Generality of the concatenated five-qubit code
aa r X i v : . [ qu a n t - ph ] O c t Generality of the concatenated five-qubit code
Long Huang, Bo You, Xiaohua Wu, ∗ and Tao Zhou † College of Physical Science and Technology, Sichuan University, Chengdu 610064, China School of Physical Science and Technology, Southwest Jiaotong University, Chengdu 610031, China (Dated: August 13, 2018)In this work, a quantum error correction (QEC) procedure with the concatenated five-qubit code is used toconstruct a near-perfect effective qubit channel (with a error below − ) from arbitrary noise channels. Theexact performance of the QEC is characterized by a Choi matrix, which can be obtained via a simple and explicitprotocol. In a noise model with five free parameters, our numerical results indicate that the concatenated five-qubit code is general: To construct a near-perfect effective channel from the noise channels, the necessary sizeof the concatenated five-qubit code depends only on the entanglement fidelity of the initial noise channels. PACS numbers: 03.67.Lx, 03.67.Pp,
I. INTRODUCTION
In quantum computation and communication, quantum er-ror correction (QEC) is necessary for preserving coherentstates from noise and other unexpected interactions. Basedon classic schemes using redundancy, Shor [1] has champi-oned a strategy where a bit of quantum information is storedin an entanglement state of nine qubits. This scheme permitsone to correct any error incurred by any of the nine qubits. Forthe same purpose, Steane [2] has proposed a protocol that usesseven quits. The five-qubit code was discovered by Bennett,DiVincenzo, Smolin and Wootters [3], and independently byLaflamme, Miquel, Paz and Zurek [4]. The QEC conditionswere proved independently by Bennett and co-authors [3], andby Knill and Laflamme [5]. The protocols above with differ-ent quantum error correction codes (QECCs) can be viewedas active error correction. There are passive error avoidingtechniques such as the decoherence-free subspaces [6–8] andnoiseless subsystem [9–11]. Recently, it has been proved thatall the active and passive QEC methods can be unified [12–14].With the known codes constructed in Refs. [2–4], the stan-dard QEC procedure is designed according to the principleof perfect correction for arbitrary single-qubit errors, whereone postulates that single-qubit errors are the dominant termsin the noise process [15]. Recently, an optimization-based ap-proach to QEC was explored. In each case, rather than correct-ing for arbitrary single-qubit errors, the error recovery schemewas adapted to model for the noise, with the goal to maximizethe fidelity of the operation [16–19]. Robust channel-adaptedQEC protocols, where the uncertainty of the noise channel isconsidered, have also been developed [20–22].For some tasks such as storing or transferring a qubit ofinformation, a near-idealized channel is usually required. Ifthe fidelity obtained from error correction is not high enough,the further increase in levels of concatenation is necessary.In the previous works [23–25], the application of QEC withthe concatenated code was discussed for the Pauli channel ∗ [email protected] † [email protected] which includes the depolarizing channel as the most impor-tant example. In general, a QEC protocol contains three steps:encoding, error evolution, and decoding. When the concate-nated code is used for encoding, there are two known meth-ods for decoding: the widespread blockwise hard decodingtechnique and the optimal decoding using a message-passingalgorithm [23]. As shown by Poulin, the Monte Carlo resultsusing the five-qubit and Steane’s code on depolarizing channelreveal significant advantages of message-passing algorithms.For the depolarizing channel, the concatenated five-qubit codeis also more efficient than the concatenated seven-qubit code.In the present work, we shall focus on the following ques-tions: For an arbitrary noise model, instead of finding theoptimal QEC protocol adapted to it, is it possible for us toconstruct a near-perfect channel with a error below − byperforming a QEC procedure with the concatenated five-qubitcode? In order to answer this question, two important meth-ods developed in the previous works are applied here. At first,following the idea in Ref. [3], we shall show