Concatenation of Error Avoiding with Error Correcting Quantum Codes for Correlated Noise Models
CConcatenation of Error Avoiding with Error Correcting Quantum Codes forCorrelated Noise Models
Carlo Cafaro and Stefano Mancini , School of Science and Technology, Physics Division,University of Camerino, I-62032 Camerino, Italy
We study the performance of simple error correcting and error avoiding quantum codes togetherwith their concatenation for correlated noise models. Specifically, we consider two error models:i) a bit-flip (phase-flip) noisy Markovian memory channel (model I); ii) a memory channel definedas a memory degree dependent linear combination of memoryless channels with Kraus decomposi-tions expressed solely in terms of tensor products of X -Pauli ( Z -Pauli) operators (model II). Theperformance of both the three-qubit bit flip (phase flip) and the error avoiding codes suitable forthe considered error models is quantified in terms of the entanglement fidelity. We explicitly showthat while none of the two codes is effective in the extreme limit when the other is, the three-qubitbit flip (phase flip) code still works for high enough correlations in the errors, whereas the erroravoiding code does not work for small correlations. Finally, we consider the concatenation of suchcodes for both error models and show that it is particularly advantageous for model II in the regimeof partial correlations. PACS numbers: quantum error correction (03.67.Pp); decoherence (03.65. Yz).
I. INTRODUCTION
It is well known that one of the most important obstacles in quantum information processing is decoherence. Itcauses a quantum computer to lose its quantum properties destroying its performance advantages over a classical com-puter. The unavoidable interaction between the open quantum processor and its environment corrupts the informationstored in the system and causes errors that may lead to wrong outputs.There are different methods for preserving quantum coherence. One possible technique exploits the redundancy inencoding information. This scheme is known as ”quantum error correcting codes” (QECCs). For a comprehensiveintroduction to QECCs, we refer to [1]. Within such scheme, information is encoded in linear subspaces (codes) ofthe total Hilbert space in such a way that errors induced by the interaction with the environment can be detectedand corrected. The QECC approach may be interpreted as an active stabilization of a quantum state in which, bymonitoring the system and conditionally carrying on suitable operations, one prevents the loss of information. Anotherpossible approach is the so-called ”decoherence free subspaces” (DFSs) (also known as error avoiding or noiselesscodes). For a comprehensive introduction to DFSs, we refer to [2]. It turns out that for specific open quantumsystems (noise models in which all qubits can be considered symmetrically coupled with the same environment), it ispossible to design states that are hardly corrupted rather than states that can be easily corrected (in this sense, DFSsare complementary to QECCs). In other words, it is possible to encode information in linear subspaces such that thedynamics restricted to such subspaces is unitary. This implies that no information is loss and quantum coherence ismaintained. DFS is an example of passive stabilization of quantum information.The formal mathematical description of the qubit-environment interaction is usually given in terms of quantumchannels. Quantum error correction is usually developed under the assumption of i.i.d. (identically and independentlydistributed) errors. These error models are characterized by memoryless communication channels Λ such that n -channel uses is given by Λ ( n ) = Λ ⊗ n . In such cases of complete independent decoherence, qubits interact withtheir own environments which do not interact with each other. However, in actual physical situations, qubits dointeract with a common environment which unavoidably introduces correlations in the noise. For instance, there aresituations 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 thermalbath of phonons [4]. The exchange of bosons between qubits causes spatial and temporal correlations that violate thecondition of error independence [5]. Memory effects introduce correlations among channel uses with the consequencethat Λ ( n ) (cid:54) = Λ ⊗ n . Recent studies try to characterize the effect of correlations on the performance of QECCs [6–10].It appears that correlations may have negative [7] or positive [8] impact on QECCs depending on the features of theerror model being considered.Devising new good quantum codes for both independent and correlated noise models is a highly non trivial problem.However, it is usually possible to manipulate existing codes to construct new ones suitable for more general errormodels and with higher performances [11, 12]. Concatenation is perhaps one of the most successful quantum codingtricks employed to produce new codes from old ones. The concept of concatenated codes was first introduced in a r X i v : . [ qu a n t - ph ] J un classical error correcting schemes by Forney [13]. Roughly speaking, concatenation is a method of combining twocodes (an inner and an outer code) to form a larger code. In classical error correction, Forney has carried on extensivestudies of concatenated codes, how to choose the inner and outer codes, and what error probability threshold valuescan be achieved. The first applications of concatenated codes in quantum error correction appear in [14, 15]. In thequantum setting, concatenated codes play a key role in fault tolerant quantum computation and in constructing gooddegenerate quantum error correcting codes. For instance, Shor’s degenerate code can be constructed by concatenatingthe three-qubit bit flip and phase flip codes.It is known that DFSs are efficient under conditions in which each qubit couples to the same environment (collectivedecoherence) while ordinary QECCs are designed to be efficient when each individual qubit couples to a differentenvironment (independent decoherence) [16]. While none of the two error correction schemes is effective in theextreme limit when the other is, QECCs will still work for correlated errors [17, 18], whereas DFSs will not work inthe independent error case [19]. Therefore, it would be of interest to study the performance of concatenated codesobtained by combining DFSs and QECCs for explicit noise models in the presence of partially correlated errors.Indeed, this will be one of the main purposes in our work.In this article, we study the performance of simple error correcting and error avoiding quantum codes togetherwith their concatenation for correlated noise models. Specifically, we consider two error models: i) a bit-flip (phase-flip) noisy Markovian memory channel (model I); ii) a memory channel defined as a memory degree dependent linearcombination of memoryless channels with Kraus decompositions expressed solely in terms of tensor products of X -Pauli( Z -Pauli) operators (model II). The performance of both the three-qubit bit flip (phase flip) and the error avoidingcodes suitable for the considered error models is quantified in terms of the entanglement fidelity. We explicitly showthat while none of the two codes is effective in the extreme limit when the other is, the three-qubit bit flip (phaseflip) code still works for high enough correlations in the errors, whereas the error avoiding code does not work forsmall correlations. Finally, we consider the concatenation of such codes for both error models and show that it isparticularly advantageous for model II in the regime of partial correlations.The layout of this article is as follows. In Section II, we briefly discuss about the concatenation technique inquantum coding, the DFSs and, the entanglement fidelity as a performance measure of quantum error correctionschemes. In Section III, we evaluate the performances of the three-qubit bit flip code and suitable DFSs for errormodels I and II. The performance of the concatenated code for both models appears in Section IV. Finally, in SectionV, we present our concluding remarks. II. ON CONCATENATION, DECOHERENCE FREE SUBSPACES AND, ENTANGLEMENT FIDELITY
In this Section, we briefly discuss about the concatenation technique in quantum coding, the DFSs and, theentanglement fidelity.
