Quantifying the Performance of Quantum Codes

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by  C. Cafaro

Quantifying the Performance of Quantum Codes
Carlo Cafaro1, Sonia L’Innocente2, Cosmo Lupo3 and Stefano Mancini4,5
1,3,4School of Science and Technology, Physics Division,
University of Camerino, I-62032 Camerino, Italy
2School of Science and Technology, Mathematics Division,
University of Camerino, I-62032 Camerino, Italy
5INFN, Sezione di Perugia, I-06123 Perugia, Italy
Westudythepropertiesoferrorcorrectingcodesfornoisemodelsinthepresenceofasymmetries
and/or correlations bymeans of the entanglement fidelity and thecode entropy. First, we consider
adephasingMarkovianmemorychannelandcharacterizetheperformanceofbotharepetitioncode
and an error avoiding code (CRC and CDFS, respectively) in terms of the entanglement fidelity.
1 We also consider the concatenation of such codes (CDFS ◦CRC) and show that it is especially
1 advantageous in the regime of partial correlations. Finally, we characterize the effectiveness of the
0 codes CDFS, CRC and CDFS◦CRC by means of the code entropy and find, in particular, that the
2 effort required for recovering such codes decreases when the error probability decreases and the

n memory parameter increases. Second, we consider both symmetric and asymmetric depolarizing
a noisyquantummemorychannelsandperformquantumerrorcorrection viathefivequbitstabilizer
J codeC . Wecharacterizethiscodebymeansoftheentanglementfidelityandthecodeentropy
[[5,1,3]]

2 asfunctionoftheasymmetricerrorprobabilitiesandthedegreeofmemory. Specifically,weuncover
1 that while the asymmetry in the depolarizing errors does not affect the entanglement fidelity of
the five qubit code, it becomes a relevant feature when the code entropy is used as a performance

] quantifier.
h
p
PACSnumbers: Quantum ErrorCorrection(03.67.Pp); Decoherence (03.65. Yz).
-
t
n
a
I. INTRODUCTION
u
q
[ Quantum error correction (QEC) is a theoretical scheme developed in quantum computing to defend quantum

coherence against environmental noise. There are different methods for preserving quantum coherence. One possible
1
techniqueexploitstheredundancyinencodinginformation. Thisschemeisknownas”quantumerrorcorrectingcodes”
v
9 (QECCs). For a comprehensive introduction to QECCs, we refer to [1]. Within such scheme, information is encoded
0 in linear subspaces (codes) of the total Hilbert space in such a way that errors induced by the interaction with the
4 environment can be detected and corrected. The QECC approach may be interpreted as an active stabilization of a
2 quantumstateinwhich,bymonitoringthesystemandconditionallycarryingonsuitableoperations,onepreventsthe
.
1 loss of information. In detail, the errors occur on a qubit when its evolution differs from the ideal one. This happens
0 by interaction of the qubit with an environment. Another possible approach is the so-called noiseless codes (also
1 knownasdecoherencefree subspaces(DFSs) or erroravoidingcodes). For a comprehensiveintroductionto DFSs, we
1 refer to [2]. It turns out that for specific open quantum systems (noise models in which all qubits can be considered
:
v symmetrically coupled with the same environment), it is possible to design states that are hardly corrupted rather
i than states that can be easily corrected (in this sense, DFSs are complementary to QECCs). In other words, it is
X
possibletoencodeinformationinlinearsubspacessuchthatthedynamicsrestrictedtosuchsubspacesisunitary. This
r
implies that no information is lost and quantum coherence is maintained. DFS is an example of passive stabilization
a
of quantum information.
The formal mathematical description of the qubit-environment interaction is usually given in terms of quantum
channels. Quantumerrorcorrectionisusuallydevelopedundertheassumptionofi.i.d. (identicallyandindependently
distributed) errors. These errormodels arecharacterizedby memorylesschannels Λ suchthat n-channeluses is given
by Λ(n) = Λ n. In such cases of complete independent decoherence, qubits interact with their own environments
⊗
which do not interact with each other. However, in many physical situations, qubits do interact with a common
environment which unavoidably introduces correlations in the noise. For instance, there are situations where qubits
in a ion trap set-up are collectively coupled to their vibrational modes [3]. In other situations, different qubits in a
quantum dot design are coupled to the same lattice, thus interacting with a common thermal bath of phonons [4].
The exchange of bosons between qubits causes spatial and temporal correlations that violate the condition of error
independence [5]. Memory effects then arise among channel uses with the consequence that Λ(n) = Λ n. Recent
⊗
6
studies tryto characterizethe effectofcorrelationsonthe performanceofQECCs[6–11]. Itappearsthatcorrelations
mayhavenegative[7]orpositive[8]impactonQECCsdependingonthefeaturesoftheerrormodelbeingconsidered.
