Performance of Five-Qubit Code for Arbitrary Error Channels

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The content of the present work is organized as follows. Ecd = c d (c, d = 0, 1) as inputs for the principle system, In Sec. II, the entanglement fidelity and the standard quan- the corresponding| ih | outputs are [15] tum process tomography of quantum channel are briefly in- † ε˜(Ecd) = TrA[ Λ ( a a c d )] troduced. In Sec. III, we will review the error correction pro- U ◦ ◦U | 0ih 0| ⊗ | ih | tocol. In Sec. IV, the analytic results are performed when er- By introducing the coeficients ror models are bit flip, bit-phase flip, phase flip, amplitude damping and generalized amplitude damping in error correc- λãb;cd = a ε˜(Ecd) b , (5) tion with five-qubit code; In Sec. V, the analytic results in h | | i error correction with seven-qubit code are obtained when the The Choi matrix of the effective channel ε˜can be expanded as errors in five-qubit code added another two depolarizing chan- 1 nels. In Sec. VI, the numerical calculation of error correc- χ(˜ε)= χãb cd ab cd , tion with five-qubit code is performed when error channels ; | ih | are generated arbitrarily. In Sec. VII, we end our paper with a,bX,c,d=0 some remarks and discussion. and the matrix elements are

χãb;cd = ab χ(˜ε) cd . (6) II. CHANNEL FIDELITY AND STANDARD QUANTUM h | | i PROCESS TOMOGRAPHY It has been shown in [41], χãb;cd can be obtained in a simple way, Before one can design the protocol of error correction, it is χ˜ = λ˜ . (7) necessary to know how well a quantun process ε can store one ab;cd ac;bd qubit information, and usually, Schumacher’s entanglement Finally, one can come to a relationship between channel fi- fidelity [39] can be used to describe it, delity and elements of Choi matrix χ(˜ε),

F = S+ ε I( S+ S+ ) S+ . (1) 1 h | ⊗ | ih | | i F (ε)= (˜χ +χ ˜ +χ ˜ +χ ˜ ) (8) 4 00;00 00;11 11;00 11;11 Here, S+ = 1/√2( 00 + 11 ) is a maximally entangled state. For| ai quantum channel| i ε|, sayi It is much easier and more accurate to measure the average fidelity than the Choi matrix of a quantum process [27–31], † ε(ρ)= EmρEm, (m =0, 1, 2, 3), and therefore, it is enough to know the fidelity in the prepara- Xm tion of error correction. the entanglement fidelity F can be expressed with the set of Kraus operators Em, m =0, 1, 2, 3 { } III. UNITARY REALIZATION OF QUANTUM ERROR CORRECTION 3 1 F = (TrE )2. (2) 4 m In this section, we will briefly introduce the realization of mX=0 QEC. Following the idea in Refs. [24, 25], the exact perfor- According to the result in Ref. [40], entanglement fidelity F mance of QEC can be quantified by the improvement of the has a beautiful relation with average fidelity F¯, channel fidelity, and this will make the calculation simplified [38]. DF +1 We start the QEC protocol with five-qubit code in Ref. [3], F¯ = . (3) D +1 1 0L = [ 00000 + 10010 + 01001 + 10100 In the present work, D = 2, and in addition, the average fi- | i 4 | i | i | i | i delity F¯ is defined as + 01010 11011 00110 11000 | i − | i − | i − | i 11101 00011 11110 01111 −| i − | i − | i − | i F¯ = dψ ψ ε(ψ) ψ , (4) 10001 01100 10111 + 00101 ], Z h | | i −| i − | i − | i | i and where dψ is the Haar measure, say dψ =1, and the integra- 1 tion is over the normalized state space.R The fidelity averaged 1L = [ 11111 + 01101 + 10110 + 01011 over the entire Hilbert space can be evaluated also by aver- | i 4 | i | i | i | i aging over a state 2-design with a set of mutually unbiased + 10101 00100 11001 00111 | i − | i − | i − | i basis [29]. 00010 11100 00001 10000 −| i − | i − | i − | i If one wants to obtain the completeinformationof the quan- 01110 10011 01000 + 11010 ]. tum process, the SQPT can be employed. The way of per- −| i − | i − | i | i forming SQPT is not limited and in the present work, the As depicted in Fig. 1 (a), the standard way to get the effec- protocol in Ref [41] will be used. For a set of operators tive noise channel contains the following steps: (i) A unitary 3