that the block-wise decoding can be carried out in a way without perform-ing the error syndrome, and the realization needs some ad-ditional quantum resources, required in the message-passingalgorithms. The other method comes from the recent workswhere several general schemes have been developed for de-scribing the exact performance of the QEC procedure basedon a N l -qubit concatenated code [24, 25]. The main ideas inthese schemes can be summarized in the following: First, theexact performance of the QEC with a N l -qubit code is de-noted by an effective Choi matrix; Secondly, the Choi matrixin the l -th level of concatenation is obtained by simulating thestandard quantum process tomography (SQPT) [15, 26–30].Based on these results, the Choi matrix in the ( l + 1) -th levelcan be obtained in a similar way. The advantage of introduc-ing the effective Choi matrix is clear: Instead of directly work-ing in the N l -dimensional Hilbert space, one could alwayssimulate the error correction in each level of concatenation ina N -dimensional system.For a general noise model, with five free parameters, ournumerical simulation indicates that the concatenated five-qubit code is general: The QEC protocol with the concate-nated five-qubit code, which is able to construct a near-perfecteffective channel from the noise depolarizing channels, is alsosufficient to complete the same task for other types of noisechannels with the same channel fidelity.The content of present work is organized as follows. InSec. II, we construct an error correction protocol with the five-qubit code [3], and a chosen unitary transformation is shownto be sufficient for correcting the errors of the principle sys-tem. In Sec. III, an explicit scheme is designed to obtain theeffective Choi matrix. For three types of channels, the de-polarizing channel, the amplitude damping channel and thebit-flip channel, the effective Choi matrices obtained by per-forming QEC with the concatenated five-qubit code is shownin Sec. IV. In Sec. V, it is shown that the concatenated seven-qubit is not general. In Sec. VI, a noise model containing fivefree parameters is constructed, and we argue that the concate-nated five-qubit code is general. Finally, we end our workwith a short discussion. II. UNITARY REALIZATION OF QUANTUM ERRORCORRECTION
The N -qubit code concatenated with itself L times yieldsa N L -qubit code, providing a better error resistance with in-creasing L . The direct simulation of the quantum dynamics,coding and encoding procedure require massive computationresources. By following the idea in Refs. [24, 25], the con-cept that the exact performance of QEC can be described byan effective channel, will make the calculation simplified.Let us consider the QEC protocol with code in Ref. [3], | L i = 14 [ | i + | i + | i + | i + | i − | i − | i − | i−| i − | i − | i − | i−| i − | i − | i + | i ] , and | L i = 14 [ | i + | i + | i + | i + | i − | i − | i − | i−| i − | i − | i − | i−| i − | i − | i + | i ] . We use H S for the two-dimensional principle system,where the basis vectors are denoted by | i and | i , while theancilla system system lies in a -dimensional Hilbert space H A with the basis {| a m i} m =0 ,..., . The standard way to getthe effective noise channel is depicted in Fig. 1(a). It containsthe following steps.(i) The encoding procedure can be realized with a unitarytransformation U , U | a i ⊗ | i → | L i , U | a i ⊗ | i → | L i . (1)(ii) The noise evolution is denoted by Λ . The five-qubitcode above is designed to correct the set of single-qubit er-rors, { E m } m =0 , ,..., . Usually, E is fixed to be identity op-erator ˆ I , and each E m ( m = 0) is one of the Pauli operators FIG. 1. (a) The way of getting the effective channel from the standardQEC protocol including encoding, noise evolution, recovery and de-coding. (b) The two processes,