A. Quantum Concatenation Trick
For the sake of simplicity we only use two layers of concatenation and consider single qubit encoding. The general-ization to arbitrary l layers of concatenation and multi-qubit encoding is relatively straightforward [20]. Assume theinner code (first layer) is a [ n , k , d ] stabilizer code C with generators G = (cid:8) g i : i = 1,..., n − k (cid:9) , and the outercode (second layer) is a [ n , 1, d ] stabilizer code C with generators G = (cid:8) g j : j = 1,..., n j − (cid:9) . The concatenatedcode C def = C ◦ C is to map k qubits into n = n n qubits, with code construction parsing the n qubits into n blocks B ( b ) ( b = 1,..., n ) each containing n qubits. In other words, given a codeword | c inner (cid:105) for the inner code C , | c inner (cid:105) = k − (cid:88) j =0 α j | φ j (cid:105) , (1)where {| φ j (cid:105)} are basis vectors for C , the concatenated code C is constructed as follows. For any codeword | c outer (cid:105) forthe outer code C , | c outer (cid:105) = (cid:88) i ,..., i n α i ,..., i n | i ... i n (cid:105) , (2)with | i ... i n (cid:105) = | i (cid:105) ⊗ ... ⊗ | i n (cid:105) , replace each basis vector | i j (cid:105) with i j = 0,..., k − j = 1,..., n by a basis vector (cid:12)(cid:12) φ i j (cid:11) in C , that is | c concatenated (cid:105) def = (cid:88) i ,..., i n α i ,..., i n | φ i (cid:105) ⊗ ... ⊗ (cid:12)(cid:12) φ i n (cid:11) . (3)Further details on the construction of the stabilizer generators of C can be found in [20]. As a final remark, we simplypoint out that the above mentioned construction produces a [ n n , k , d ] code with d ≥ d d . As an illustrativeexample, consider a concatenated code that is based on l layers of concatenation of the same code, for instance theseven-qubit [7, 1, 3] CSS code. Here, an unencoded qubit is encoded into a block of seven qubits. Next each qubit isitself encoded into a block of seven qubits. Repeating this process l times produces the concatenated code in whichone logical qubit is recursively encoded into the state of 7 l physical qubits. Thus, such concatenated code is a (cid:2) l , 1, d (cid:3) code with d ≥ l . Recall that a distance d QECC can correct t errors, where t = (cid:104) ( d − (cid:105) and [ x ] is the integer part of x . Since d ≥ l , we see that the number of errors that a concatenated code can correct grows exponentially with thenumber of layers of concatenation l . Therefore, concatenation can be a good coding trick as long as the error modelat each encoding level has the same form, that is, the same Kraus error operators with possibly different amplitude. B. Decoherence Free Subspaces
Following [2], we mention few relevant properties of DFSs. Consider the dynamics of a closed system composed ofa quantum system Q coupled to a bath B . The unitary evolution of the closed system is described by the combinedsystem-bath Hamiltonian H tot , H tot = H Q ⊗ I B + H B ⊗ I Q + H int , H int = (cid:88) α E α ⊗ B α . (4)The operator H Q ( H B ) is the system (bath) Hamiltonian, I Q ( I B ) is the identity operator of the system (bath), E α arethe error generators acting solely on Q while B α act on the bath. The last term in (4) is the interaction Hamiltonian.A subspace H DFS of the total system Hilbert space H is a decoherence free subspace (unitary evolution in H DFS for all possible bath states) if and only if:1. E α | ψ (cid:105) = c α | ψ (cid:105) with c α ∈ C , for all states | ψ (cid:105) spanning H DFS , and for every error operator E α in H int . In otherwords, all basis states spanning H DFS are degenerate eigenstates of all the error generators E α ;2. Q and B are initially decoupled;3. H Q | ψ (cid:105) has no overlap with states in the subspace orthogonal to H DFS .To establish a direct link between QECCs and DFSs, it is more convenient to present an alternative formulationof DFSs in terms of the Kraus operator sum representation. Within such description, the evolution of the system Q density matrix is written as, ρ Q ( t ) = T r B (cid:2) U ( ρ Q ⊗ ρ B ) U † (cid:3) = (cid:88) k A k ρ Q (0) A † k , (5)where U = e − i (cid:126) H tot t is the unitary evolution operator for the system-bath closed system and the Kraus operator A k (satisfying the normalization condition) are given by, A k = √ n (cid:104) m | U | n (cid:105) with (cid:88) k A † k A k = I Q , (6)where k = ( n , m ), | m (cid:105) and | n (cid:105) are bath states. It turns out that a N DFS -dimensional subspace H DFS of H is a DFSif and only if all Kraus operators have an identical unitary representation (in the basis where the first N DFS statesspan H DFS ) upon restriction to it, up to a multiplicative constant, A k = (cid:18) g k U (DFS) Q
00 ¯ A k (cid:19) , (7)where g k = √ n (cid:104) m | U c | n (cid:105) and U c = e − i (cid:126) H c t with H c = H B + H int . Furthermore, ¯ A k is an arbitrary matrix that acts on H ⊥ DFS (with H = H DFS ⊕ H ⊥ DFS ) and may cause decoherence there; U (DFS) Q is U Q restricted to H DFS . Now recall thatin ordinary QECCs, it is possible to correct the errors induced by a given set of Kraus operators { A k } if and only if, R r A k = (cid:18) λ rk I C B rk (cid:19) , ∀ r and k , (8)or, equivalently, A † k A k (cid:48) = (cid:18) γ kk (cid:48) I C
00 ¯ A † k ¯ A k (cid:48) (cid:19) , (9)where { R r } are the recovery operators. The first block in the RHS of (8) acts on the code space C while the matrices B rk act on C ⊥ where H = C ⊕ C ⊥ . From (7) and (8), it follows that DFS can be viewed as a special class of QECCs,where upon restriction to the code space C , all recovery operators R r are proportional to the inverse of the system Q evolution operator, R r ∝ (cid:16) U (DFS) Q (cid:17) † . (10)Assuming that (10) holds, from (7) and (9) it also turns out that, A k ∝ U (DFS) Q , (11)upon restriction to C . Furthermore, from (7) and (9), it follows that γ kk (cid:48) = g ∗ k g k (cid:48) . However, while in the QECCs case γ kk (cid:48) is in general a full-rank matrix (non-degenerate code), in the DFSs case this matrix has rank 1. In conclusion, aDFS can be viewed as a special type of QECC, namely a completely degenerate quantum error correcting code whereupon restriction to the code subspace all recovery operators are proportional to the inverse of the system evolutionoperator. As a side remark, in view of this last observation we point out that it is not unreasonable to quantify theperformance of both active and passive QEC schemes by means of the same performance measure. C. Entanglement Fidelity