A part from being correlated, noise errors may also be asymmetric. Most of the quantum computing devices [12]
are characterizedby relaxationtimes (τ ) that are one-two orders of magnitude largerthan the corresponding
relaxation
2
dephasing times (τ ). Relaxation leads to both bit-flip and phase-flip errors, whereas dephasing (loss of
dephasing
phase coherence) only leads to phase-flip errors. Such asymmetry between τ and τ translates to an
relaxation dephasing
asymmetry in the occurrence probability of bit-flip (p ) and phase-flip errors (p ). The ratio pZ is known as the
X Z pX
channelasymmetry. Quantumerrorcorrectionschemesshouldbe designedinsucha waythatno resources(time and
qubits)arewastedinattemptingtodetectandcorrecterrorsthatmayberelativelyunlikelytooccur. Quantumcodes
should be designed in order to exploit this asymmetry and provide better performance by neglecting the correction
of less probable errors [13–15]. Indeed, examples of efficient quantum error-correcting codes taking advantage of
this asymmetry are given by families of codes of the Calderbank-Shor-Steane(CSS) type (asymmetric stabilizer CSS
codes) [16, 17].
Devising new goodquantum codes for independent, correlatedandasymmetric noise models is a highly non trivial
problem. However, it is usually possible to manipulate in a smart way existing codes to construct new ones suitable
for more general error models and with higher performances [18, 19]. Concatenation is perhaps one of the most
successful quantum coding tricks employed to produce new codes from old ones. The concept of concatenated codes
wasfirstintroducedinclassicalerrorcorrectingschemesbyForney[20]. Roughlyspeaking,concatenationisamethod
of combining two codes (an inner and an outer code) to form a larger code. In classical error correction, Forney has
carriedonextensivestudiesofconcatenatedcodes,howtochoosetheinnerandoutercodes,andwhaterrorprobability
threshold values can be achieved. The first applications of concatenatedcodes in quantum error correctionappear in
[21, 22]. In the quantum setting, concatenated codes play a key role in fault tolerant quantum computation and in
constructing good degenerate quantum error correcting codes.
Obviously enough, the choice of a code depends on the error model. However, in selecting a code rather than
another,one shouldtake into accounttheir performanceonthe errormodel. Since there is no single quantity holding
all information on a code, different quantum code performance quantifiers have appeared into the QEC literature
[23]. For instance, in [24] it is suggested that the entanglement fidelity [25] is the important quantity to maximize
in schemes for quantum error correction. In [26], it is proposed that the code entropy is an important quantitative
measureofthequantumcorrectionschemesbydeterminingahierarchicalstructureinthesetofprotectedspaces[27].
Motivatedbyallthe abovementionedconsiderations,inthis workwestudy the propertiesoferrorcorrectingcodes
for noise models in the presence of asymmetries and/or correlations by means of the entanglement fidelity and the
code entropy. First, we consider a dephasing Markovianmemory channel and characterizethe performance of both a
repetition code and an error avoiding code ( and , respectively) in terms of the entanglement fidelity. We
RC DFS
C C
also consider the concatenationof suchcodes ( ) and show that it is especially advantageousin the regime
DFS RC
C ◦C
of partial correlations. Finally, we characterize the effectiveness of the codes , and by means
DFS RC DFS RC
C C C ◦C
of the code entropy. We show that the effort required for recovering such codes decreases when the error probability
decreases and/or the memory parameter increases. Second, we consider an asymmetric depolarizing noisy quantum
memory channel and perform quantum error correction via the five qubit stabilizer code . We characterize
[[5,1,3]]
C
thiscodebymeansoftheentanglementfidelityandthecodeentropyasfunctionoftheasymmetricerrorprobabilities
andthedegreeofmemory. Specifically,weuncoverthatwhiletheasymmetryinthedepolarizingerrorsdoesnotaffect
the entanglementfidelity ofthe five qubit code, it becomes a relevantfeature when quantifying the code performance
by means of the code entropy.
The layout of this article is as follows. In Section II, we present few theoretical tools employed in the rest of
the article. In particular, we emphasize the conceptual and operational meanings of the entanglement fidelity and
code entropy, the two code performance quantifiers employed here. In Section III, we study a dephasing Markovian
memorychannelandcharacterizetheperformanceofbotharepetitioncodeandanerroravoidingcode( and ,
RC DFS
C C
respectively)intermsoftheentanglementfidelity. Wealsoconsidertheconcatenationofsuchcodes( )and
DFS RC
C ◦C
show that it is especially advantageousin the regime of partialcorrelations. Finally, we characterizethe effectiveness
ofthecodes , and bymeansofthecodeentropy. InSectionIV,weanalyzebothsymmetricand
DFS RC DFS RC
C C C ◦C
asymmetric depolarizing noisy quantum memory channels and perform quantum error correction via the five qubit
stabilizer code . We characterize this code by means of the entanglement fidelity and the code entropy as
[[5,1,3]]
C
function of the asymmetric error probabilities and the degree of memory. We find, among other things, that while
asymmetric depolarizing errors do not affect the entanglement fidelity of the five qubit code, they do affect its code
entropy. A summary of the uncovered results and our comments on the usefulness of both the entanglement fidelity
and code entropy as suitable code performance quantifiers for the error models considered appear in Section V.
II. PRELIMINARIES
In this Section, we briefly describe some basic features of quantum error correcting codes and decoherence free
subspaces. Finally, we point out the conceptual and operational meanings of the entanglement fidelity and code
entropy, the two code performance quantifiers employed in this work.