R ◦ U † and U † ◦ ˜ R , are equivalent.(c) When U † for decoding is fixed according Eq. (6), the process ˜ R can be decomposed as ˜ R SA = ˜ R A ⊗ I S . Certainly, it can be movedaway. (d) Our protocol where the chosen unitary transformation issufficient to correct the errors of the principle system. The errors ofthe ancilla system is left to be uncorrected. ˆ σ ij ( i = 1 , ..., , j = x, y, z ) . With the logical codes, one couldintroduce a set of normalized states | m, + i = E m | L i , | m, −i = E m | L i . (2)Since the set of errors, { E m } m =0 ,..., , could be perfectlycorrected, there should be ˆ P C E † m E n ˆ P C = δ mn ˆ P C , where ˆ P C = | L ih L | + | L ih L | . Therefore, one may easily ver-ify that {| m, ±i} m =0 form an orthogonal basis.(iii) With the denotation | a m , i i = | a m i ⊗ | i i , the re-covery operation can be described by a process R such that R ( ρ SA ) = P m =0 R m ρ SA R † m , where the Kraus operators R m are [24], R m = | a m , ih m, + | + | a m , ih m, −| . (3)(iv) The decoding is realized by U † , the Hermite conjugateof U , U U † = I ⊗ , and the effective channel ¯ ε is ˜ ε ( ρ S ) = Tr A [ U † ◦ R ◦ Λ ◦ U ( | a ih a | ⊗ ρ S )] . (4)The above is the standard QEC protocol, and the unitarytransformation U ( U † ) used for encoding (decoding) is notunique. In this work, however, the unitary transformation isfixed as U | a m i ⊗ | i → | m, + i , U | a m i ⊗ | i → | m, −i , (5)or in the equivalent form U † | m, + i → | a m i ⊗ | i , U † | m, −i → | a m i ⊗ | i , (6)Certainly, | L i = | , + i , | L i = | , −i . [It should be empha-sized that the U † introduced above is nothing else but the U used in the Eq. (87) of the original work in Ref. [3].] As ithas been argued in Ref. [3], the recovery process R is not anecessary step, since the U † defined in Eq. (6) is sufficient forcorrecting the errors of the principle system. One can observethat, the following two processes are equivalent R ◦ U † ≡ U † ◦ ˜ R , (7)where the process ˜ R is defined as ˜ R = U ◦ R ◦ U † . (8)Furthermore, it can be expressed with a more explicit way, ˜ R ( ρ SA ) = P m =0 ˜ R m ( ρ SA ) ˜ R † m , where the Kraus operators ˜ R m take the form ˜ R m = U R m U † . By some simple algebra,one may get ˜ R m = | a ih a m | ⊗ I , and one can easily ver-ify that: After an arbitrary state ρ SA of the jointed system issubjected to the process ˜ R , the state of the principle systemremains unchanged, say, ρ S = Tr A [ ρ SA ] = Tr A [ ˜ R ( ρ SA )] . (9)According to this analysis, the process ˜ R can be moved away.It has been shown in Fig. 1(d) that a simplified protocol toobtain the effective channel is defined as, ˜ ε ( ρ S ) = Tr A [ U † ◦ Λ ◦ U ( | a ih a | ⊗ ρ S )] . (10) III. THE STANDARD QUANTUM PROCESSTOMOGRAPHY
To get the complete information about the effective chan-nel, we shall introduce a convenient tool where a bounded ma-trix in H is related to a vector in the enlarged Hilbert space H ⊗ . Let A be a bounded matrix in the -dimensional Hilbertspace H , with A ij = h i | A | j i the matrix elements for it, andan isomorphism between A and a -dimensional vector | A ii is defined as | A ii = √ A ⊗ I | S + i = X i,j =0 A ij | ij i , (11)where | S + i is the maximally entangled state for H ⊗ , and | S + i = √ P k =0 | kk i with | ij i = | i i ⊗ | j i . This isomor-phism provides a one-to-one mapping between the matrix andits vector form.For a quantum process ε , the Kraus operators { A m } can bedescribed by a corresponding Choi matrix, χ ( ε ) = X m | A m iihh A m | . (12)Via the isomorphism above, this matrix can also be rewrit-ten as χ ( ε ) = 2 ¯ ρ , with ¯ ρ = ε ⊗ I( | S + ih S + | ) . For the nor-malized state ¯ ρ , Schumacher’s entangling fidelity is definedas F = h S + | ¯ ρ | S + i , and it provides a measure for how wellthe entanglement is preserved by the quantum process ε [31].Certainly, one may calculate the entangling fidelity F ( ε ) = 12 h S + | χ ( ε ) | S + i . (13) With χ known, one can derive the Kraus operators of ε .This can be completed through the following simple protocol:The eigenvalues λ m and the corresponding eigenvectors | Φ m i of χ can be easily calculated, say, χ = P m λ m | Φ m ih Φ m | .Suppose that | Φ m i can be expanded as | Φ m i = P ij c mij | ij i ,with c mij = h ij | Φ m i the expanding coefficients. Then, theKraus operators A m can be expressed as A m = √ λ m X i,j =0 c mij | i ih j | . (14)With these operators, one may verify that the relation inEq. (12) is recovered.The effective channel can be obtained via the performingthe SQPT [15]. Here, it should be mentioned that the way ofperforming SQPT is not limited. In the present work, we shallapply the protocol presented in Ref. [32]. For convenience, abrief review of this protocol