Hilbert space, and its inverse V † is the decoding process. 63 The set of correctable errors Em m=0 consists of the iden- ˆ⊗7 { } tity operator E0 = I2 , all the rank-one Pauli operators j σi (i = x,y,z,j = 1, 2, ..., 7) and a number of 42 rank-two 1 2 5 3 operators such as σx σy, σz σx, ..., etc. With the deﬁnition⊗ ⊗

FIG. 1. (a) The way of getting the effective channel from the stan- m, + = Em 0L , m, = Em 1L , | i | i | −i | i dard QEC protocol includs encoding, noise evolution, recovery, and 63 decoding. (b) Our protocol where the chosen unitary transformation the set of normalized vectors m, constitutes a ba- {| ±i}m=0 is sufficient to correct the errors of the principle system. The errors sis of the 27-dimensional Hilbert space, and for the ancilla 63 of the ancilla system are left uncorrected. system, the basis is denoted by am m=0. Then, from the requirements {| i} transformation U for encoding process ; (ii) The noise evo- V am 0 = m, + = Em 0L , U | i ⊗ | i | i | i lution denoted by Λ; (iii) The recovery operation described by V am 1 = m, = Em 1L , SA 15 SA † | i ⊗ | i | −i | i a process such that (ρ ) = m=0 Rmρ Rm, with R R † one can obtain the unitary transformation as Rm the Kraus operators; (iv) The decodingP process real- V ized by U †. U 63 When designing the QEC protocol in quantum computa- V = ( m, + am, 0 + m, am, 1 ). tion, one can have | ih | | −ih | mX=0 † † ˜, R◦U ≡U ◦ R Finally, the effective channel for seven-qubit is ˜ † with a new process = . According to analysis in S † S R U◦R◦U˜ ε˜(ρ ) = TrA[ Λ ( a0 a0 ρ )]. (11) Ref. [3], the recovery process is not necessary and can be V ◦ ◦V | ih |⊗ moved away. If the protocol isR applied in quantum information storage and transmission, the recovery of auxiliary qubits In the selection of the correctable states, the orthogonal- can be abandoned. In Fig.1 (b), a simplified protocol to obtain ity of correctable states in the process is required to perform the effective channel is defined as effective correction capability for the seven-qubit code. The maxmum number for all the correctable errors is m g, where S † S m is the number of the selected correctable states,∗ also equal ε˜(ρ ) = TrA[ Λ ( a0 a0 ρ )]. (9) U ◦ ◦U | ih |⊗ to the dimension of the Hilbert space divided by 2, and g is In the protocol above, and † are the most important the number of generators for Steane code. processes, where not onlyU encodingU and decoding, but also error correction is implemented. As designed in Ref. [3], IV. EXACT PERFORMANCE OF FIVE-QUBIT CODE FOR U am 0 = m, + = Em 0L , COMMON ERROR CHANNELS | i ⊗ | i | i | i U am 1 = m, = Em 1L , (10) | i ⊗ | i | −i | i In this section, to describe how the five-qubit code works where m = 0, 1, ..., 15, E0 is the identity operator Iˆ, and for i against common error models, we take the depolarizing chan- m = 0, Em is one of the Pauli operators σ (i = 1, ...5, j = 6 j nel as the contrast and then compare the performancebetween x,y,z). common error channels and the depolarizing channels. The In the following, the Steane code [2] will be used in Sec. V common errors considered here are bit flip error, bit-phase flip as a contrast, and the logic states are error, phase flip error, amplitude damping error and general- 1 ized amplitude damping error for five-qubit code. The initial 0L = [ 0000000 + 1010101 + 0110011 channel fidelity F is the probability that errors do not hap- | i 2√2 | i | i | i 0 pen, and we set F = p here. The Kraus operators can be + 1100110 + 0001111 + 1011010 0 | i | i | i expressed as, + 0111100 + 1101001 ], | i | i (1) (2) E = √pIˆ , E = 1 pσˆx and BF 2 BF − p 1 for bit flip channel, 1L = [ 1111111 + 0101010 + 1001100 | i 2√2 | i | i | i (1) (2) + 0011001 + 1110000 + 0100101 E = √pIˆ2, E = 1 pσˆy | i | i | i BPF BPF − + 1000011 + 0010110 ]. p | i | i for bit-phase flip channel, In the error correction with the steane code, encoding pro- 7 (1) (2) cess is a unitary transformation V in a 2 -dimensional E = √pIˆ , E = 1 pσˆz. V PF 2 PF − p 4 for phase flip channel,