We recall the concept of entanglement fidelity as a useful performance measure of the efficiency of quantum errorcorrecting codes. Entanglement fidelity is a quantity that keeps track of how well the state and entanglement of asubsystem of a larger system are stored, without requiring the knowledge of the complete state or dynamics of thelarger system. More precisely, the entanglement fidelity is defined for a mixed state ρ = (cid:80) i p i ρ i =tr H R | ψ (cid:105) (cid:104) ψ | interms of a purification | ψ (cid:105) ∈ H ⊗ H R to a reference system H R . The purification | ψ (cid:105) encodes all of the informationin ρ . Entanglement fidelity is a measure of how well the channel Λ preserves the entanglement of the state H with itsreference system H R . The entanglement fidelity is defined as follows [21], F ( ρ , Λ) def = (cid:104) ψ | (Λ ⊗ I H R ) ( | ψ (cid:105) (cid:104) ψ | ) | ψ (cid:105) , (12)where | ψ (cid:105) is any purification of ρ , I H R is the identity map on M ( H R ) and Λ ⊗ I H R is the evolution operator extendedto the space H ⊗ H R , space on which ρ has been purified. If the quantum operation Λ is written in terms of its Krausoperator elements { A k } as, Λ ( ρ ) = (cid:80) k A k ρA † k , then it can be shown that [22], F ( ρ , Λ) = (cid:88) k tr ( A k ρ ) tr (cid:16) A † k ρ (cid:17) = (cid:88) k | tr ( ρA k ) | . (13)This expression for the entanglement fidelity is very useful for explicit calculations. Finally, assuming thatΛ : M ( H ) (cid:51) ρ (cid:55)−→ Λ ( ρ ) = (cid:88) k A k ρA † k ∈ M ( H ) , dim C H = N (14)and choosing a purification described by a maximally entangled unit vector | ψ (cid:105) ∈ H ⊗ H for the mixed state ρ = C H I H , we obtain F (cid:18) N I H , Λ (cid:19) = 1 N (cid:88) k | tr A k | . (15)The expression in (15) represents the entanglement fidelity when no error correction is performed on the noisy channelΛ in (14). III. THREE-QUBIT BIT FLIP CODE AND DFS
In this Section, we consider two correlated error models. Error correction is performed by means of the three-qubitbit flip code and a suitable decoherence free subspace. The code performance measure used is the entanglementfidelity.
Model I . The first model is a bit flip noisy quantum Markovian memory channel Λ ( n ) ( ρ ) (model I). In explicit terms,we consider n qubits and Markovian correlated errors in a bit flip quantum channel,Λ ( n ) ( ρ ) def = (cid:88) i ,..., i n =0 p i n | i n − p i n − | i n − ... p i | i p i ( A i n ⊗ ... ⊗ A i ) ρ ( A i n ⊗ ... ⊗ A i ) † , (16)where A = I , A = X are Pauli operators. Furthermore the conditional probabilities p i k | i j are given by, p i k | i j = (1 − µ ) p i k + µδ i k , i j , p i k =0 = 1 − p , p i k =1 = p , (17)with, (cid:88) i ,..., i n =0 p i n | i n − p i n − | i n − ... p i | i p i = 1. (18)To simplify our notation, we may choose to omit the symbol of tensor product ” ⊗ ” in the future, A i n ⊗ ... ⊗ A i ≡ A i n ... A i . Furthermore, we may choose to omit the bar ” | ” in p i k | i j and simply write the conditional probabilities as p i k i j . Model II . The second quantum communication channel Λ ( n ) µ ( ρ ) (model II) that we consider is a memory quantumchannel defined in terms of a linear combination of simple memoryless channels with Kraus decompositions expressedin terms of bit-flip error operators. The coefficients of such combination are dependent on the memory parameter µ .In explicit terms, we consider the following channel Λ ( n ) µ ( ρ ),Λ ( n ) µ ( ρ ) def = a ( µ ) Λ ( n )0 ( ρ ) + a ( µ ) Λ ( n )1 ( ρ ) , (19)where a ( µ ) def = 1 − µ , a ( µ ) def = µ , Λ ( n )0 ( ρ ) ≡ Λ ( n ) ( ρ ) in (16) in the limiting case of µ = 0 ( n -uses of a memoryless bitflip channel). Finally, Λ ( n )1 ( ρ ) describes n -uses of a memoryless channel whose Kraus decomposition is characterizedonly by weight-0 and weight- n error operators with amplitudes (1 − p ) n and [1 − (1 − p ) n ] , respectively,Λ ( n )1 ( ρ ) def = (1 − p ) n ρ + [1 − (1 − p ) n ] X X ... X n − X n ρX X ... X n − X n . (20)In this Section, QEC is performed via the three-qubit bit flip code and a suitable decoherence free subspace. Althoughthe error models considered are not truly quantum in nature, from this preliminary work we hope to gain usefulinsights for extending error correction techniques to quantum error models in the presence of partial correlations. Theperformance of quantum error correcting codes is quantified by means of the entanglement fidelity as function of theerror probability p and degree of memory µ . A. The three-qubit bit flip code
1. Model I
Error Operators . In the simplest example, we consider the limiting case of (16) with n = 3,Λ (3) ( ρ ) def = (cid:88) i , i , i =0 p i | i p i | i p i (cid:104) A i A i A i ρA † i A † i A † i (cid:105) , with (cid:88) i , i , i =0 p i | i p i | i p i = 1. (21)Substituting (17) in (21), it follows that the error superoperator A associated to channel (21) is defined in terms ofthe following error operators, A ←→ { A (cid:48) ,.., A (cid:48) } with Λ (3) ( ρ ) def = (cid:88) k =0 A (cid:48) k ρA (cid:48)† k and, (cid:88) k =0 A (cid:48)† k A (cid:48) k = I × . (22)In an explicit way, the error operators { A (cid:48) ,.., A (cid:48) } are given by, A (cid:48) = (cid:113) ˜ p (3)0 I ⊗ I ⊗ I , A (cid:48) = (cid:113) ˜ p (3)1 X ⊗ I ⊗ I , A (cid:48) = (cid:113) ˜ p (3)2 I ⊗ X ⊗ I , A (cid:48) = (cid:113) ˜ p (3)3 I ⊗ I ⊗ X , A (cid:48) = (cid:113) ˜ p (3)4 X ⊗ X ⊗ I , A (cid:48) = (cid:113) ˜ p (3)5 X ⊗ I ⊗ X , A (cid:48) = (cid:113) ˜ p (3)6 I ⊗ X ⊗ X , A (cid:48) = (cid:113) ˜ p (3)7 X ⊗ X ⊗ X , (23)where the coefficients ˜ p (3) k for k = 1,.., 7 are given by,˜ p (3)0 = p p , ˜ p (3)1 = p p p , ˜ p (3)2 = p p p , ˜ p (3)3 = p p p ,˜ p (3)4 = p p p , ˜ p (3)5 = p p p , ˜ p (3)6 = p p p , ˜ p (3)7 = p p , (24)with, p = (1 − p ) , p = p , p = ((1 − µ ) (1 − p ) + µ ) , p = (1 − µ ) (1 − p ) , p = (1 − µ ) p , p = ((1 − µ ) p + µ ) . (25) Encoding Operator . Consider a three-qubit bit flip code that encodes 1 logical qubit into 3-physical qubits. Thecodewords are given by, | (cid:105) tensoring −→ | (cid:105) ⊗ | (cid:105) = | (cid:105) def = | L (cid:105) , | (cid:105) tensoring −→ | (cid:105) ⊗ | (cid:105) = | (cid:105) U ⊗ I −→ | (cid:105) U ⊗ I −→ | (cid:105) def = | L (cid:105) . (26)The operator U ij CNOT is the CNOT gate from qubit i to j defined as, U ij CNOT def = 12 (cid:2)(cid:0) I i + Z i (cid:1) ⊗ I j + (cid:0) I i − Z i (cid:1) ⊗ X j (cid:3) . (27)Finally, the encoding operator U enc such that U enc | (cid:105) = | (cid:105) and U enc | (cid:105) = | (cid:105) is defined as, U enc def = (cid:0) U ⊗ I (cid:1) ◦ (cid:0) U ⊗ I (cid:1) . (28) Correctable Errors and Recovery Operators . The set of error operators satisfying the detectability condi-tion [23], P C A (cid:48) k P C = λ A (cid:48) k P C , where P C = | L (cid:105) (cid:104) L | + | L (cid:105) (cid:104) L | is the projector operator on the code subspace C = Span {| L (cid:105) , | L (cid:105)} is given by, A detectable = { A (cid:48) , A (cid:48) , A (cid:48) , A (cid:48) , A (cid:48) , A (cid:48) , A (cid:48) } ⊆ A . (29)The only non-detectable error is A (cid:48) . Furthermore, since all the detectable errors are invertible, the set of correctableerrors is such that A † correctable A correctable is detectable. It follows then that, A correctable = { A (cid:48) , A (cid:48) , A (cid:48) , A (cid:48) } ⊆ A detectable ⊆ A . (30)The action of the correctable error operators A correctable on the codewords | L (cid:105) and | L (cid:105) is given by, | L (cid:105) → A (cid:48) | L (cid:105) = (cid:113) ˜ p (3)0 | (cid:105) , A (cid:48) | L (cid:105) = (cid:113) ˜ p (3)1 | (cid:105) , A (cid:48) | L (cid:105) = (cid:113) ˜ p (3)2 | (cid:105) , A (cid:48) | L (cid:105) = (cid:113) ˜ p (3)3 | (cid:105)| L (cid:105) → A (cid:48) | L (cid:105) = (cid:113) ˜ p (3)0 | (cid:105) , A (cid:48) | L (cid:105) = (cid:113) ˜ p (3)1 | (cid:105) , A (cid:48) | L (cid:105) = (cid:113) ˜ p (3)2 | (cid:105) , A (cid:48) | L (cid:105) = (cid:113) ˜ p (3)3 | (cid:105) . (31)The two four-dimensional orthogonal subspaces V L and V L of H generated by the action of A correctable on | L (cid:105) and | L (cid:105) are given by, V L = Span (cid:8)(cid:12)(cid:12) v L (cid:11) = | (cid:105) , (cid:12)(cid:12) v L (cid:11) = | (cid:105) , (cid:12)(cid:12) v L (cid:11) = | (cid:105) , (cid:12)(cid:12) v L (cid:11) = | (cid:105) (cid:9) , (32)and, V L = Span (cid:8)(cid:12)(cid:12) v L (cid:11) = | (cid:105) , (cid:12)(cid:12) v L (cid:11) = | (cid:105) , (cid:12)(cid:12) v L (cid:11) = | (cid:105) , (cid:12)(cid:12) v L (cid:11) = | (cid:105) (cid:9) , (33)respectively. Notice that V L ⊕ V L = H . The recovery superoperator R ↔ { R l } with l = 1,..,4 is defined as [18], R l def = V l (cid:88) i =0 (cid:12)(cid:12) v i L l (cid:11) (cid:10) v i L l (cid:12)(cid:12) , (34)where the unitary operator V l is such that V l (cid:12)(cid:12) v i L l (cid:11) = | i L (cid:105) for i ∈ {
0, 1 } . Substituting (32) and (33) into (34), itfollows that the four recovery operators { R , R , R , R } are given by, R = | L (cid:105) (cid:104) L | + | L (cid:105) (cid:104) L | , R = | L (cid:105) (cid:104) | + | L (cid:105) (cid:104) | , R = | L (cid:105) (cid:104) | + | L (cid:105) (cid:104) | , R = | L (cid:105) (cid:104) | + | L (cid:105) (cid:104) | . (35)Using simple algebra, it turns out that the 8 × R l ] with l = 1,..,4 of the recovery operatorsis given by, [ R ] = E + E , [ R ] = E + E , [ R ] = E + E , [ R ] = E + E , (36)where E ij is the 8 × ij -position and it equals1. It follows that R ↔ { R l } is indeed a trace preserving quantum operation since, (cid:88) l =1 R † l R l = I × . (37)The action of this recovery operation R on the map Λ (3) ( ρ ) in (22) leads to,Λ (3)recover ( ρ ) ≡ (cid:16) R◦ Λ (3) (cid:17) ( ρ ) def = (cid:88) k =0 4 (cid:88) l =1 ( R l A (cid:48) k ) ρ ( R l A (cid:48) k ) † . (38) Entanglement Fidelity . We want to describe the action of R◦ Λ (3) restricted to the code subspace C . Therefore, wecompute the 2 × R l A (cid:48) k ] |C of each R l A (cid:48) k with l = 1,.., 4 and k = 0,.., 7 where,[ R l A (cid:48) k ] |C def = (cid:18) (cid:104) L | R l A (cid:48) k | L (cid:105) (cid:104) L | R l A (cid:48) k | L (cid:105)(cid:104) L | R l A (cid:48) k | L (cid:105) (cid:104) L | R l A (cid:48) k | L (cid:105) (cid:19) . (39)Substituting (31) and (35) into (39), it turns out that the only matrices [ R l A (cid:48) k ] |C with non-vanishing trace are givenby, [ R A (cid:48) ] |C = (cid:113) ˜ p (3)0 (cid:18) (cid:19) , [ R A (cid:48) ] |C = (cid:113) ˜ p (3)1 (cid:18) (cid:19) ,[ R A (cid:48) ] |C = (cid:113) ˜ p (3)2 (cid:18) (cid:19) , [ R A (cid:48) ] |C = (cid:113) ˜ p (3)3 (cid:18) (cid:19) . (40)Therefore, the entanglement fidelity F (3)bit ( µ , p ) defined as, F (3)bit ( µ , p ) def = F (3) (cid:18) I × , R◦ Λ (3) (cid:19) = 1(2) (cid:88) k =0 4 (cid:88) l =1 (cid:12)(cid:12)(cid:12) tr (cid:16) [ R l A (cid:48) k ] |C (cid:17)(cid:12)(cid:12)(cid:12) , (41)results, F (3)bit ( µ , p ) = ˜ p (3)0 + ˜ p (3)1 + ˜ p (3)2 + ˜ p (3)3 . (42)The expression for F (3)bit ( µ , p ) in (41) represents the entanglement fidelity quantifying the performance of the errorcorrection scheme provided by the three-qubit bit flip code here considered. The quantum operation R◦ Λ (3) appearingin (41) is defined in equation (38) and the recovery operators R l are explicitly given in (35). The action of R l A (cid:48) k in(41) is restricted to the code space C defined in (26).Substituting (24) and (25) into (42), we finally obtain F (3)bit ( µ , p ) = µ (cid:0) p − p + p (cid:1) + µ (cid:0) − p + 6 p − p (cid:1) + (cid:0) p − p + 1 (cid:1) (model I). (43)Notice that for a vanishing degree of memory µ , the entanglement fidelity becomes, F (3)bit (0, p ) = 2 p − p + 1. (44) Remarks on the coding for phase flip memory channels . The code for the phase flip channel has the same character-istics as the code for the bit flip channel. These two channels are unitarily equivalent since there is a unitary operator,the Hadamard gate H , such that the action of one channel is the same as the other, provided the first channel ispreceded by H and followed by H † [24],Λ phase ( ρ ) def = (cid:0) H ◦ Λ bit ◦ H † (cid:1) ( ρ ) = (1 − p ) ρ + pZρZ , (45)where, Λ bit ( ρ ) def = (1 − p ) ρ + pXρX and, H def = 1 √ (cid:18) − (cid:19) . (46)These operations may be trivially incorporated into the encoding and error-correction operations. The encoding forthe phase flip channel is performed in two steps: i) first, we encode in three qubits exactly as for the bit flip channel;ii) second, we apply a Hadamard gate to each qubit, | (cid:105) tensoring −→ | (cid:105) U bitenc −→ | (cid:105) H ⊗ −→ | L (cid:105) def = | + + + (cid:105) , | (cid:105) tensoring −→ | (cid:105) U bitenc −→ | (cid:105) H ⊗ −→ | L (cid:105) def = |− − −(cid:105) , (47)where, (cid:18) | + (cid:105)|−(cid:105) (cid:19) = 1 √ (cid:18) − (cid:19) (cid:18) | (cid:105)| (cid:105) (cid:19) . (48)The unitary encoding operator U phaseenc is given by U phaseenc def = H ⊗ ◦ U bitenc ,with U bitenc defined in (28). Furthermore, inthe phase flip code, the recovery operation is the Hadamard conjugated recovery operation from the bit flip code, R phase k def = H ⊗ R bit k H ⊗ .
2. Model II
We consider Λ ( n ) µ ( ρ ) in (19) with n = 3. Technical details will be omitted. They can be obtained by following theline of reasoning presented for the three-qubit bit flip error correction scheme applied to the model I. It turns outthat, F (3)bit ( µ , p ) = µ (cid:0) − p + 6 p − p (cid:1) + (cid:0) p − p + 1 (cid:1) (model II). (49) B. DFS for bit flip noise
1. Model I
Error Operators . Consider the limiting case of (16) with n = 2 qubits and correlated errors in a bit flip quantumchannel, Λ (2) ( ρ ) def = (cid:88) i , i =0 p i | i p i ( A i ⊗ A i ) ρ ( A i ⊗ A i ) † , (50)The error superoperator A associated to channel (50) is defined in terms of the following error operators, A ←→ { A (cid:48) ,.., A (cid:48) } with Λ (2) ( ρ ) def = (cid:88) k =0 A (cid:48) k ρA (cid:48)† k and, (cid:88) k =0 A (cid:48)† k A (cid:48) k = I × . (51)In an explicit way, the error operators { A (cid:48) ,.., A (cid:48) } are given by, A (cid:48) = (cid:113) ˜ p (2)0 I ⊗ I , A (cid:48) = (cid:113) ˜ p (2)1 X ⊗ I , A (cid:48) = (cid:113) ˜ p (2)2 I ⊗ X , A (cid:48) = (cid:113) ˜ p (2)3 X ⊗ X , (52)where the coefficients ˜ p (2) k for k = 0,.., 3 are given by,˜ p (2)0 = p p , ˜ p (2)1 = p p , ˜ p (2)2 = p p , ˜ p (2)3 = p p . (53) Encoding Operator . We encode our logical qubit with a simple decoherence free subspace of two qubits given by [2], | (cid:105) −→ | L (cid:105) = | + −(cid:105) and, | (cid:105) −→ | L (cid:105) = |− + (cid:105) , (54)with |±(cid:105) defined in (48). As a side remark, we point out that a suitable DFS encoding for a phase flip Markoviancorrelated noise model is given by | (cid:105) −→ | L (cid:105) = | (cid:105) and, | (cid:105) −→ | L (cid:105) = | (cid:105) . Correctable Errors and Recovery Operators . The set of error operators satisfying the detectability condi-tion [23], P C A (cid:48) k P C = λ A (cid:48) k P C , where P C = | L (cid:105) (cid:104) L | + | L (cid:105) (cid:104) L | is the projector operator on the code subspace C = Span {| L (cid:105) , | L (cid:105)} is given by, A detectable = { A (cid:48) , A (cid:48) } ⊆ A . (55)Furthermore, since all the detectable errors are invertible, the set of correctable errors is such that A † correctable A correctable is detectable. It follows then that, A correctable = A detectable ⊆ A . (56)The action of the correctable error operators A correctable on the codewords | L (cid:105) and | L (cid:105) is given by, | L (cid:105) → A (cid:48) | L (cid:105) = (cid:113) ˜ p (2)0 | + −(cid:105) , A (cid:48) | L (cid:105) = − (cid:113) ˜ p (2)3 | + −(cid:105) , | L (cid:105) → A (cid:48) | L (cid:105) = (cid:113) ˜ p (2)0 |− + (cid:105) , A (cid:48) | L (cid:105) = − (cid:113) ˜ p (2)3 |− + (cid:105) .(57)The two one-dimensional orthogonal subspaces V L and V L of H generated by the action of A correctable on | L (cid:105) and | L (cid:105) are given by, V L = Span (cid:8)(cid:12)(cid:12) v L (cid:11) = | + −(cid:105) (cid:9) and, V L = Span (cid:8)(cid:12)(cid:12) v L (cid:11) = |− + (cid:105) (cid:9) . (58)Notice that V L ⊕ V L (cid:54) = H . This means that the trace preserving recovery superoperator R is defined in terms ofone standard recovery operator R and by the projector R ⊥ onto the orthogonal complement of (cid:76) i =0 V i L , i. e. thepart of the Hilbert space H which is not reached by acting on the code C with the correctable error operators. Inthe case under consideration, R = | + −(cid:105) (cid:104) + −| + |− + (cid:105) (cid:104)− + | , R ⊥ = (cid:88) s =1 | r s (cid:105) (cid:104) r s | , (59)where {| r s (cid:105)} is an orthonormal basis for (cid:0) V L ⊕ V L (cid:1) ⊥ . A suitable basis B ( V L ⊕V L ) ⊥ is given by, B ( V L ⊕V L ) ⊥ = { r = | ++ (cid:105) , r = |−−(cid:105)} . (60)Therefore, R ↔ { R , R ⊥ } is indeed a trace preserving quantum operation, R † R + R †⊥ R ⊥ = I × . (61)0The action of this recovery operation R with R ≡ R ⊥ on the map Λ (2) ( ρ ) in (51) yields,Λ (2)recover ( ρ ) ≡ (cid:16) R◦ Λ (2) (cid:17) ( ρ ) def = (cid:88) k =0 2 (cid:88) l =1 ( R l A (cid:48) k ) ρ ( R l A (cid:48) k ) † , (62) Entanglement Fidelity . We want to describe the action of R◦ Λ (2) restricted to the code subspace C . Therefore, wecompute the 2 × R l A (cid:48) k ] |C of each R l A (cid:48) k with l = 1, 2 and k = 0,.., 3 where,[ R l A (cid:48) k ] |C def = (cid:18) (cid:104) L | R l A (cid:48) k | L (cid:105) (cid:104) L | R l A (cid:48) k | L (cid:105)(cid:104) L | R l A (cid:48) k | L (cid:105) (cid:104) L | R l A (cid:48) k | L (cid:105) (cid:19) . (63) FIG. 1: Threhold curves for model I: concatenated code (dashed line), DFS (thin solid line).