is organized in following: Intro-ducing the set of operators, say, E cd = | c ih d | ( c, d = 0 , ,we take them as the inputs for the principle system, and for agiven E cd , the corresponding output is ˜ ε ( E cd ) = Tr A [ U † ◦ Λ ◦ U ( | a ih a | ⊗ | c ih d | )] . (15)Then, one may introduce the coefficients ˜ λ ab ; cd = h a | ˜ ε ( E cd ) | b i , (16)and ˜ ε ( E cd ) can be expanded as ˜ ε ( E cd ) = P a,b =0 ˜ λ ab ; cd | a ih b | . The Choi matrix of the effectivechannel ˜ ε in Eq. (10), can be expanded as χ (˜ ε ) = X a,b,c,d =0 ˜ χ ab ; cd | ab ih cd | , (17)with ˜ χ ab ; cd its matrix elements, ˜ χ ab ; cd = h ab | χ (˜ ε ) | cd i . (18)It has been shown in Ref. [32] that the ˜ χ ab ; cd can be obtainedin a simple way, ˜ χ ab ; cd = ˜ λ ac ; bd . (19) IV. EXACT PERFORMANCE OF THE CONCATENATEDFIVE-QUBIT CODE
Now, we shall restrict our attention to the uncorrelated er-rors. We use ε : { A m } to denote the quantum process of eachtwo-dimensional subsystem. Formally, Λ = ε ⊗ . Let us con-sider the lowest level of error correction for the depolarizingchannel ε DP , A = p F ˆ I , A i = r − F σ i (20)with i = 1 , , and F its channel fidelity. We shall take it asan explicit example to show how the SQPT is completed.(a) Let | ih | the input of the principle system. The corre-sponding output is denoted by ˜ ε ( | ih | ) . With the equation ˜ ε ( | ih | ) = Tr A [ U † ◦ Λ ◦ U ( | a ih a | ⊗ | ih | ] , one has ˜ ε ( | ih | ) = (cid:18) a
00 1 − a (cid:19) , where a = (1 + 2 F ) (37 − F + 144 F − F ) .Based on the definition in Eq. (16), the four matrix elements, ˜ λ ab ;00 ( a, b = 0 , ), are ˜ λ = a, ˜ λ = ˜ λ = 0 , ˜ λ = 1 − a. (b) Similarly with step (a), one may also obtain ˜ ε ( | ih | ) = (cid:18) a −
10 0 (cid:19) , ˜ ε ( | ih | ) = (cid:18) a − (cid:19) , ˜ ε ( | ih | ) = (cid:18) − a a (cid:19) , and therefore, ˜ λ = ˜ λ = ˜ λ = 0 , ˜ λ = 2 a − , ˜ λ = ˜ λ = ˜ λ = 0 , ˜ λ = 2 a − , ˜ λ = 1 − a, ˜ λ = ˜ λ = 0 , ˜ λ = a. (c) With all the matrix elements, ˜ λ ab ; cd , the so-called matrix ˜ λ can be orgnized as ˜ λ = a − a a − a − − a a . According to the one-to-one relation in Eq. (19), the effectiveChoi matrix ˜ χ is ˜ χ ( ε DEP ) = a a −
10 1 − a − a a − a . (21)(d) The entangling fidelity of the effective channel ˜ χ is de-noted by F , and now Eq. (13) can be rewritten as F = 14 ( ˜ χ + ˜ χ + ˜ χ + ¯ χ ) . Based on it, there should be F ( ε DEP ) = 127 (5+20 F − F +40 F +160 F − F ) . (22)A parameter p can be used to characterize the depolarizingchannel, say, A = p − p/ I , A i = √ p/ σ i , and withthe relation F = 1 − p/ , Eq. (22) can be rewritten as F ( ε DEP ) = 1 − p + 758 p − p + 98 p . (23) This is the same as the result by Reimpell and Werner [16].Furthermore, by requiring that F ≥ F , we have the thresh-old p < . (or F > . ), the condition under which thefive-qubit code works for the depolarizing channel.(e) By some simple algebra, the eigenvalues λ m and thecorresponding eigenvectors | Φ m i can be derived, λ = 2 F , λ = λ = λ = 2(1 − F )3 , | Φ i = 1 √ | i + | i ) , | Φ i = 1 √ | i + | i ) , | Φ i = 1 √ | i − | i ) , | Φ i = 1 √ | i − | i ) . With Eq. (14), one may verify that the effective Choi matrixcan be also expressed by a set of Kraus operators ˜ A m , ˜ χ = P m | ˜ A m iihh ˜ A m | , where ˜ A = p F ˆ I , ˜ A i = r − F σ i (24)with i = 1 , , and F the entangling fidelity. Obviously, theeffective channel is also a depolarizing channel [24, 25].(f) With the Kraus operators above, a new supper operator ˜ ε : ˜ ε ( ρ ) = P ˜ A m ρ ˜ A † m can be defined. Let Λ = ˜ ε ⊗ andfollow the steps from (a) to (e), we can obtain the effectivechannel by the -qubit concatenated code. By repeating theargument above, we can get the effective channel from the L -qubit concatenated code.The protocol developed above, used for the depolarizingchannel ˜ ε DEP , can be easily generalized for other cases.For instance, the amplitude damping channel ε AD , has beenwidely discussed in previous works, and the Kraus operatorsare now A = (cid:18) √ − γ (cid:19) , A = (cid:18) √ γ (cid:19) , (25)where γ is the damping parameter. The entangling fidelity of itis F = (1 + √ − γ ) . When the five-qubit code is appliedfor correcting the amplitude damping errors, the entanglingfidelity of the effective channel is F ( γ ) = 14 [1 + 14 (1 − γ ) (4 + 8 