(1) 1 0 (2) 0 P2 EAD = , EAD = . 0 P1 0 0 for amplitude damping channel,

(1) 1 0 (2) 0 P2 EGAD = √p , EGAD = √p , 0 P1 0 0

(3) P1 0 (4) 0 0 EGAD = 1 p , EGAD = 1 p . − 0 1 − P2 0 p p for generalized amplitude damping channel, with P1 = 2√p 1 , P2 = 4(√p p), and | − | − p 1 p E(1) = √pIˆ , E(2) = − σˆ , DEP 2 DEP r 3 x 1 p 1 p E(3) = − σˆ , E(4) = − σˆ . DEP r 3 y DEP r 3 z for depolarizing channel. 5 ⊗5 Concretely, we choose the noisy channel as Λ0 = εDEP for ﬁve-qubit code, and after error correction, the ﬁdelity of the ′ effective channel F 5 is Λ0

′ 1 2 3 4 5 FIG. 2. (color online). (a) The fidelity gap of the effective channels F 5 = (5+20p 70p +40p + 160p 128p ). (12) Λ0 27 − − between common errors and depolarizing errors after error correction for five-qubit (or seven-qubit) code. The numerical results for five- This is the same as the result by Reimpell and Werner [16]. In qubit code in Eq. (15) are shown with the solid line, while the ones 5 the next, we choose quantum channels Λ1 = εBF εBPF for seven-qubit code in Eq. (19) are shown with the dashed line. (b) ⊗ ⊗ ∆R ∆R εPF εAD εGAD as a contrast, and after error correction, The relative deviations of fidelity gaps Λ5 and Λ7 . The solid ⊗ ⊗ ′ 1 1 the fidelity of the effective channel F 5 becomes line shows the relative deviation of fidelity gap for five-qubit code in Λ1 Eq. (16), while the dashed line shows the relative deviation of fidelity ′ 3 2 5 3 F 5 =3p 4p 2 3p +4p 2 +5p gap for seven-qubit code in Eq. (20). Λ1 − − 7 9 8p 2 +4p4 +8p 2 8p5. (13) − − ′ Then, define the fidelity gap ∆F 5 between specific com- Λ1 V. EXACT PERFORMANCE OF SEVEN-QUBIT CODE mon errors and depolarizing errors in QEC as FOR COMMON ERROR CHANNELS ′ ′ ′ ∆F 5 = F 5 F 5 , (14) Λ1 Λ1 − Λ0 and one can come to We will describe how seven-qubit code works against common errors in this section. We take the depolarizingchannel as ′ 1 3 2 5 ∆F 5 = ( 5+61p 108p 2 11p + 108p 2 a contrast and then compare the performance of a common er- Λ1 27 − − − ror channels with that of depolarizing channels. Similarly, the 3 7 4 9 5 +95p 216p 2 52p + 216p 2 88p ).(15) common errors chosen are bit flip error, bit-phase flip error, − − − phase flip error, amplitude damping error, generalized ampli- If one chooses the initial fidelity F0 = p =0.92, it is shown in Fig. 2 that as p is growing in the interval [0.92, 1], the fidelity tude damping error, and another two depolarizing channels for ′ seven-qubit code. gap ∆F 5 is always greater than 0 and decreasing nearly to 0 Λ1 from 1.05533 10−4. For the initial channel fidelity F0 = p, take the noisy chan- × 7 ⊗7 In order to further compare the performances of five-qubit nel Λ0 = εDEP as a contrast for seven-qubit code, and after ′ error correction, the fidelity of the effective channel F 7 is code between common error channels and depolarizing chan- Λ0 5 nels, the relative deviation of fidelity gap ∆RΛ1 should firstly be defined ′ ′ ′ ′ 1 2 3 ∆F 5 F 5 F 5 7 Λ1 Λ1 Λ0 FΛ = (154 + 350p 1491p + 2296p 5 − 0 729 − ∆RΛ1 = ′ = ′ , (16) F 5 F0 F 5 F0 4 5 6 7 Λ0 Λ0 +140p 4368p + 8512p 4864p ). (17) − − − − and this quantity represents the relative deviation of fidelity gap from the improvementof fidelity in depolarizing channels 7 for five-qubit code. The numerical results for Eq. (16) are For the noisy evolution Λ1 = εBF εBPF εPF εAD shown in Fig. 2. ε ε ε , the fidelity⊗ of the effective⊗ ⊗ channel⊗ GAD ⊗ DEP ⊗ DEP 5