Substituting (52) and (59) into (63), it turns out that the only matrices [ R l A (cid:48) k ] |C with non-vanishing trace are givenby, [ R A (cid:48) ] |C = (cid:113) ˜ p (2)0 (cid:18) (cid:19) , [ R A (cid:48) ] |C = − (cid:113) ˜ p (2)3 (cid:18) (cid:19) (64)Therefore, the entanglement fidelity F (2) DF S ( µ , p ) defined as, F (2) DF S ( µ , p ) def = F (2) (cid:18) I × , R◦ Λ (2) (cid:19) = 1(2) (cid:88) k =0 2 (cid:88) l =1 (cid:12)(cid:12)(cid:12) tr (cid:16) [ R l A (cid:48) k ] |C (cid:17)(cid:12)(cid:12)(cid:12) , (65)results, F (2) DF S ( µ , p ) = ˜ p (2)0 + ˜ p (2)3 . (66)The expression for F (2) DF S ( µ , p ) in (65) represents the entanglement fidelity quantifying the performance of the errorcorrection scheme provided by the error avoiding code here considered. The quantum operation R◦ Λ (2) appearing in(65) is defined in equation (62) and the recovery operators R l are explicitly given in (59). The action of R l A (cid:48) k in (65)is restricted to the code space C defined in (54).Substituting (25) and (53) into (66), we finally obtain F (2) DF S ( µ , p ) = µ (cid:0) − p + 2 p (cid:1) + (cid:0) p − p + 1 (cid:1) (model I). (67)We point out that error correction schemes improve the transmission accuracy only if the failure probability P ( µ , p )is strictly less than the error probability p [20],1 P ( µ , p ) def = 1 − F ( µ , p ) < p . (68)In view of (68), we can determine threshold curves ¯ µ ( p ) that allow to select the two-dimensional parametric regionwhere error correction schemes are useful. For instance, considering the model I in absence of correlations, it followsthat the three-qubit code is effective only if p < .
5. However for such values of the error probability, the DFSconsidered does not work for µ approaching zero. The threshold curve for the DFS for the model I appear in Figure1. The DFS works only in the parametric region above the thin solid line in Figure 1, while the three-qubit bit-flipcode works for all values of the memory degree µ when the error probability is less than 0 .
2. Model II
We consider Λ ( n ) µ ( ρ ) in (19) with n = 2. Technical details will be omitted. They can be obtained by following theline of reasoning presented when studying the error avoiding code applied to the model I. It turns out that, F (2) DF S ( µ , p ) = µ (cid:0) − p + 2 p (cid:1) + (cid:0) p − p + 1 (cid:1) (model II). (69)Notice that the entanglement fidelities in (67) and (69) are equal. Finally, the threshold curves for the three-qubitbit flip and DFS for the model II appear in Figure 2. The DFS works only in the parametric region above the dashedline while the three-qubit code works only below the thin solid line. FIG. 2: Threshold curves for model II: DFS (dashed line), three-qubit bit flip code (thin solid line), concatenated code (thicksolid line).
