizedbyevolutionunderaMarkovianmasterequation
d
ρ =L(ρ).
dt
HeretheLiouvillianhasLindbladform
X 1 n o
L(ρ)= L ρL†− L†L ,ρ
j j 2 j j
j
with Lindblad operators L acting locally on a constant number of qubits. We ask whether
j
the completely positive trace-preserving map (CPTPM) etL generated by L is an (approximate)
2
NewJ.Phys.16(2014)013023 JDengisetal
Figure 1. This figure indicates the relevant qubits in theorem 1, and their appropriate
initializationforencoding:qubits A ,A areinitializedinthestatetobeencoded.Each
1 2
qubitinB∪B0 isinthestate|0i,whilequbitsinC∪C0 areinthestate|+i.Thestateof
theremainingqubitsD canbe anarbitrary(mixed)state. Wealsoillustratethe support
of possible realizations for logical operators (X¯ ,Z¯ ) and (X¯ ,Z¯ ) associated with the
1 1 2 2
first and second logical qubit, respectively. Our encoding procedure requires choosing
realizations for each logical Pauli generator such that they overlap only at the initial
unencodedqubits.
encoder for sufficiently large times t, i.e. whether it can realize the map (1). Our result is the
following.
Theorem1. Let Q⊂(C2)⊗2L2 be the toric code consisting of 2L2 qubits on the edges
of a periodic L×L lattice. Consider the partition of the qubits into disjoint sets
A∪B∪B0∪C∪C0∪D showninfigure1.Thatis
• A={A , A } are two neighboring qubits having a common adjacent vertex v and
1 2 ∗
plaquette p ;
∗
• B ={B ,...,B } and C ={C ,...,C } are located along a vertical, respectively
1 L−1 1 L−1
horizontal,linepassingthrough A ;
1
• B0 ={B0,...,B0 } and C0 ={C0,...,C0 } are located along a horizontal, respectively
1 L−1 1 L−1
vertical,linepassingthrough A and
2
• D aretheremaining2(L−1)2 qubits.
There is a geometrically local Liouvillian L (with four-qubit Lindblad operators) such that the
following holds: for any state ρ on D∼=(C2)⊗2(L−1)2 and |9i∈(C2)⊗2, and for any (cid:15) >0, we
D
have
(cid:13) (cid:16) (cid:17) (cid:13)
(cid:13)etL |9ih9| ⊗|+ih+|⊗2(L−1)⊗|0ih0|⊗2(L−1)⊗ρ −|9ih9|(cid:13) 6(cid:15)
(cid:13) A BB0 CC0 D (cid:13)
1
whenever
t >(4ln(2))L+2ln(16(cid:15)−2). (2)
√
Inthisexpression,weusethetracenormkAk =tr A†A forHermitianoperators.
1
3
NewJ.Phys.16(2014)013023 JDengisetal
WewillgiveadetaileddescriptionoftherelevantLiouvillianinsection2.TheevolutionetL
implements a continuous-time version of a local error correction process somewhat analogous
to Toom’s rule: excitations move toward a single plaquette/vertex where they annihilate. An
analogousground-statepreparationschemeformoregeneralfrustration-freeHamiltonianswas
discussedin[18].However,guaranteesaboutencodedinformationappeartobehardertoobtain
in their generic scheme. Furthermore because their construction requires the injectivity [14]
propertyoftheassociatedprojectedentangledpairstate(PEPS)description(andhenceblocking
of sites), the resulting locality of the Lindblad operators will be slightly worse. In contrast,
our scheme directly exploits the stabilizer structure of the underlying code, resulting in a
comparatively simple Liouvillian. Indeed, our construction is optimal in terms of locality, i.e.
the number of particles involved in each Lindblad term [12]. Related locality constraints for
pure steady states were derived4 in [15, 16]. In our setup, the initial product state determines in
atransparentwaywhichcodestateisprepared.Ourworkgoesbeyondthemerecharacterization
of steady states by adding two key ingredients: the consideration of logical observables (which
arepreserved)andananalysisoftheconvergencetowardthegroundspaceofthetoriccode.