′ after error correction F 7 becomes and another set of operators Am can be introduced as Λ1 { } † 1 3 5 ¯ ′ 1 2 Am = U2(θ, φ)AmU2 (θ, φ), (21) F 7 = (3+5p 2 15p 4p 2 + 46p 28p 2 Λ1 9 − − − 7 9 where θ and φ are two free parameters. With three free pa- 3 2 4 2 5 136p + 157p + 347p 618p 48p rameters α, β, and γ, the four operators A¯ can be explicitly − 11 13− − m +576p 2 244p6 16p 2 16p7). (18) expressed as − − − 7 Meanwhile, for the noisy evolution Λ = εBF εBPF cos α 0 0 0 1 A¯1 = , A¯2 = , ε ε ε ε ε we have chosen,⊗ similar⊗ 0 sin β cos γ sin α sin γ 0 PF ⊗ AD ⊗ GAD ⊗ DEP ⊗ DEP to Eq. (14), the fidelity gap between the specific common 0 sin β sin γ sin α cos γ 0 A¯ = , A¯ = . errors and depolarizing errors after error correction is defined 3 0 0 4 0 cos β as The detailed discussions about this model can be found in ′ 1 1 3 2 ∆F 7 = (89 + 405p 2 1565p 324p 2 + 5217p Ref. [38]. Λ1 729 − − 5 7 As a constrain, fidelity for each qubit channel is cho- 2268p 2 13312p3 + 12717p 2 + 27967p4 sen to be the same, and the free parameters θ, φ, α, β are − 9− 11 50058p 2 + 480p5 + 46656p 2 28276p6 arbitrary. Then, each channel for the five qubits is gen- − 13 − 7 erated independently from the arbitrary error model over 1296p 2 + 3568p ). (19) 5 − 10 times. The N-th error process can be written as , and If one chooses the initial fidelity F = p =0.92, it is shown in Λ(N) = ε1(N) ε2(N) ε3(N) ε4(N) ε5(N) 0 ⊗ ⊗ ⊗are generated⊗ indepen- Fig. 2 that as p is growing in the interval [0.92, 1], the fidelity ε1(N),ε2(N),ε3(N),ε4(N),ε5(N) ′ ′ dentlybasedonEq. (21). If five-qubitcode works well against gap ∆F 7 is smaller than 0 and its absolute value ∆F 7 is Λ1 | Λ1 | arbitrary errors, concatenated five-qubit code can be efficient decreasing nearly to 0 from 1.11788 10−3. × either, and therefore, we just need to perform the first level In order to further compare the performance of seven-qubit error correction here. We choose the initial channel fidelity code between common error channels and depolarizing chan- F0 = 0.9, 0.91, 0.92, 0.93, 0.94, 0.945, 0.95, 0.96, 0.97, 0.98, nels, the relative deviation of fidelity gap 7 can also be ∆RΛ1 0.99, 0.992, 0.9993. defined After error correction with five-qubit code, based on Eq. ′ (14), we employ the minmum fidelity of the effective channel ∆F 7 Λ1 ′ ′ 7 | | Fmin to define the fidelity gap ∆Fmin, which shows the worst ∆RΛ1 = ′ , (20) F 7 F0 performance in error correction for