IV. CONCATENATED CODESA. Model I
Error Operators . Consider the limiting case of (16) with n = 6 qubits and correlated errors in a bit flip quantumchannel, Λ (6) ( ρ ) def = (cid:88) i ,..., i =0 p i | i p i | i p i | i p i | i p i | i p i ( A i A i A i A i A i A i ) ρ ( A i A i A i A i A i A i ) † , (70)The error superoperator A associated to channel (70) is defined in terms of the following error operators, A ←→ { A (cid:48) ,.., A (cid:48) } with Λ (6) ( ρ ) def = − (cid:88) k =0 A (cid:48) k ρA (cid:48)† k and, − (cid:88) k =0 A (cid:48)† k A (cid:48) k = I × . (71)2The error operators in the Kraus decomposition (71) are 2 = 64, (cid:88) k =0 (cid:18) k (cid:19) = 2 , (72)where (cid:0) k (cid:1) is the cardinality of weight- k error operators. Encoding Operator . We encode our logical qubit with a concatenated subspace obtained by combining the deco-herence free subspace in (54) (inner code, C DF S = C inner ) with the three-qubit bit flip code in (26) (outer code, C bit = C outer ). We obtain that the codewords of the concatenated code C = C DF S ◦ C bit are given by, | L (cid:105) = 12 ( | (cid:105) − | (cid:105) + | (cid:105) − | (cid:105) ) , | L (cid:105) = 12 ( | (cid:105) + | (cid:105) − | (cid:105) − | (cid:105) ) . (73)As a side remark, we point out that a suitable concatenated code for the case of a phase flip Markovian correlatednoise model is given by | (cid:105) −→ | L (cid:105) = | + + + − −−(cid:105) and, | (cid:105) −→ | L (cid:105) = |− − − + ++ (cid:105) . Correctable Errors and Recovery Operators . Recall that the detectability condition is given by P C A (cid:48) k P C = λ A (cid:48) k P C where the projector operator on the code space C is P C = | L (cid:105) (cid:104) L | + | L (cid:105) (cid:104) L | . Observe that, P C A (cid:48) k P C = (cid:104) L | A (cid:48) k | L (cid:105) | L (cid:105) (cid:104) L | + (cid:104) L | A (cid:48) k | L (cid:105) | L (cid:105) (cid:104) L | + (cid:104) L | A (cid:48) k | L (cid:105) | L (cid:105) (cid:104) L | + (cid:104) L | A (cid:48) k | L (cid:105) | L (cid:105) (cid:104) L | . (74)Therefore, it turns out that for detectable error operators we must have, (cid:104) L | A (cid:48) k | L (cid:105) = (cid:104) L | A (cid:48) k | L (cid:105) and, (cid:104) L | A (cid:48) k | L (cid:105) = (cid:104) L | A (cid:48) k | L (cid:105) = 0. (75)In the case under consideration, it follows that the only error operators (omitting for the sake of simplicity the propererror amplitudes) not fulfilling the above conditions are proportional to, X X X and X X X . (76)For such operators, we get (cid:10) L | X X X | L (cid:11) = 1, (cid:10) L | X X X | L (cid:11) = −
1, and, (cid:10) L | X X X | L (cid:11) = − (cid:10) L | X X X | L (cid:11) = 1. (77)Therefore X X X and X X X are not detectable. Thus, the cardinality of the set of detectable errors A detectable is62. Furthermore, recall that the set of correctable errors A correctable is such that A † correctable A correctable is detectable (inthe hypothesis of invertible error operators). Therefore, after some reasoning, we conclude that the set of correctableerrors is composed by 32 error operators. The correctable weight-0, 1 and 2 correctable error operators are (omittingthe proper error amplitudes), { I } weight-0 , (cid:8) X , X , X , X , X , X (cid:9) weight-1 , (78)and, (cid:8) X X , X X , X X , X X , X X , X X , X X , X X , X X (cid:9) weight-2 , (79)respectively. The correctable weight-4 errors are, X X X X , X X X X , X X X X , X X X X , X X X X , X X X X , X X X X , X X X X , X X X X weight-4 . (80)Finally, weight-5 and weight-6 error operators are given by, (cid:8) X X X X X , X X X X X , X X X X X , X X X X X , X X X X X , X X X X X (cid:9) weight-5 ,(81)and, (cid:8) X X X X X X (cid:9) weight-6 , (82)3respectively. The action of the correctable errors on the codewords in (73) is such that the Hilbert space H can bedecomposed in two 32-dimensional orthogonal subspaces V L and V L . In other words, H = V L ⊕ V L where V L = Span (cid:12)(cid:12) v L k +1 (cid:11) = 1 (cid:113) ˜ p (6) k A (cid:48) k | L (cid:105) and, V L = Span (cid:12)(cid:12) v L k +1 (cid:11) = 1 (cid:113) ˜ p (6) k A (cid:48) k | L (cid:105) , (83)with A (cid:48) k ∈ A correctable ∀ k = 0,..., 31 (numbering the correctable error operators from 0 to 31). Notice that (cid:68) v i L k | v j L k (cid:48) (cid:69) = δ kk (cid:48) δ ij , with k , k (cid:48) ∈ { } and i , j ∈ {
0, 1 } since, (cid:68) v i L k | v j L k (cid:48) (cid:69) = (cid:42) i L | A (cid:48)† k − (cid:113) ˜ p (6) k A (cid:48) k (cid:48) − (cid:113) ˜ p (6) k (cid:48) | j L (cid:43) = 1 (cid:113) ˜ p (6) k ˜ p (6) k (cid:48) (cid:68) i L | A (cid:48)† k A (cid:48) k (cid:48) | j L (cid:69) = 1 (cid:113) ˜ p (6) k ˜ p (6) k (cid:48) α (cid:48) kk (cid:48) δ ij = δ kk (cid:48) δ ij , (84)where we have used the fact that the square (Hermitian) matrix α (cid:48) kk (cid:48) equals (cid:113) ˜ p (6) k ˜ p (6) k (cid:48) δ kk (cid:48) . The recovery superoperator R ↔ { R l } with l = 1,.., 32 is defined as [18], R l def = V l (cid:88) i =0 (cid:12)(cid:12) v i L l (cid:11) (cid:10) v i L l (cid:12)(cid:12) , (85)where the unitary operator V l is such that V l (cid:12)(cid:12) v i L l (cid:11) = | i L (cid:105) for i ∈ {
0, 1 } . Notice that, R l def = V l (cid:88) i =0 (cid:12)(cid:12) v i L l (cid:11) (cid:10) v i L l (cid:12)(cid:12) = | L (cid:105) (cid:10) v L l (cid:12)(cid:12) + | L (cid:105) (cid:10) v L l (cid:12)(cid:12) . (86)If turns out that the 32 recovery operators are given by, R l +1 = R A (cid:48) l (cid:112) ˜ p (cid:48) l = ( | L (cid:105) (cid:104) L | + | L (cid:105) (cid:104) L | ) A (cid:48) l (cid:112) ˜ p (cid:48) l , (87)with l ∈ { } . Notice that R ↔ { R l } is a trace preserving quantum operation since, (cid:88) l =1 R † l R l = (cid:88) l =1 (cid:32)(cid:88) i L | i L (cid:105) (cid:10) v i L l (cid:12)(cid:12)(cid:33) † (cid:88) j L | j L (cid:105) (cid:68) v j L l (cid:12)(cid:12)(cid:12) = (cid:88) l =1 (cid:88) i L , j L (cid:12)(cid:12) v i L l (cid:11) (cid:104) i L | j L (cid:105) (cid:68) v j L l (cid:12)(cid:12)(cid:12) = (cid:88) l =1 (cid:88) i L (cid:12)(cid:12) v i L l (cid:11) (cid:10) v i L l (cid:12)(cid:12) = I × ,(88)since (cid:8)(cid:12)(cid:12) v i L l (cid:11)(cid:9) with l = 1,..., 32 and i L ∈ {