3
A. On quantum error correction schemes
1. Quantum error correcting codes
The formal mathematical description of a system-environment interaction is usually given in terms of quantum
channels [23]. When a quantum system in a state ρ (belonging to the set of density operators σ( ) defined over the
H
Hilbert space of dimension d associated to the system) is exposed to the interaction with the environment, the
H
resultis the noisystate describablebythe actionofacompletely positiveandtracepreservingmapΛ:σ( ) σ( )
H → H
with,
Λ(ρ)= AkρA†k. (1)
k
X
The operators A : (Kraus operators) define the errors affecting the system’s state ρ. It is well known [23]
k
H → H
that a standardquantumerrorcorrectionschemeis characterizedby a subspace suchthat forsome Hermitian
C ⊆ H
andpositivematrixofcomplexscalarsΓ=(γ )thecorrespondingprojectionoperatorP fulfillstheerrorcorrection
lm
condition, C
P A†lAmP =γlmP , (2)
C C C
for l, m = 1,..., d2. The matrix Γ is also known as the error correction matrix. The action of the correctable error
operators A A on vectors spanning (codewords) allows to define the (completely positive and
correctable k
{ } ⊆ { } C
trace preserving) recovery map R . The composition of the recovery map with the error map gives the
l
R ↔ { } R A
recoveredchannel,
def
Λrec.(ρ) = ( Λ)(ρ)= (RlAk)ρ(RlAk)†. (3)
R◦
l k
XX
For a comprehensive introduction to QECCs, we refer to [1].
2. Decoherence Free Subspaces
Following [2], we mention few relevant properties of DFSs. Consider the dynamics of a closed system composed of
a quantum system coupled to a bath . The unitary evolution of the closed system is described by the combined
Q B
system-bathHamiltonianH , H =H I +H I +H , H = E B . The operatorH (H )is the
system (bath) Hamiltonian,tIot (Ito)t is thQe⊗idenBtity oBp⊗eraQtor ofintthe syinsttem (bαathα)⊗, Eαare the error geneQratorBs acting
α
solely on while B act on tQhe bBath. The last term in H is the interacPtion Hamiltonian.
α tot
Q
Asubspace ofthetotalsystemHilbertspace isadecoherencefreesubspaceifandonlyif: i)E ψ =c ψ
DFS α α
withc C,foHrallstates ψ spanning ,andforHeveryerroroperatorE inH . Inotherwords,all|baisissta|tesi
α DFS α int
∈ | i H
spanning aredegenerateeigenstatesofalltheerrorgeneratorsE ;ii) and areinitiallydecoupled;iii)H ψ
DFS α
has nooveHrlapwith states inthe subspace orthogonalto . To establisQha dirBectlink betweenQECCsandDQF|Ssi,
DFS
H
itismoreconvenienttopresentanalternativeformulationofDFSsintermsoftheKrausoperatorsumrepresentation.
Withinsuchdescription,the evolutionofthe system densitymatrixiswrittenas, ρ (t)=Tr U(ρ ρ )U =
†
Q Q B Q⊗ B
kAkρ (0)A†k, where U = e−iHtott is the unitary evolution operator for the system-bath clos(cid:2)ed system and (cid:3)the
initial baQth density matrix ρ equals λ n n. The KrausoperatorsA (satisfying the normalizationcondition)
P B n n| ih | k
aoruetgthivaetnabNyAk =-di√mλennshimon|aUl|snuibwspitahcePkA†kAokf =IisQawDitFhSkif=an(dn,omnl)yaifndallwKherareus|mopiearnadto|rnsihaarveebaanthidsetnattiecsa.lIutntiutarnrys
DFS DFS
PH H
representation (in the basis where the first N states span ) upon restriction to it, up to a multiplicative
DFS DFS
H
constant,
g U(DFS) 0
A = k , (4)
k Q0 A¯
(cid:18) k (cid:19)
where g =√n mU n and U =e iHct with H =H +H . Furthermore, A¯ is an arbitrarymatrix that acts on
k c c − c int k
h | | i B (DFS)
(with = ) and may cause decoherence there; U is U restrictedto . Now recallthat
iHnD⊥oFrSdinary QHECCHsD,FitSi⊕s pHoD⊥ssFiSble to correct the errors induced by a gQiven set oQf Kraus operatoHrsDFAS if and only if,
k
{ }
λ I 0
RrAk = r0k C B , (5)
rk
(cid:18) (cid:19)
4
r and k, or equivalently,
∀
γ I 0
kk′
A†kAk′ =(cid:18) 0 C A¯†kA¯k′ (cid:19), (6)
where R are the recoveryoperators. The first block in the RHS of (5) acts on the code space while the matrices
r
{ } C
B act on where = . From (4) and (5), it follows that DFS can be viewed as a special class of QECCs,
rk ⊥ ⊥
C H C⊕C
where upon restriction to the code space , all recoveryoperatorsR are proportionalto the inverse of the system
r
C Q
evolution operator, R U(DFS) †. Assuming that R is proportional to the dagger of U(DFS) , from (4) and (6) it
r r
∝
Q Q
also turns out that A (cid:16)U(DFS) u(cid:17)pon restriction to . Furthermore, from (4) and (6), it follows that γ = g g .