five-qubit code, Λ0 − ′ ′ ′ and this quantity represents the relative deviation of fidelity ∆F = F F 5 . (22) min min − Λ0 gap from improvement of fidelity in depolarizing channels for ′ seven-qubit code. The numerical results for Eq. (20) are de- Since ∆Fmin < 0, in Fig. 3 (a) and Table I we have taken ′ ′ its absolute value ∆F for convenience. Here F 5 can be picted in Fig. 2. min Λ0 5 | | From Fig. 2, it is aware that for the noisy evolutions Λ1 calculated by Eq. (12). 7 ′ ′ and Λ1 above, the fidelity gap ∆F 7 for seven-qubit code is Also based on Eq. (14), another fidelity gap ∆F avg Λ1 | | | ′ | can be defined to show the average disparity of five-qubit about 10 times greater than ∆F 5 for five-qubit code. Mean- | Λ1 | code when applied in arbitrary error channels and depolariz- while, relative deviation of fidelity gap ∆R 7 for seven-qubit Λ1 ing channels. Since the values of N-th fidelity gap ∆F ′ = 5 N code is about 100 times greater than ∆RΛ1 for five-qubit code. ′ ′ FN F 5 can be both positive and negative, we take the ab- Therefore, one can believe that five-qubit code is more effec- − Λ0 solute value of the fidelity gap ∆F ′ as tive than seven-qubit code in the case above. However, if the | |avg errors order change and become even arbitrary, does five-qubit ′ 1 ′ ′ code still work well? This is our following discussion. ∆F avg = FN F 5 . (23) | | N | − Λ0 | XN where N is the time the calculation has be performed, and VI. FIVE-QUBIT CODE AGAIST ARBITRARY ERROR 5 ′ CHANNELS can reach over 10 , FN means the N-th fidelity of the effec- ′ tive channel after error correction, and F 5 can be obtained in Λ0 In this section, the numerical calculation for five-qubit code Eq. (12) either. The numerical results for these quantities are is carried out where error channels are arbitrary, and this is shown in Fig. 3 (a) and listed in Table I. different from Ref. [38], where the error on each qubit is the Meanwhile, when initial fidelity F0 = 0.92, 0.93, 0.94, 0.945, 0.95, 0.96, 0.97, 0.98, 0.99, 0.992, 0.9993, based on same. Let A¯m be a set of Kraus operators, and introduce an { } Eq. (19), the absolute values of fidelity gap for seven-qubit arbitrary 2 2 unitary transformation ′ × code ∆F 7 are listed in Table I. Λ1 θ θ | | cos 2 sin 2 exp[ iφ] In the following, to show the relative deviation of fidelity U2(θ, φ)= − , sin θ exp[iφ] cos θ gap for five-qubit code, according to Eq. (16) and Eq. (22), − 2 2 6