γ − γ + γ )+ 12 p − γ (4 + 2 γ − γ + 5 γ )] . (26)Another important case is the bit-flip channel ε BF with A = p F (cid:18) (cid:19) , A = p − F (cid:18) (cid:19) , (27)where F is the entangling fidelity. For a reason whichwill be clear soon, here we first introduce the denotations, ε DP ≡ ε ( ω , F ) , ε AD ≡ ε ( ω , F ) and ε BF ≡ ε ( ω , F ) ,where F is the fidelity of the initial channel without per-forming QEC. After performing QEC with the l -qubit code,the effective channel is denoted by ε l ( ω, F ) with the setting ( ω, F ) indicating that the effective channel is originated fromthe initial ε ( ω, F ) . With the Choi matrix χ ( ε l ( ω, F )) , theentangling fidelity of the effective channel can be calculatedas F l ( ω, F ) = 12 h S + | χ ( ε l ( ω, F )) | S + i . (28)In the present work, a quantum channel is called near-perfect if the entangling fidelity has a value above − − .For a given initial channel ε ( ω, F ) , we introduce the quan-tity L ( ω,F ) and let L ( ω,F be the minimum size of the con-catenated code sufficient for constructing a near-perfect chan-nel, F L ( ω,F ≥ − − . Correspondingly, a L ( ω,F -qubitcode is said to be general if L ( ω,F ) is independent of ω , whichis used for denoting the actual type of the error model.When the concatenated code is applied, the effective χ ( ε l ( ω, F )) ( ω = ω ) usually does not have an analyticalform. Noting that χ ( ε l ( ω , F )) always represents a depolar-izing channel, we suppose that χ ( ε l ( ω, F )) can be approxi-mated by χ ( ε l ( ω , F )) , and the error of the approximation ischaracterized by the distance measure D l ( ω, F ) = 14 | χ ( ε l ( ω, F )) − χ ( ε l ( ω , F )) | , (29)with | A | = √ A † A [15]. Especially, if D l ( ω, F ) ≪ ,the entangling fidelities of the two channels, ε l ( ω, F ) and ε l ( ω , F ) , almost have the same value.Under the condition that the entangling fidelity F ( F ≥ . ) is fixed, besides the steps from (a) to (f) for getting theeffective channel, we added another two ones:(g) With the Choi matrix χ ( ε ( ω, F )) corresponding to theeffective channel ε ( ω, F ) , the entangling fidelity F ( ω, F ) is decided by Eq. (28), and the distance D ( ω, F ) is calcu-lated according to Eq. (29).(h) With a simple program, the Kraus operators of the effec-tive channel can be decided, and the calculation for the effec-tive channel in the second level starts from step (a). The cal-culation for a given ε ( ω, F ) can be terminated if F l ( ω, F ) ≥ − − .Based on the iterative protocol developed above, the effec-tive channels in each level of the concatenation can be workedout. For the typical case where F = 0 . , as shown in Ta-ble I, we have two observations that: (I) For all the possiblechannels, a perfect effective channel (with a error below − )can be constructed by using the same concatenated five-qubitcode; (II) Meanwhile, the resulted effective channel can beapproximated by the depolarizing channel. As shown in TableII, the error of the approximation approaches to zero when l ,the level of concatenation, is increased.The effective channel for other cases, where F takes dif-ferent values, have also been calculated. Our calculation in-dicates that the two observations, (I) and (II) above, are inde-pendent of the choice of F . TABLE I. The fidelity F l ( ω, F ) l F l ( ω ) F l ( ω ) F l ( ω ) l D l ( ω , . D l ( ω , . . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − V. EXACT PERFORMANCE OF THE CONCATENATEDSEVEN-QUBIT CODE
In this section, we will show that the concatenated seven-qubit code is not general, or in other word, the number oflevels for the concatenation, which is necessary for construct-ing a near-perfect channel, is dependent on the actual typeof the noise. For the seven-qubit code [2], we can also de-fine a unitary transformation V in a -dimensional Hilbertspace for encoding, and its inverse V † is applied for decod-ing. Let | L i and | L i be the logical codes, select a set ofcorrectable errors { E m } m =0 including: The identity operator E = I ⊗ ), all the rank-one Pauli operator σ ni ( i = x, y, z , n = 1 , , ..., , and a number of 44 rank-two operators like σ ⊗ σ , σ ⊗ σ , ..., etc. , and with the definition | m, + i = E m | L i , | m, −i = E m | L i , we find that the set of normalized vectors {| m, ±i} m =0 form the basis of the -dimensional Hilbert space. Use {| a m i} m =0 to denote the basis of ancilla system, and with thedenotations | a m , i = | a m i ⊗ | i and | a m , i = | a m i ⊗ | i ,the unitary transformation V is V = X m =0 (cid:0) | m, + ih a m , | + | m, −ih a m , | (cid:1) . Finally, define