However, while in thekQ∝ECCQs case γ is in general aCfull-rank matrix (non-degenerate code), in the DFkSks′casek∗thki′s
kk′
matrix has rank 1. In conclusion, a DFS can be viewed as a special type of QECC, namely a completely degenerate
quantum errorcorrectingcode where upon restrictionto the code subspace allrecoveryoperatorsareproportionalto
the inverse ofthe system evolutionoperator. As a side remark,in view of this lastobservationwe point out that it is
reasonable to quantify the performance of both active and passive QEC schemes by means of the same performance
measure. In what follows, we will briefly describe the conceptual and operational meanings of the entanglement
fidelity and code entropy, respectively.
B. On code performance quantifiers: entanglement fidelity and code entropy
1. Entanglement Fidelity
Entanglement fidelity is a useful measure of the efficiency of QECCs. It is a quantity that keeps track of how
well the state and entanglement of a subsystem of a larger system are stored, without requiring the knowledge of
the complete state or dynamics of the larger system. More precisely, the entanglement fidelity is defined for a mixed
state ρ= p ρ =tr ψ ψ in terms of a purification ψ to a reference system . The purification
ψ encodes iallioif theHinRf|ormihat|ion in ρ. Entanglement fide|litiy∈isHa⊗mHeaRsure of how well the chaHnRnel Λ preserves the
| i P
entanglement of the state with its reference system . The entanglement fidelity is formally defined as follows
R
H H
[25],
def
(ρ, Λ) = ψ (Λ I )(ψ ψ ) ψ , (7)
F h | ⊗ HR | ih | | i
where ψ is any purificationof ρ, I is the identity mapon σ( ) and Λ I is the evolutionoperatorextended
to the|spaice ,spaceonwhicHhRρ hasbeenpurified. Ifthe qHuRantumop⊗eraHtiRonΛ iswritteninterms ofits Kraus
R
H⊗H
operator elements {Ak} as, Λ(ρ)= kAkρA†k, then it can be shown that the operational expression of (7) becomes
[24],
P
F(ρ, Λ)= tr(Akρ)tr A†kρ = |tr(ρAk)|2. (8)
Xk (cid:16) (cid:17) Xk
This expression for the entanglement fidelity is very useful for explicit calculations. Finally, assuming that
Λ:σ(H)∋ρ7−→Λ(ρ)= AkρA†k ∈σ(H), dimCH=N (9)
k
X
and choosing a purification described by a maximally entangled unit vector ψ for the mixed state ρ =
1 I , we obtain | i ∈ H⊗H
dimCH H
1 1
I , Λ = trA 2. (10)
F N H N2 | k|
(cid:18) (cid:19) k
X
Theexpressionin(10)representstheentanglementfidelitywhennoerrorcorrectionisperformedonthenoisychannel
Λ in (9).
5
2. Code entropy
Aconvenientquantifierofthe actionofthe mapΛin (1)ona initialquantumstate ρ is representedby the entropy
change (Λ(ρ)) (ρ). It can be shown that [28],
S −S
0 (Λ(ρ)) (σ) (ρ) (Λ(ρ))+ (σ), (11)
≤|S −S |≤S ≤S S
where σ =σ(Λ, ρ)is the so-calledLindbladmatrix,anauxiliaryquantumstatewith matrixelements σ defined as,
lm
def
σlm = Tr ρA†lAm , (12)
(cid:16) (cid:17)
with l, m = 1, 2,.., d2. The bounds provided in (11) are the so-called Lindblad bounds [28]. The quantity (σ) is
S
known as the entropy exchange of the operation Λ. From (11), it follows that if ρ is pure, (ρ) = 0 and therefore
S
(σ)= (Λ(ρ)).
S S
Aspointedoutearlier,astandardquantumerrorcorrectionschemeforagivenquantumoperationΛischaracterized
by a subspace such that for some Hermitian and positive error correction matrix of complex scalars Γ=(γ ) the
lm
C
corresponding projection operator fulfills the relation (2). A quantum error correcting code for Λ is determined by
the subspace relatedto P . The rankof Γ is bounded aboveby the dynamicalChoimatrix associatedwith Λ(ρ),
Λ
C D
def
DΛ = Tr A†lAm . (13)
(cid:16) (cid:17)
For orthogonal Kraus operators A , the Choi matrix equals dδ with 0 d representing the eigenvalues of .
l l lm l Λ
{ } ≤ D
The rank of equals the minimal number of Kraus operators needed to describe Λ. Given a code for Λ and
Λ
D C
assuming that the initial quantum state ρ belongs to the code subspace (that is, P ρP = ρ), it turns out that
σ = γ [26]. In other words, the error correction matrix Γ equals the Lindblad matCrixCσ(Λ, ρ) provided that the
lm lm
initial quantum state ρ belongs to the code subspace. Motivated by these considerations,given a quantum operation
Λ with Kraus operators A and a code with error correction matrix Γ, Kribs and coworkers named the von
l
{ } C
Neumann entropy (Λ, ),
S C
def
(Λ, ) = (Γ)= Tr(ΓlogΓ), (14)
S C S −
the entropy of relative to Λ [26]. They show that,
C
0 (Λ, ) logD, (15)
≤S C ≤
where (Λ, ) = 0 iff is a unitarily correctable code for Λ and (Λ, ) = logD iff is a non-degenerate code for
S C C S C C
Λ. From a conceptual point of view, the entropy of a code relative to a quantum noisy channel Λ can be viewed
C
as a measure quantifying the nearness of the givenerror correctingcode to a decoherence free subspace C . The
DFS
C
closer is to a C , the smaller is the code entropy (Λ, ). The smaller is the code entropy, the smaller is the
DFS
C S C
amount of effort required to recover the quantum state corrupted by the noise by means of a suitable error recovery
operation. The simplest scenario occurs for codes with zero entropy. Such codes are known, as we said, as unitarily
correctable codes and can be recovered with a single unitary operation. In particular decoherence free subspaces are
a special class of unitarily correctable codes where the recovery is given by the trivial identity operation.