TABLE II. The relative deviation of the worst fidelity gap and the average fidelity gap in five-qubit code, and the relative deviation fidelity gap in seven-qubit code.

F ∆R ∆R ∆R 7 0 min avg Λ1 0.9 9.52501 × 10−2 6.45269 × 10−4 None 0.91 5.99347 × 10−2 4.33414 × 10−4 None 0.92 3.84932 × 10−2 2.66134 × 10−4 1.45506 0.93 2.47053 × 10−2 1.75141 × 10−4 0.119470 0.94 1.55591 × 10−2 1.08709 × 10−4 5.16485 × 10−2 0.945 1.21850 × 10−2 8.82583 × 10−5 3.74590 × 10−2 0.95 9.42051 × 10−3 7.07914 × 10−5 2.79574 × 10−2 0.96 5.32710 × 10−3 3.97459 × 10−5 1.61721 × 10−2 0.97 2.67621 × 10−3 2.09959 × 10−5 9.30149 × 10−3 0.98 1.07176 × 10−3 9.45914 × 10−6 4.93263 × 10−3 FIG. 3. (Color online). (a) The yellow line with dots shows the 0.99 2.43240 10−4 3.28571 10−6 2.01035 10−3 average fidelity gap of five-qubit code against arbitrary errors and × × × depolarizing errors. The red line with dots shows the minmum fi- 0.992 1.52812 × 10−4 2.53247 × 10−6 1.54728 × 10−3 delity gap of five-qubit code against arbitrary errors and depolarizing −6 −7 −4 errors. The blue line with dots shows the fidelity gap of seven-qubit 0.9993 2.80508 × 10 6.06365 × 10 1.18156 × 10 code against common chosen errors and depolarizing errors. (b) The yellow line with dots shows the relative deviation of the average fidelity gap when five-qubit code against arbitrary errors and depolar- one can define the relative deviation of the min fidelity gap izing errors. The red line with dots shows the relative deviation of ∆Rmin, the min fidelity gap when five-qubit code against arbitrary errors and ′ depolarizing errors. The blue line with dots shows the relative de- ∆Fmin ∆Rmin = | ′ | . (24) viation of the fidelity gap when seven-qubit code against common 5 FΛ F0 chosen errors and depolarizing errors. 0 − The numerical results for ∆Rmin are shown in Fig. 3 (b) and listed in Table II. Similarly, according to Eq. (23), the relative deviation of TABLE I. The worst performance fidelity gap and the average fidelity the average fidelity gap ∆R can be defined as gap of five-qubit code between an arbitrary error channel and a de- avg polarizing channel, and the fidelity gap of seven-qubit code between ′ ∆F avg common error channels and depolarizing channels. ∆Ravg = | ′ | , (25) F 5 F0 Λ0 ′ ′ ′ − F0 |∆Fmin| |∆F |avg |∆F 7 | Λ1 and the results are also shown in Fig. 3 (b) and listed in −3 −5 0.9 1.95185 × 10 1.32228 × 10 None Table II. Moreover, when initial fidelity F0 = 0.92, 0.93, 0.94, 0.945, 0.95, 0.96, 0.97, 0.98, 0.99, 0.992, 0.9993, via 0.91 1.44212 × 10−3 1.04286 × 10−5 None 7 Eq. (20), the relative deviation for seven-qubit code ∆RΛ1 is 0.92 1.02643 × 10−3 7.09649 × 10−6 1.11788 × 10−3 listed in Table II. 0.93 6.96773 × 10−4 4.93955 × 10−6 8.58032 × 10−4 As shown in Fig. 3 (a) and listed in Table I, when the initial fidelity F0 =0.9, the worst performance fidelity gap between 0.94 4.44576 × 10−4 3.10616 × 10−6 6.30686 × 10−4 ′ arbitrary errors and depolarizing errors ∆Fmin =1.95185 −4 −6 −4 −3 | | ′ × 0.945 3.44677 × 10 2.49656 × 10 5.29647 × 10 10 , the average performance fidelity gap ∆F avg = −5 | | 0.95 2.60648 10−4 1.95867 10−6 4.37225 10−4 1.32228 10 , and they are both decreasing to nearly 0 × × × with almost× the same speed as the initial fidelity increases, 0.96 1.35187 10−4 1.00864 10−6 2.78688 10−4 ′ × × × and in the process, ∆Fmin keeps nearly 100 times greater ′ | | −5 −7 −4 than ∆F . As shown in Fig. 3 (b) and listed in Table II, 0.97 5.77680 × 10 4.53211 × 10 1.55730 × 10 | |avg when the initial fidelity F0 =0.9, the relative deviation of the 0.98 1.73357 × 10−5 1.53001 × 10−7 6.85675 × 10−5 −2 min fidelity gap ∆Rmin = 9.52501 10 , the relative de- × −4 0.99 2.19452 × 10−6 2.96438 × 10−8 1.69308 × 10−5 viation of the average fidelity gap ∆Ravg =6.45269 10 , and they are both decreasing to nearly with almost the× same 0.992 1.12642 10−6 1.86676 10−8 1.08047 10−5 0 × × × speed as the initial fidelity increases. 