Λ = ε ⊗ and | a i = | i , and the effec-tive channel, which is obtained by performing QEC procedurewith the seven-qubit code, can be obtained as ˜ ε ( ρ S ) = Tr A [ V † ◦ Λ ◦ V ( | a ih a | ⊗ ρ S )] . (30)To make sure that our program works in a perfect way, weconsider a scenario where the seven-qubit code is applied forthe amplitude damping in Eq. (26). The analytical expression FIG. 2. (Color online) The entangling fidelities of the effective chan-nel when the five-qubit code and the seven-qubit code are appliedagainst the amplitude damping errors. The exact function in Eq. (26)is shown with the solid line while the one in Eq. (31) for the seven-qubit code, is in the dash line.TABLE III. The fidelity obtained for the concatenated Steane’s code1 F l ( ε DEP ) F l ( ε AD ) F l ( ε BF ) > − − > − − > − − > − − > − − > − − > − − for the entangling fidelity of the effective channel is F ( γ ) = 14 [1 + p − γ (2 + γ )+ (1 − γ ) γ − γ + 21 γ − γ ) (31) + √ − γ
16 ( − γ + 180 γ − γ + 39 γ )] . One can easily check that this result recovers the numericalone given in Ref. [19], and the entangling fidelities of the ef-fective channel when the five-qubit code and the seven-qubitcode are used against the amplitude errors are compared inFig. 2.As shown in the above section, one can get the exact perfor-mance of the QEC protocol based on the concatenated Steanecode. For three types of noise models, the depolarizing chan-nel ε DEP , the amplitude damping channel ε AD , and the bit flipchannel ε BF , the corresponding entangling fidelities F l havebeen calculated under the condition that the fidelity of the un-corrected channels is fixed to be F = 0 . . Our results, listedin Table III, demonstrate that the necessary numbers of theconcatenation are dependend on the actual types of the error. VI. THE GENERAL NOISE MODEL
In this section, we shall show that the observation, whichhas been observed for the amplitude damping and bit flip channels, is general. Let { ¯ A m } to be a set of Kraus-operators,introduce an arbitrary × unitary transformation, U ( θ, φ ) = (cid:18) cos θ sin θ exp {− iφ }− sin θ exp { iφ } cos θ (cid:19) , and another set of operators { A m } can be defined as A m = U ( θ, φ ) ¯ A m U † ( θ, φ ) , (32)where θ and φ are two free parameters. Three free parameters, α , β , and γ , are used to define the following four operators, ¯ A = (cid:18) cos α
00 sin β cos γ (cid:19) , ¯ A = (cid:18) α sin γ (cid:19) , ¯ A = (cid:18) β sin γ (cid:19) , ¯ A = (cid:18) sin α cos γ
00 cos β (cid:19) . (33)Now, let us recall some discussions about using the Blochsphere representation to describe the single-qubit chan-nel [15]. With ~r the Bloch vector for an input state ρ , and ~ ¯ r for the output state, ¯ ε ( ρ ) = P m ¯ A m ρ ( ¯ A m ) † , on can obtaina map ~r → ~ ¯ r = M~r + ~δ, (34)where M is a × real matrix, and ~δ is a constant vector. Thisis an affine map, mapping the Bloch vector into itself. The setof operators in Eq. (33) can be described as ¯ r x ¯ r y ¯ r z = η ⊥ η ⊥
00 0 η z r x r y r z + δ z , (35)with the coefficients η ⊥ = sin( α + β ) cos γ,η z = 1 − (sin α + sin β ) sin γ, (36) δ z = (sin β − sin α ) sin γ. This affine map can be roughly classified into the followingtwo cases: the centered map with δ z = 0 (if α = β ) and thenon-centered one with δ z = 0 .The centered map of Eq. (35) is equivalent with the Pauli-channel, √ p ˆ I , √ p x ˆ σ x , √ p y ˆ σ y , √ p z ˆ σ z , and the parameters p i are given by p x = p y = sin α sin γ , p z = (cos α − sin α cos γ ) , and p = 1 − p x − p y − p z . Certainly, it alsocontains the following two important situations: If α is fixedas cos α = cos γ + sin γ √ γ , sin α = 1 √ γ , one can obtain a depolarizing channel, while for γ = 0 and anarbitrary α , we have the phase-flip channel.With the U ( θ, φ ) introduced above, our noise modelshould also contain the bit-flip channel and bit-phase flip chan-nel. For the non-centered case, let α = 0 and β = π , and wecan come to the amplitude damping channel, ¯ A = (cid:18) γ (cid:19) , ¯ A = (cid:18) γ (cid:19) . For the case where the constraint α + β = π holds, we havethe so-called generalized amplitude damping channel [15], ¯ A = cos α (cid:18) γ (cid:19) , ¯ A = sin α (cid:18) γ (cid:19) , ¯ A = cos α (cid:18) γ (cid:19) , ¯ A = sin α (cid:18) cos γ