We recallthatinstandardactiveerrorcorrectionschemes,the actionof the recoveryoperationpushesallthe noise
into the ancillary qubits, so that errors are eliminated when the ancilla is traced out [29]. That said, we emphasize
that the numerical value of the code entropy quantifies the number of ancilla qubits needed to perform a recovery
operation[26] and the rank of Γ gives the number of Kraus operatorsnecessary to describe the action of Λ restricted
to and therefore the number of Kraus operators necessary for a recoveryoperation.
C
III. MODEL I: DEPHASING MARKOVIAN MEMORY CHANNEL
In this Section, we consider a dephasing Markovian memory channel and characterize the performance of both a
repetition code and an error avoiding code ( and , respectively) in terms of the entanglement fidelity. In
RC DFS
C C
particular, we consider the concatenation of such codes ( ) and show that it is especially advantageous in
DFS RC
C ◦C
the regime of partial correlations. Finally, we characterize the effectiveness of the codes , and
DFS RC DFS RC
C C C ◦C
by means of the code entropy. We show that the effort required for recovering such codes decreases when the error
probability decreases and the memory parameter increases.
6
A. Entanglement fidelity-based analysis
Repetitioncodeforcorrelatedphaseflips. ThemodelconsideredisadephasingquantumMarkovianmemorychannel
Λ(n)(ρ). In explicit terms, we consider n qubits and Markoviancorrelated errors in a dephasing quantum channel,
1
Λ(n)(ρ)d=ef pin|in−1pin−1|in−2...pi2|i1pi1(Ain ⊗...⊗Ai1)ρ(Ain ⊗...⊗Ai1)†, (16)
i1,..X.,in=0
def def
where A = I, A = Z are Pauli operators. Furthermore the conditional probabilities p are given by,
0 1 ik|ik−1
p =(1 µ)p +µδ , p =1 p,p =p, (17)
ik|ik−1 − ik ik,ik−1 ik=0 − ik=1
with,
1
p p ...p p =1. (18)
in|in−1 in−1|in−2 i2|i1 i1
i1,..X.,in=0
To simplify our notation, we may choose to omit the symbol of tensor product ” ” in the future, A ... A
⊗ in⊗ ⊗ i1 ≡
A ...A . Furthermore, we may choose to omit the bar ”” in p and simply write the conditional probabilities
asinp i.1 QEC is performed via the three-qubit repetition|code.ikA|iljthough the error model considered is not truly
ikij
quantum, we can gain useful insights for extending error correction techniques to fully quantum error models in the
presence of partial correlations. The performance of quantum error correcting codes is quantified by means of the
entanglement fidelity as function of the error probability p and degree of memory µ. By considering the case of (16)
with n =3, it follows that the error superoperator associated to channel is defined in terms of the following error
A
operators,
7 7
A←→{A′0,.., A′7} with Λ(3)(ρ)d=ef A′kρA′k† and, A′k†A′k =I8×8. (19)
k=0 k=0
X X
In an explicit way, the error operators A ,.., A are given by,
{ ′0 ′7}
A = p˜(3)I1 I2 I3, A = p˜(3)Z1 I2 I3, A = p˜(3)I1 Z2 I3,
′0 0 ⊗ ⊗ ′1 1 ⊗ ⊗ ′2 2 ⊗ ⊗
q q q
A = p˜(3)I1 I2 Z3, A = p˜(3)Z1 Z2 I3, A = p˜(3)Z1 I2 Z3,
′3 3 ⊗ ⊗ ′4 4 ⊗ ⊗ ′5 5 ⊗ ⊗
q q q
A = p˜(3)I1 Z2 Z3, A = p˜(3)Z1 Z2 Z3, (20)
′6 6 ⊗ ⊗ ′7 7 ⊗ ⊗
q q
where the coefficients p˜(3) for k =1,.., 7 read,
k
p˜(3) = p2 p , p˜(3) =p p p , p˜(3) =p p p , p˜(3) =p p p ,
0 00 0 1 00 10 0 2 01 10 0 3 00 01 1
p˜(3) = p p p , p˜(3) =p p p , p˜(3) =p p p , p˜(3) =p2 p , (21)
4 10 11 0 5 01 10 1 6 01 11 1 7 11 1
with,
p = (1 p), p =p, p =((1 µ)(1 p)+µ),
0 1 00
− − −
p = (1 µ)(1 p), p =(1 µ)p, p =((1 µ)p+µ). (22)
01 10 11
− − − −
Then consider a repetition code that encodes 1 logical qubit into 3-physical qubits. The codewords are given by,
def def
0 0 = +++ , 1 1 = , (23)
L L
| i→| i | i | i→| i |−−−i
where |±i d=ef |0i√±2|1i. The set of error operators satisfying the detectability condition [30], PCA′kPC = λA′kPC, where
P = 0 0 + 1 1 is the projector operator on the code subspace =Span 0 , 1 is given by,