0, 1 } is an orthonormal basis for H . Finally, the action of this recoveryoperation R on the map Λ (6) ( ρ ) in (71) leads to,Λ (6)recover ( ρ ) ≡ (cid:16) R◦ Λ (6) (cid:17) ( ρ ) def = − (cid:88) k =0 32 (cid:88) l =1 ( R l A (cid:48) k ) ρ ( R l A (cid:48) k ) † . (89) Entanglement Fidelity . We want to describe the action of R◦ Λ (6) restricted to the code subspace C . Recalling that A (cid:48) l = A (cid:48)† l , it turns out that, (cid:104) i L | R l +1 A (cid:48) k | j L (cid:105) = 1 (cid:112) ˜ p (cid:48) l (cid:104) i L | L (cid:105) (cid:68) L | A (cid:48)† l A (cid:48) k | j L (cid:69) + 1 (cid:112) ˜ p (cid:48) l (cid:104) i L | L (cid:105) (cid:68) L | A (cid:48)† l A (cid:48) k | j L (cid:69) . (90)We now need to compute the 2 × R l A (cid:48) k ] |C of each R l A (cid:48) k with l = 0,.., 31 and k = 0,.., 2 − R l +1 A (cid:48) k ] |C def = (cid:18) (cid:104) L | R l +1 A (cid:48) k | L (cid:105) (cid:104) L | R l +1 A (cid:48) k | L (cid:105)(cid:104) L | R l +1 A (cid:48) k | L (cid:105) (cid:104) L | R l +1 A (cid:48) k | L (cid:105) (cid:19) . (91)For l , k = 0,.., 31, we note that [ R l +1 A (cid:48) k ] |C becomes,[ R l +1 A (cid:48) k ] |C = (cid:68) L | A (cid:48)† l A (cid:48) k | L (cid:69) (cid:68) L | A (cid:48)† l A (cid:48) k | L (cid:69) = (cid:113) ˜ p (cid:48) l δ lk (cid:18) (cid:19) , (92)4while for any pair ( l , k ) with l = 0,.., 31 and k >
31, it follows that, (cid:104) L | R l +1 A (cid:48) k | L (cid:105) + (cid:104) L | R l +1 A (cid:48) k | L (cid:105) = 0. (93)We conclude that the only matrices [ R l A (cid:48) k ] |C with non-vanishing trace are given by [ R l +1 A (cid:48) l ] |C with l = 0,.., 31 where,[ R l +1 A (cid:48) l ] |C = (cid:113) ˜ p (cid:48) l (cid:18) (cid:19) . (94)Therefore, the entanglement fidelity F (6)conc ( µ , p ) defined as, F (6)conc ( µ , p ) def = F (6)conc (cid:18) I × , R◦ Λ (6) (cid:19) = 1(2) − (cid:88) k =0 32 (cid:88) l =1 (cid:12)(cid:12)(cid:12) tr (cid:16) [ R l A (cid:48) k ] |C (cid:17)(cid:12)(cid:12)(cid:12) , (95)becomes, FIG. 3: Entanglement fidelity vs. memory parameter µ with p = 10 − for model II: concatenated code (dashed line), DFS(thin solid line) and three-qubit bit flip code (thick solid line). F (6)conc ( µ , p ) = p p + 2 p p p + p p p (4 p + p ) + 4 p p p p + p p p p p + 3 p p p p ++4 p p p p + 3 p p p p + p p p p p + p p p ( p + 4 p ) + 2 p p p + p p . (96)Substituting (25) into (96), we finally get F (6)conc ( µ , p ) = µ (cid:0) − p + 24 p − p + 8 p (cid:1) + µ (cid:0) p − p + 130 p − p + 10 p (cid:1) ++ µ (cid:0) − p + 240 p − p + 128 p − p + 2 p (cid:1) ++ µ (cid:0) p − p + 252 p − p + 10 p + 2 p (cid:1) ++ µ (cid:0) − p + 120 p − p + 24 p + 12 p − p (cid:1) + (cid:0) p − p + 18 p + 4 p − p + 1 (cid:1) . (97)The threshold curves for the concatenated code defined in (73) for the model I appear in Figure 1. It turns out thatthe concatenated code does not work in the region delimited by the two dashed lines. We emphasize that in view ofequations (43), (67) and (97), it turns out that the concatenated code outperforms the bit-flip code in regions withvery high memory parameter values and outperforms the DFS in regions with low memory parameter values. In5particular, the relevance of the concatenation trick shines where the weaknesses of the inner and outer codes (yet notconcatenated) are more pronounced, that is in regions with both high error probability and low memory parametervalues. It is within this area that we can identify the region where the concatenated code outperforms both the innerand the outer codes. However, there is a big parametric region characterized by intermediate values of the degree ofmemory (see Figure I) where the concatenation trick does not work well for model I. On the contrary, we will showthat the quantum coding trick is particularly useful for the model II in the presence of partial correlations. B. Model II
We consider Λ ( n ) µ ( ρ ) in (19) with n = 6. Technical details will be omitted. They can be obtained by following theline of reasoning presented when studying the concatenated code applied to the model I. It turns out that, F (6)conc ( µ , p ) = µ (cid:0) − p + 24 p − p − p + 6 p (cid:1) + (cid:0) p − p + 18 p + 4 p − p + 1 (cid:1) . (98)The threshold curve for the concatenated code defined in (73) for the model II appears in Figure 2. It follows thatwhile none of the two codes is effective in the extreme limit when the other is, the three-qubit bit flip (phase flip) codestill works for correlated errors, whereas the error avoiding code does not work in the absence of correlations. Theconcatenated code works everywhere except below the thick solid curve in Figure 2. Here there is a parametric regioncharacterized by partial correlations and delimited by the dashed and thin solid lines where only the concatenated codeis effective. Furthermore, from (49), (69) and (98), it follows that the concatenated code is especially advantageousfor the model II for partially correlated error operators. For the sake of clarity, in Figure 3 we plot the entanglementfidelities (69) (thin solid line), (98) (dashed line) and (49) (tick solid line) for p = 10 − . For such value of the errorprobability, the error avoiding code only works for µ (cid:38) .
44 (threshold value obtained from Figure 2), the three-qubitbit flip code only works for µ (cid:46) .
30 (threshold value obtained from Figure 2) while the concatenated code withentanglement fidelity given in (98) is is efficient for any value of µ ∈ [0, 1]. V. FINAL REMARKS
In this article, we studied the performance of simple error correcting and error avoiding quantum codes together withtheir concatenation for correlated noise models. For model I, a bit-flip (phase-flip) noisy Markovian memory channel,we have applied both the three-qubit bit flip (phase flip) and a suitable error avoiding code. The performance of thecodes was quantified in terms of the entanglement fidelities in (43) and (67). The performance of the concatenatedcode applied to model I appears in (97). In Figure 1, we have plotted the parametric regions where error correctionis effective. We have presented a similar analysis for the model II, a memory channel defined as a memory degreedependent linear combination of memoryless channels with Kraus decompositions expressed solely in terms of tensorproducts of X -Pauli ( Z -Pauli) operators (model II). The performance of the codes was quantified in terms of theentanglement fidelities in (49) and (69). The performance of the concatenated code applied to model II appears in(98). In Figure 2, we have plotted the parametric regions where error correction schemes are effective.Our analysis explicitly shows that while none of the two codes is effective in the extreme limit when the other is,the three-qubit bit flip (phase flip) code still works for correlated errors, whereas the error avoiding code does notwork in the absence of correlations. Finally, our final finding leads to conclude that the concatenated code in (73) isparticularly advantageous for model II in the regime of partial correlations (see Figure 3). Acknowledgments
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