Thebound O(L)ontheconvergencetimeestablishedbytheorem1improvesonthe O(L2)
upperboundpredictedfortheanalogousconstructionin[18],withoutaguaranteeonthelogical
information. In fact, it is tight: there are initial states ρ which thermalize slowly, i.e. etL(ρ) is
far away from the code space Q for any time t (cid:28) L. In a separate work [10], we provide a
general no-go theorem in this direction: dissipative state preparation of topologically ordered
states requires at least a linear amount of time in L if the Liouvillian is local. Combined with
theorem 1, this implies that the construction presented here is optimal in terms of preparation
timeamongtheentireclassoflocalLiouvillians.
In summary, our work shows that dissipative processes can be used to implement an
encoder for the toric code. Intriguingly, this encoder is more time-efficient than the best known
unitary circuit. We stress, however, that both types of encoders need to be supplemented with
additionalmechanismsinthepresenceofnoise,especiallyiftheencodedinformationisfurther
processed. As discussed in [13], local Liouvillians such as the one considered here are not
suitableforthepreservationofencodedinformation.
2. Description of the Liouvillian
2.1. A Liouvillian for a general stabilizer code
We first describe a generic Liouvillian associated with a stabilizer code Q with stabilizer
generators {S } . We will subsequently specialize this to the toric code. Let P± = 1(I ±S )
j j∈S j 2 j
bethe projectionsontothe±1eigenspaces ofthestabilizer S .The codespaceQisthe ground
j
spaceoftheHamiltonian
 
1 X X
H = 2 |S|·I − Sj= P−j
j j
(the global energy shift is introduced for convenience). Our goal is to implement a local error-
correction strategy by dissipative evolution. Concretely, we associate a unitary Pauli correction
4 Intermsof[16,definition2],ourconstructioniscapableofpreparinganystateingroundspaceofthetoriccode
asaconditionallyasymptoticallystablestateofaLindbladdynamics.
4
NewJ.Phys.16(2014)013023 JDengisetal
operatorC witheachstabilizergenerator S suchthatC and S anticommute,i.e.{C ,S }=0.
j j j j j j
WedefinetheCPTPMs
T (ρ)=P+ρP++C P−ρP−C†.
j j j j j j j
−
Notethatbydefinition,T lowerstheenergyofthetermP intheHamiltonian,i.e.
j j
tr(T (ρ)P−)6tr(ρP−) foranystateρ. (3)
j j j
While after application of T , the stabilizer constraint defined by S is satisfied, its application
j j
maycreateanon-trivialsyndrome(excitation)foraneighboringstabilizer S ,k 6= j.Bydesign
k
(i.e.thechoiceofcorrectionoperatorsforthetoriccodediscussedbelow),repeatedapplication
ofall{T } eventuallyremovesallexcitations,resultinginastatesupportedonthecodespace
j j∈S
Q.ItwillbeconvenienttointroducetheaveragedCPTPM
1 X
T (ρ)= T (4)
av |S| j
j
whichrandomlychoosesasyndromeandappliestheassociatedcorrectionmap.
Bythecorrespondencediscussedin[20],eachlocalCPTPMT definesalocalLiouvillian
j
L =T −id. We are interested in the evolution under L=P L , which, by definition, is a
j j j j
sumofconstant-strengthlocalterms.ObservethatL=|S|(T −id).
av
2.2. Construction for the toric code
We now consider the toric code with qubits on the edges of an L×L (periodic) square lattice.
WeseparatetheHamiltonianinto
 
!
1 X 1 X
H = H(p)+H(v) where H(p) = L2I − Sp and H(v) = L2I − Sv ,
2 2
p∈S(p) v∈S(v)
where the former includes all plaquette and the latter includes all vertex terms. Here, we have
taken S and S tocorrespondtoplaquetteandvertexstabilizers,respectively,
p v
S = Z⊗4 = S = X⊗4 = .
p v
In other words, S is the product of Z-type Pauli operators acting on the edges bounding
p
plaquette p ∈S(p) and S is the product of Pauli operators acting on the qubits incident to
v
vertex v ∈S(v). To define the associated Pauli correction operators {Cv}v∈S(v) and {Cp}p∈S(p), let
uspartitionthesetofverticesandplaquettesinto
S(p) ={p }∪S(p)∪S(p) and S(v) ={v }∪S(v)∪S(v).