L L L L L L
C | ih | | ih | C {| i | i}
= A , A , A , A , A , A , A . (24)
Adetectable { ′0 ′1 ′2 ′3 ′4 ′5 ′6}⊆A
7
The only non-detectable error is A . Furthermore, since all the detectable errorsare invertible, the set of correctable
′7
errors is such that A†correctableAcorrectable is detectable. It follows then that,
= A , A , A , A . (25)
Acorrectable { ′0 ′1 ′2 ′3}⊆Adetectable⊆A
The action of the correctable error operators on the codewords 0 and 1 is given by,
correctable L L
A | i | i
(3) (3) (3) (3)
|0Li → A′0|0Li= p˜0 |+++i, A′1|0Li= p˜1 |−++i, A′2|0Li= p˜2 |+−+i, A′3|0Li= p˜3 |++−i
q q q q
1 A 1 = p˜(3) , A 1 = p˜(3) + , A 1 = p˜(3) + , A 1 = p˜(3) +(2.6)
| Li → ′0| Li 0 |−−−i ′1| Li 1 | −−i ′2| Li 2 |− −i ′3| Li 3 |−− i
q q q q
The two four-dimensional orthogonal subspaces V0L and V1L of H23 generated by the action of Acorrectable on |0Li
and 1 result,
L
| i
0L =Span v0L = +++ , v0L = ++ , v0L = + + , v0L = ++ , (27)
V 1 | i 2 |− i 3 | − i 4 | −i
and, (cid:8)(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:9)
(cid:12) (cid:12) (cid:12) (cid:12)
1L =Span v1L = , v1L = + , v1L = + , v1L = + , (28)
V 1 |−−−i 2 | −−i 3 |− −i 4 |−− i
respectively. Notice that V0(cid:8)L(cid:12)(cid:12)⊕V(cid:11)1L =H23. The(cid:12)(cid:12) rec(cid:11)overy superop(cid:12)(cid:12)erat(cid:11)or R↔{Rl} w(cid:12)(cid:12)ith(cid:11)l =1,..,4 is(cid:9)defined as [31],
1
R d=efV viL viL , (29)
l l l l
i=0
X(cid:12) (cid:11)(cid:10) (cid:12)
(cid:12) (cid:12)
where the unitary operator V is such that V viL = i for i 0, 1 . Substituting (27) and (28) into (29), it
l l l | Li ∈ { }
follows that the four recovery operators R , R , R , R are given by,
1 2 3 4
{ (cid:12) (cid:11) }
(cid:12)
R = 0 0 + 1 1 , R = 0 ++ + 1 + ,
1 L L L L 2 L L
| ih | | ih | | ih− | | ih −−|
R = 0 + + + 1 + , R = 0 ++ + 1 + . (30)
3 L L 4 L L
| ih − | | ih− −| | ih −| | ih−− |
Using simple algebra, it turns out that the 8 8 matrix representation [R ] with l = 1,..,4 of the recovery operators
l
×
is given by,
[R ]=E +E , [R ]=E +E , [R ]=E +E , [R ]=E +E , (31)
1 11 88 2 12 87 3 13 86 4 14 85
where E is the 8 8 matrix where the only non-vanishingelement is the one locatedin the ij-positionandit equals
ij
×
1. The action of this recovery operation on the map Λ(3)(ρ) in (19) leads to,
R
7 4
Λ(r3ec)over(ρ)≡ R◦Λ(3) (ρ)d=ef (RlA′k)ρ(RlA′k)†. (32)
(cid:16) (cid:17) Xk=0Xl=1
We want to describe the action of Λ(3) restricted to the code subspace . Therefore, we compute the 2 2 matrix
R◦ C ×
representation [RA ] of each R A with l=1,.., 4 and k =0,.., 7 where,
l ′k l ′k
|C
[RlA′k] d=ef h10L|RRlAA′k|00Li h10L|RRlAA′k|11Li . (33)
|C (cid:18)h L| l ′k| Li h L| l ′k| Li(cid:19)
Substituting (26) and (30) into (33), it turns out that the only matrices [R A ] with non-vanishing trace are given
l ′k
by, |C
[R A ] = p˜(3) 1 0 , [R A ] = p˜(3) 1 0 ,
1 ′0 0 0 1 2 ′1 1 0 1
|C q (cid:18) (cid:19) |C q (cid:18) (cid:19)
[R A ] = p˜(3) 1 0 , [R A ] = p˜(3) 1 0 . (34)
3 ′2 2 0 1 4 ′3 3 0 1
|C q (cid:18) (cid:19) |C q (cid:18) (cid:19)
8
Therefore, the entanglement fidelity (3) (µ, p) defined as,
Fphase
7 4
1 1 2
Fp(3h)ase(µ, p)d=efF(3)(cid:18)2I2×2, R◦Λ(3)(cid:19)= (2)2 kX=0Xl=1(cid:12)tr(cid:16)[RlA′k]|C(cid:17)(cid:12) , (35)
(cid:12) (cid:12)
results, (cid:12) (cid:12)
(3) (µ, p)=µ2 2p3 3p2+p +µ 4p3+6p2 2p + 2p3 3p2+1 . (36)
Fphase − − − −
DFS for correlated phase flips. By co(cid:0)nsidering the(cid:1)case o(cid:0)f (16) with n=2(cid:1)qub(cid:0)its it follows t(cid:1)hat the error superop-
erator associated to channel is defined in terms of the following error operators,
A
3 3
A←→{A′0,.., A′3} with Λ(2)(ρ)d=ef A′kρA′k† and, A′k†A′k =I4×4. (37)
k=0 k=0
X X
In an explicit way, the error operators A ,.., A are given by,
{ ′0 ′3}