00 1 (cid:19) . From the discussions above, it can be seen that nearly all thenoise channel listed in Ref. [15] are included here. Therefore,our noise model, defined in Eqs. (32) and (33), is general.Now, let us consider a typical case where the entanglingfidelity of the uncorrected channel is fixed as F = 0 . . Forthe depolarizing channel, using the result in Eq. (22), we canget the entangling fidelities in each level of the concatenation: F ( ω ) = 0 . , F ( ω ) = 0 . ,F ( ω ) = 0 . , F ( ω ) = 0 . ,F ( ω ) = 0 . , F ( ω ) = 0 . . (37)Therefore, to construct a near perfect channel from the depo-larizing channel with entangling fidelity F = 0 . , the five-qubit code should be concatenated with itself L = 6 times.From Eq. (21) and the known F l ( ω ) , the effective choi matrix χ ( ε ( ω , F )) can be easily calculated.Then, under the condition that F is fixed, F = 0 . , wecan design a program to generate an arbitrary setting for thefive free parameters introduced above. The generated channelis denoted by ε ( ω , F ) . Let Λ = ε ( ω , F ) ⊗ , the effec-tive channel in each l -th level of concatenation can be decidedby following the same method as the amplitude channel inSec. IV. As the result of this run of calculation, we can get theexact values F l ( ω , F ) and D l ( ω , F ) with l = 1 , , ..., .After the calculation of the first one is completed, anotherchannel with a fidelity of . will be generated and denotedby ε ( ω , F ) . Similarly, we have the results F l ( ω , F ) and D l ( ω , F ) with l = 1 , , ..., . Usually, in the m -th run ofcalculation, the generated channel is denoted by ε ( ω m , F ) with F = 0 . . After performing QEC with the -qubit con-catenated code, we can get a series of exact values F l ( ω m , F ) and D l ( ω m , F ) with l = 1 , , ..., . For a fixed value of F ,there is about M ( M ≥ ) examples of noise channel thatwill be generated. Based on the numerical data in each level ofconcatenation, a distance measure D max l ( F ) can be definedto denote the maximum value of error for the approximationdefined in Eq. (29), D max l ( F ) = max { D l ( ω m , F ) } , ≤ l ≤ , ≤ m ≤ M, (38)and the minimum fidelity F min l ( F ) = min { F l ( ω m , F ) } Mm =0 , ≤ l ≤ , ≤ m ≤ M. (39)For the case F = 0 . , our numerical calculation gives F min1 ( F ) = 0 . , F min2 ( F ) = 0 . ,F min3 ( F ) = 0 . , F min4 ( F ) = 0 . ,F min5 ( F ) = 0 . , F min6 ( F ) = 0 . . (40) FIG. 3. (Color online) Numerical results for the maximum distancemeasure D max l ( F ) defined in Eq. (38). The numerical results for D max l ( F ) are given in Fig. 3,where the initial fidelity take different values, 0.9, 0.92, 0.945,0.97, 0.97, 0.992, and 0.9993. For each given initial fidelity,the values of the distance measure D max l ( F ) are given in thedomain ≤ l ≤ L ( ω ,F ) . Our numerical results indicatethat: In each level of the concatenation, the effective channelcan be approximated by a corresponding depolarizing chan-nel. The error of the approximation, which is characterizedby the distance measure D max l ( F ) , approaches zero as thelevel of concatenation is increased.Based on the results in Eq. (37) and Eq. (40), one can ob-serve that: (a) In each level of concatenation, F l ( ω ) − F min l ≥ . When l is increased, this difference will approach to zero.(b) Since that F min6 ( F ) ≥ − − , the concatenated five-qubit code, which is able to construct a near perfect channelfrom the depolarizing channel, is also suitable for the arbi-trary channel with the same initial fidelity as the depolarizingchannel. For the other cases where the initial fidelity takesdifferent values, 0.92, 0.945, 0.97, 0.97, 0.992, our numericalresults, which are depicted in Fig. 4, show that the properties(a) and (b) can be also observed.Therefore, as a direct consequence of the observations, weargue that the concatenated five-qubit is general: Under thecondition that the fidelity is fixed, a L ( ω ,F -qubit code,which is the minimal-sized one for constructing a near-perfectqubit-channel from the depolarizing channels, can be used tocomplete the same task for the general noise channels with thesame initial fidelity F . VII. DISCUSSIONS AND REMARKS