∗ → ↑ ∗ ← ↓
Here p isasingleplaquette,S(p) consistsofthe L−1plaquetteslyingonafundamentalcycle
∗ →
of the torus (which we refer to as the ‘equator’, running ‘horizontally’ or ‘east–west’ along the
torus) on which p is located, whereas S(p) are the remaining L2−L plaquettes. The vertex v
∗ ↑ ∗
aswellasthesetsS(v) andS(v) aredefinedsimilarlyontheduallattice,seefigure2.
← ↓
The subscript associated with these sets indicates the direction of movement of an
excitation under application of the local correction map. For example, a magnetic excitation
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NewJ.Phys.16(2014)013023 JDengisetal
(a) Correction operations for plaquettes (b) Correction operations for vertices
Figure2.ThisfiguredefinestheLiouvillianL(seetext).Inparticular,(a)definesthesets
S(p)andS(p).TheassociatedcorrectionoperationC consistsofasingle-qubitPauli-X.
→ ↑ p
It moves magnetic (plaquette-type) excitations to the neighboring plaquette according
(v) (v)
to the indicated arrow. Similarly, (b) defines the sets S and S . Electric (vertex-
← ↓
type)excitationsaremovedfromonevertextothenextaccordingtothesearrows.The
associated correction operator Cv is a single-qubit Pauli-Z. The qubits A1A2 carrying
the logical information are indicated in red. They are both part of the special plaquette
p∗,andincidenttothevertexv∗.Nocorrectionoperationactsonthequbits A1A2.
(caused by an X-error) on a plaquette p ∈S(p) will move to the neighboring plaquette to the
↑
northof p underapplicationofthecorrectionmapC .Thatis,wedefine
p
C = for p ∈S(p), C = for p ∈S(p)
p ↑ p →
C = forv ∈S(v), C = forv ∈S(v).
v ↓ v ←
Wewillset
C =C = I,
v∗ p∗
corresponding to a trivial correction operation (this is simply done for convenience). The
LiouvillianListhendefinedasinsection2.1.
3. Proof of theorem 1
The proof of theorem 1 relies on two basic statements. The first one concerns the logical
information encoded in the state: for suitable initial states, this information is preserved during
theevolution.Recallthatthetoriccodehastwoencodedqubits.Let
! !
X¯ = OX ⊗X , Z¯ = OZ ⊗Z , Y¯ =iX¯ Z¯
1 b A1 1 c A1 1 1 1
b∈B c∈C
! !
X¯ = OX ⊗X , Z¯ = OZ ⊗X , Y¯ =iX¯ Z¯
2 b A2 2 c A2 2 2 2
b∈B0 c∈C0
bethetwo-qubitlogicalPaulioperatorsdefinedbyfigure1,andletusset
P¯ =(cid:8)P¯ P¯ | P¯ ∈{I,X¯ ,Y¯ ,Z¯ }, P¯ ∈{I,X¯ ,Y¯ ,Z¯ }(cid:9),
2 1 2 1 1 1 1 2 2 2 2
where I istheidentityoperator.Theseoperatorsplayacrucialroleinthefollowingstatement.
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NewJ.Phys.16(2014)013023 JDengisetal
Lemma1(Preservationoflogicalinformation). For any two-qubit Pauli operator P =
P ⊗ P , P ∈{I,X ,Y ,Z },let P = P P , P ∈P¯ beitscorrespondinglogicalcounterpart.
1 2 j j j j 1 2 j 2
Consideraninitialstateoftheform
ρ =|9ih9| ⊗|+ih+|⊗2(L−1)⊗|0ih0|⊗2(L−1)⊗ρ , (5)
0 A BB0 CC0 D
wherethestateρ isarbitraryon(C2)⊗2(L−1)2.Then
D
tr(PetL(ρ ))=h9|P|9i forallt andall P ∈P . (6)
0 2
Proof.Let P = P ⊗ P bearbitrary.Observefirstthattherhsof(6)isequalto
1 2
h9|P|9i=tr(Pρ ) (7)
0
becauseofthedefinitionof P (cffigure2)andtheformofρ ,i.e.thefactthatthequbitsinBB0
0
0
and CC are +1-eigenstates of the single-qubit X and Z operators, respectively. In other words,
it suffices to show the expectation value tr(PetL(ρ )) is time independent for initial states ρ of
0 0
theform(5).Weclaimthatanevenstrongerstatementholds:wehave
(etL)†(P)= P forany P ∈P , (8)
2
i.e.anyobservablesoftheform P donotevolveintheHeisenbergpicture.