A′0 = p˜(02)I1⊗I2, A′1 = p˜(12)Z1⊗I2, A′2 = p˜(22)I1⊗Z2, A′3 = p˜(32)Z1⊗Z2, (38)
q q q q
where the coefficients p˜(2) for k =0,.., 3 read,
k
p˜(2) =p p , p˜(2) =p p , p˜(2) =p p , p˜(2) =p p . (39)
0 00 0 1 10 0 2 01 1 3 11 1
We encode our logical qubit with a simple decoherence free subspace of two qubits given by [2],
def def
0 0 = 01 and, 1 1 = 10 . (40)
L L
| i−→| i | i | i−→| i | i
The set of error operators satisfying the detectability condition [30], P A P = λ P , where P = 0 0 +
C ′k C A′k C C | Lih L|
1 1 is the projector operator on the code subspace =Span 0 , 1 is given by,
L L L L
| ih | C {| i | i}
Adetectable ={A′0, A′3}⊆A. (41)
Furthermore, since all the detectable errors are invertible, the set of correctable errors is such that
A†correctableAcorrectable is detectable. It follows then that,
= . (42)
correctable detectable
A A ⊆A
The action of the correctable error operators on the codewords 0 and 1 is given by,
correctable L L
A | i | i
0 A 0 = p˜(2) 01 , A 0 = p˜(2) 01 , 1 A 1 = p˜(2) 10 , A 1 = p˜(2) 10 . (43)
| Li→ ′0| Li 0 | i ′3| Li − 3 | i | Li→ ′0| Li 0 | i ′3| Li − 3 | i
q q q q
The two one-dimensional orthogonal subspaces V0L and V1L of H22 generated by the action of Acorrectable on |0Li
and 1 are given by,
L
| i
0L =Span v0L = 01 and, 1L =Span v1L = 10 . (44)
V 1 | i V 1 | i
Notice that V0L ⊕V1L 6= H22. This mea(cid:8)n(cid:12)(cid:12)s th(cid:11)at the t(cid:9)race preserving recov(cid:8)e(cid:12)(cid:12)ry s(cid:11)uperope(cid:9)rator R is defined in terms of
1
one standard recovery operator R1 and by the projector R onto the orthogonal complement of iL, i. e. the
⊥ i=0 V
part of the Hilbert space 2 which is not reached by acting on the code with the correctable erLror operators. In
H2 C
the case under consideration,
2
def
R = 01 01 + 10 10 , R = r r , (45)
1 s s
| ih | | ih | ⊥ | ih |
s=1
X
where {|rsi} is an orthonormalbasis for V0L ⊕V1L ⊥. A suitable basis B( 0L 1L)⊥ is given by,
V ⊕V
(cid:0) (cid:1)
= r = 00 , r = 11 . (46)
B( 0L 1L)⊥ { 1 | i 2 | i}
V ⊕V
9
FIG. 1: Threshold curves for code effectiveness: concatenated code (dashed line), DFS (solid line). The concatenated code
works in the parametric region below the dashed line. The DFS works in the region above the solid line. The repetition code
works for any valueof µwhen p≤0.5.
The action of this recovery operation with R R on the map Λ(2)(ρ) in (37) yields,
2
R ≡ ⊥
3 2
Λ(r2ec)over(ρ)≡ R◦Λ(2) (ρ)d=ef (RlA′k)ρ(RlA′k)†. (47)
(cid:16) (cid:17) Xk=0Xl=1
We want to describe the action of Λ(2) restricted to the code subspace . Therefore, we compute the 2 2 matrix
R◦ C ×
representation [RA ] of each R A with l=1, 2 and k =0,.., 3 where,
l ′k l ′k
|C
[RlA′k] d=ef h10L|RRlAA′k|00Li h10L|RRlAA′k|11Li . (48)
|C (cid:18)h L| l ′k| Li h L| l ′k| Li(cid:19)
Substituting (38)and(45)into(48),itturnsoutthattheonlymatrices[RA ] withnon-vanishingtracearegiven
l ′k
by, |C
1 0 1 0
[R1A′0] = p˜(02) 0 1 , [R1A′3] =− p˜(32) 0 1 (49)
|C q (cid:18) (cid:19) |C q (cid:18) (cid:19)
Therefore, the entanglement fidelity (2) (µ, p) defined in (10) results,
FDFS
(2) (µ, p)=µ 2p2+2p + 2p2 2p+1 . (50)
FDFS − −
We point out that error correctionschemes improve(cid:0)the transm(cid:1)issio(cid:0)n accuracy o(cid:1)nly if the failure probability (µ, p)
P
is strictly less than the error probability p [32],
def
(µ, p) = 1 (µ, p)<p. (51)
P −F
The thresholdcurvesfor code effectiveness µ¯(p)are defined by the relation1 (µ¯(p), p)=0. They allow to select
−F
the two-dimensional parametric region where error correction schemes are useful. The threshold curve for the DFS
for the model appears in Figure 1. The DFS works only in the parametric region above the thin solid line in Figure
1, while the repetition code works for all values of the memory degree µ when the error probability p is less than 0.5.