0.9993 1.94983 × 10−9 4.21489 × 10−10 8.17661 × 10−8 Meanwhile, when comparing the worst performance of five-qubit code with the performance of seven-qubit code in 7 common error channels Λ7 = ε ε ε ε results can be summarized: (1) In the preparation of QEC, 1 BF ⊗ BPF ⊗ PF ⊗ AD ⊗ εGAD εDEP εDEP, it is found that even the worst perfor- only the channel fidelity is required in the construction of the mance⊗ of five-qubitcode⊗ is better than that (not the worst one) QECCs. (2) When five-qubit code is employed in bit-flip er- of seven-qubit code for both ∆F ′ and ∆R. Furthermore, ror channels, where the initial channel fidelity is below 0.992, the worst performance fidelity| gap is| the fidelity gap between one can take the fidelity of effective channel as the threshold bit-flip error channels and depolarizing error channels except in the construction of QEC. (3) When designing QEC with an anomalous data in Table I where initial fidelity is 0.9993. concatenated five-qubit code, it is not necessary to adjust the [With the same initial fidelity 0.9993, the fidelity gap between protocol even when errors in each qubit are time-varying or a bit-flip error and a depolarizing errors is 7.61556 10−10]. level-varying. However, this does not matter if the fidelity of effective× chan- In recent works, suppression of coherent error in multi- nel after error correction in bit-flip error channels is taken as qubit entangling gates in trapped ion systems has been in- the threshold for five-qubit code, where the initial channel fi- troduced [42], and the effect of coherent errors on the log- delity is required to be below 0.992. ical error rate of the Steane [[7, 1, 3]] QECC has also been studied [43]. Quite recently, failure distributions of coherent and stochastic error models in QEC have been compared [44]. VII. REMARKS AND DISCUSSION Therefore, the application of five-qubit code in suppressionof coherent error will be our future consideration. When pro- In section IV and section V, it is shown that five-qubit code tecting a quantum system from coherent error, the effective is more efficient than seven-qubit code in a complicated noise channel of every qubit in the system may not be the same in environment via an analytic example in this work, and in sec- general case, and based on the results obtained in this work, tion VI, the numerical calculation for five-qubit code has been the five-qubit code may be a good choice for the preservation carried out when error models are arbitrary. We find the worst of the system. performance fidelity gap between arbitrary error channels and depolarizing channels is sufficient small, and it is a little better than the performance of seven-qubit code in error chan- ACKNOWLEDGEMENTS nels. Meanwhile, the worst performance fidelity gap is the fidelity gap between bit-flip error channels and depolarizing This work was supported by the National Natural Science error channels in most cases, say when initial channel fidelity Foundation of China under Grant No. 11405136, and the Fun- below 0.992. damental Research Funds for the Central Universities under Based on the analysis in the present work, the following Grant No. 2682016CX059.

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