Toprove(8),observethattheone-parameterfamilyofunitalmaps{(etL)†} isgenerated
t>0
bytheadjointL† oftheLiouvillian,hence(8)isequivalentto
L†(P)=0.
Since L=P L is a sum of Liouvillians L =T −id associated with stabilizers j, it suffices
j j j j
toverifythatforevery j,wehaveT†(P)= P or
j
P+PP++P−C†PC P− = P. (9)
j j j j j j
Because P isalogicaloperator,itcommuteswitheachstabilizer,andhencealsotheprojections
P±. Equality (9) is in fact implied by [C†, P]=0, which we can verify by a case-by-case
j j
analysisforsingle-qubit(logical)operators P ∈{X¯ ,Z¯ ,X¯ ,Z¯ }(thegeneralcasethenfollows
1 1 2 2
sinceaproduct P P commuteswithC† ifeachfactor P does).Considerforexamplethecase
1 2 j j
where P = Z¯ .Wethenhavetoconsidertwocases:
1
(i) The correction j = p is associated with a plaquette. In this case C is a Pauli-X (or
p
¯
the identity). However, inspection of the support of Z (see figure 1) and the location
1
of the correction operators (see figure 2(a)) reveals that no correction operation C acts
p
onthesupportof Z¯ ,hence[C†,Z¯ ]=0asdesired.
1 p 1
(ii) Thecorrection j =visassociatedwithavertex.InthiscaseC isaPauli-Z (ortheidentity),
v
hence[C†,Z¯ ]=0holdstrivially. ut
v 1
Thesecondfundamentalstatementisaboutconvergencetothegroundspace.
Lemma2(Convergencetime). Let Q be the projection onto the code space of the toric code,
and let Q⊥ = I −Q be the projection onto the orthogonal complement. Let ρ be an arbitrary
initialstate.Thenwehave
tr(Q⊥etL(ρ))6(cid:15) forallt >(4ln(2))L+2ln(1/(cid:15)). (10)
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NewJ.Phys.16(2014)013023 JDengisetal
We have not optimized the constants in this bound as we are interested in the overall (linear)
scaling in L. The proof strategy is different from the arguments in [18] and may be of
independentinterest.
Proof.Considerthefunction f:S\{v , p }→N∪{0}definedasfollows:
∗ ∗
foraplaquette p:f(p)+1=lengthofapathmovingnorth-eastfrom p to p∗
foravertexv: f(v)+1=lengthofapathmovingsouth-westfromv tov∗.
Inotherwords,thefunctionexpressestheaxialdistancetov∗ and p∗,respectively.Foravertex
v (plaquette p), the quantity f(v) (f(p)) is the number of vertices (plaquettes) traversed by
an electric (magnetic) excitation on the primary (dual) lattice before reaching v (p ) along a
∗ ∗
path (v =v ,v ,...,v ,v ) (or (p = p , p ,..., p , p )). For the special vertex v∗ and
1 2 f(v) ∗ 1 2 f(p) ∗
the plaquette p∗, we set f(v∗)= f(p∗)=−1 for convenience (alternatively, we could omit
the discussion of the corresponding trivial correction operations altogether as the stabilizer
generatorsarelinearlydependent).
Thekeypropertyofthefunction f isthefactthatitiscompatiblewiththewayexcitations
are propagated under the correction operations. More precisely, for each stabilizer S , let
j
Pred(j)bethesetofcorrectionoperationsthatanticommutewithit,i.e.