Concatenated Code. Following the line of reasoning presented in [33], we consider the concatenation of the two
codes just described. In the case of (16) with n=6 qubits and correlated errors in a dephasing quantum channel,
1
Λ(6)(ρ)d=ef pi6i5pi5i4pi4i3pi3i2pi2i1pi1(Ai6Ai5Ai4Ai3Ai2Ai1)ρ(Ai6Ai5Ai4Ai3Ai2Ai1)†, (52)
| | | | |
i1,..X.,i6=0
10
The error superoperator associated to channel (52) is defined in terms of the following error operators,
A
26 1 26 1
A←→{A′0,.., A′63} with Λ(6)(ρ)d=ef − A′kρA′k† and, − A′k†A′k =I64×64. (53)
k=0 k=0
X X
Theerroroperatorsinthe Krausdecomposition(53)are 6 6 =26 =64,where 6 isthe cardinalityofweight-k
k=0 k k
error operators.
P (cid:0) (cid:1) (cid:0) (cid:1)
We encode our logicalqubit with a concatenatedsubspace obtainedby combining the decoherencefree subspace in
(40) (inner code, = ) with the repetition code in (23) (outer code, = ). We obtain that the
DFS inner phase outer
C C C C
codewords of the concatenated code = are given by,
DFS phase
C C ◦C
def def
0 = +++ , 1 = +++ . (54)
L L
| i | −−−i | i |−−− i
Recall that the detectability condition is given by P A P =λ P where the projector operator on the code space
C ′k C A′k C
is P = 0 0 + 1 1 . Observe that,
L L L L
C C | ih | | ih |
PCA′kPC =h0L|A′k|0Li|0Lih0L|+h0L|A′k|1Li|0Lih1L|+h1L|A′k|0Li|1Lih0L|+h1L|A′k|1Li|1Lih1L|. (55)
Therefore, it turns out that for detectable error operators we must have,
h0L|A′k|0Li=h1L|A′k|1Li and, h0L|A′k|1Li=h1L|A′k|0Li=0. (56)
In the case under consideration,it follows that the only erroroperator (omitting for the sake of simplicity the proper
error amplitudes) not fulfilling the above conditions is proportional to,
Z1Z2Z3Z4Z5Z6. (57)
For such operator, we get
0 Z1Z2Z3Z4Z5Z6 1 =1=0 and, 1 Z1Z2Z3Z4Z5Z6 0 =1=0 . (58)
L L L L
| | 6 | | 6
Therefore Z1Z2Z3Z4(cid:10)Z5Z6 is not detectable.(cid:11)Thus, the card(cid:10)inality of the set of det(cid:11)ectable errors is 63.
detectable
A
Furthermore, recall that the set of correctable errors Acorrectable is such that A†correctableAcorrectable is detectable (in
the hypothesis of invertible error operators). Therefore, after some reasoning,we conclude that the set of correctable
errorsis composed by 32 erroroperators. The correctableweight-0,1 and 2 correctableerroroperators are (omitting
the proper error amplitudes),
I , Z1, Z2, Z3, Z4, Z5, Z6 , (59)
{ }weight-0 weight-1
and, (cid:8) (cid:9)
Z1Z2, Z1Z3, Z1Z4, Z1Z5, Z1Z6, Z2Z3, Z2Z4, Z2Z5, Z2Z6, Z3Z4, Z3Z5, Z3Z6, Z4Z5, Z4Z6, Z5Z6 ,
weight-2
(60)
(cid:8) (cid:9)
respectively. The correctable weight-3 errors are,
Z1Z3Z5, Z1Z3Z6, Z1Z4Z5, Z1Z4Z6, Z1Z5Z6, Z2Z3Z4, Z2Z3Z5, Z2Z3Z6, Z2Z4Z5, Z2Z4Z6 . (61)
weight-3
The(cid:8)re are no weight-4, 5 and 6 correctable error operators. The action of the correctable errors on t(cid:9)he codewords in
(54) is such that the Hilbert space 6 can be decomposed in two 32-dimensionalorthogonalsubspaces 0L and 1L.
H2 V V
In other words, 6 = 0L 1L where
H2 V ⊕V
1 1
V0L =Span vk0L+1 = p˜(6)A′k|0Li and, V1L =Span vk1L+1 = p˜(6)A′k|1Li, (62)
(cid:12) (cid:11) k  (cid:12) (cid:11) k 
(cid:12) q (cid:12) q
withA k =0,...,31(numberingthecorrectableerroroperatorsfrom0to31). Noticethat viL vjL =
′k ∈Acorrectable∀ k | k′
δkk′δij, with k, k′ 0,..., 31 and i, j 0, 1 since, D E
∈{ } ∈{ }
vkiL|vkjL′ =*iL|A′kp˜†−(61) A′kp˜′−(61)|jL+= p˜(61)p˜(6) iL|A′k†A′k′|jL = p˜(61)p˜(6)α′kk′δij =δkk′δij, (63)
D E k k′ k k′ D E k k′
q q q q