Our present work is based on the assumption that the noiseof quantum channel comes from the interaction between thephysical qubit and the environment. The apparatus, which areused for state preparation, encoding and decoding, are sup-posed to be perfect. The main results of this work are: (I) Asimple and explicit scheme is developed to get the effectiveChoi matrix resulted by the QEC procedure with the concate-nated five-qubit code. (II) Based on this scheme, we haveshown that the QEC procedure, which is optimal for the noisedepolarizing channels, also offers an efficient way to constructa near-perfect channel, where the noise of the physical qubit is
FIG. 4. (Color online) (a) For the general noise error model, the nu-merical results of the minimum fidelity F min l ( F ) defined in Eq. (39)are depicted; (b) Numerical results for the entangling fidelity whenthe initial channels are fixed to be the depolarizing ones. The re-sults show the generality of the five-qubit code. Though the numeralresults look similar in the two viewgraphs, they are not strictly iden-tical. More specifically, for the case F = 0 . , the entangling fideli-ties and minimum fidelities in each level of the concatenation givenin Eq. (37) and Eq. (40) respectively, are not strictly the same. a general one including the amplitude damping. Within a gen-eral noise model, though not a strict proof, our numerical re-sults indicate that the concatenated five-qubit code is general:To construct an effective channel with a error below − , thenecessary number of the levels for concatenation is decidedby the fidelity of the initial channels and it does not depend onthe actual types of noise models.Naturally, there is still an open question: Is the concate-nated five-qubit general for an arbitrary noise channel? Forthe single-qubit case, such a trace-preserving model requiresabout twelve parameters. Under the constraint that the fidelityis fixed, how to effectively generate an arbitrary setting for allthese parameters is still an unsolved problem. From the results of the present work, we guess that the concatenated five-qubitis general. Our argument is based on the following two facts.First, a general QEC protocol does exist, and in the workof Horodeckis’ [33], the so-called twirling procedure hasbeen introduced. Performing twirling on an arbitrary chan-nel ε ( ω, F ) , one may obtain a depolarizing channel ε ( ω , F ) with the unchanged channel fidelity, and before performingQEC with the concatenated five-qubit code, all the initial (un-known) channels may be transferred into the depolarizingchannels through twirling. We may call the so-constructedQEC protocol, which is general for the arbitrary noise models,the term twirling-assisted QEC scheme . As a basic property,the effective channel in each level of concatenation should al-ways be a depolarizing channel.Second, within the general noise model, our numerical re-sults indicate that the QEC protocol based on the concate-nated five-qubit code can be viewed as an approximate real-ization of the twirling-assisted QEC scheme . As shown, theeffective channel in each level can be approximated by thedepolarizing channel while the error of the approximation ap-proaches to zero as the level of concatenation is increased.The blockwise decoding protocol, which is suboptimal, isused in our work. From the work of Poulin [23], it is knownthat the message-passing decoding algorithm is optimal, andby jointing the twirling stage and the message-passing decod-ing protocol together, one can have an optimal way to con-struct a near perfect channel for the arbitrary noise. However,from the results in present work, it is shown that twirling isnot a necessary step if the blockwise decoding is applied. Ac-tually, one may guess that the twirling is also not necessaryfor the QEC with the message-passing decoding algorithm.In other word, the optimal QEC protocol for the depolarizingchannel, which has been developed by Poulin, may also begeneral. We expect this guess could lead to further theoreticalor experimental consequences. ACKNOWLEDGEMENTS
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