Pred(j):={k ∈S |k 6= j and{C ,S } =0}.
k j +
Then f hasthepropertythat
k ∈Pred(j)implies f(k)> f(j)+1. (11)
Namely, whenever an excitation is created by C at S , one can be certain that a higher valued
k j
excitationhasbeenremovedat S .Wewillarguethatforanyα >1,wehave
k
tr(Q⊥etL(ρ))6e−(1−α−1m)t Xαf(j), (12)
j∈S
where m :=max |{j|k ∈Pred(j)}| is the maximal number of stabilizers a single correction
k∈S
operator can excite. In the case of the toric code, m =1, hence we obtain tr(Q⊥etL(ρ))6
e−t/222L (implyingtheclaim)bychoosingα =2andobservingthat
X XL−1 (cid:18)αL −1(cid:19)2 2α2L−1 (cid:12)
αf(j) =2α−1 αr+s =2α−1 < (cid:12) =22L.
α−1 (α−1)2(cid:12)α=2
j∈S r,s=0
Toprove(12),considertheobservable
X
D = αf(j)P−.
j
j∈S\{v∗,p∗}
Clearly D >Q⊥ foranyα >1,henceitsufficestoshowthattheexpectationvaluetr(cid:0)DetL(ρ)(cid:1)
isupperboundedbytherhsof(12),i.e.
tr(DetL(ρ))6e−(1−α−1m)t Xαf(j). (13)
j∈S
ConsidertheHeisenbergevolutionoftheprojectionoperators{P } .SincePaulioperatorseither
j j
commuteoranticommute,astraightforwardcalculationgives
(cid:26)0 if[C ,S ]=0,
L†k(P−j )= (I −P−)P−−P−P− if{Ck,Sj}=0.
j k j k k j
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NewJ.Phys.16(2014)013023 JDengisetal
In particular, the expectation values behave classically under the designed Liouvillian, i.e.
(writinghXi =tr(XetL(ρ))forbrevity)
t
dhP−j it = −hP−i + X h(I −P−)P−i −hP−P−i
dt j t j k t k j t
k∈Pred(j)
X
6 −hP−i + hP−i . (14)
j t k t
k∈Pred(j)
Accordingto(14),wehave
dhDi X X
t 6 −hDi + αf(j) hP−i
dt t k t
j k∈Pred(j)
X X
6 −hDi +α−1 αf(k)hP−i 1,
t k t
k j:k∈Pred(j)
where we used property (11) on the function f. According to the definition of m and D, this
implies
dhDi
t 6−(1−α−1m)hDi ,
t
dt
i.e. the expectation value decays exponentially. The claim (13) then follows from the fact that
P− 6 I forall j ∈S sincetheseareprojections,hencetr(Dρ)=hDi 6P αf(j).
j 0 j∈S ut
Withlemmas1and2,wearereadytoproveourmainresult.
Proofoftheorem1.Consideraninitialstateρ oftheform(5)andassumethatt >(4ln(2))L+
0
2ln((cid:15)−1). By lemma 2, we have tr(Q⊥etL(ρ ))6(cid:15). With the gentle measurement lemma (see
0
e.g.[19,lemma9.4.1]),thisimplies
√
ketL(ρ )−ρ¯0k 62 (cid:15), (15)
0 t 1
whereρ¯0 isdefinedasρ¯0 := QetL(ρ0)Q .Observethatthestateρ¯0 issupportedentirelyonthecode
t t tr(QetL(ρ0)) t
spaceQ.
Notethatforanytwostatesρ,σ wehavekρ−σk =max tr(P(ρ−σ))andtherefore
1 kPk61
|tr(P(ρ−σ))|6kρ−σk for any normalized operator P. Hence (15) implies that for any
1
normalizedlogicaloperator P,wehave
√
|tr(cid:0)P(etL(ρ )−ρ¯0)(cid:1)|62 (cid:15). (16)
0 t
Let|9i∈Qbethetargetencodedstate,i.e.thestatesatisfying(cf(7)and(8))
h9|P|9i=h9|P|9i=tr(Pρ )=tr(PetL(ρ )) foranylogicaloperator P.
0 0
Combining this with (16) shows that |9ih9| and ρ¯0 have approximately the same expectation
t
values for logical operators. Since both states are supported on the ground space Q for which
9