A Coding and Automatic Error-Correction Circuit Based on the Five-Particle Entangled State

Xi Chen,Pei Zhang,and Xiaoqing Zhou

(Collegeof Physicsand Mechatronics Engineering,Jishou University,Jishou 416000,China)

Abstract In this paper,we discuss the concepts of quantum coding and error correction for a five-particle entangled state.Error correction can correct bit-reverse or phase-flip errors of one and two quantum states and is no longer limited to only one quantum state.We encode a single quantum state into a five-particle entangled state before being transferred to the sender.We designed an automatic error-correction circuit to correct errors caused by noise.We also simplify the design process for a multiple quantum error-correction circuit.We compare error-correction schemes for five and three entan?gled particlesin termsof efficiency and capabilities.Theresultsshow that error-correction efficiency and fidelity areim?proved.

Keyw ords quantumcommunication;channel coding;five-particleentangled state;fidelity

1 Introduction

Q uantum information transmitted from sender to re?ceiver is affected by the environment and decays exponentially over time[1].De-coherence is as?sociated with the development of quantum theory and has had a negative impact on quantum theory[2]-[5].To reduce and correct errors caused by noise interfer?ence in a channel,channel coding has been introduced on the sender side.A single quantum state can be encoded into a five-particle entangled state before transmission,and correc?tions can be made automatically during transmission.With a decoder at the receiver,a quantum state with no more than two error conditions can be corrected.This eases retransmission pressure in the information channel.The best way to reduce the probability of errors in quantum coding is to introduce quantum channel coding error correction[3].In 1995,the Shore coding scheme was designed.In 1996,Stenae,Calder?bank and Shor created CSScode[6]-[9].The same year,Ben?nett et al.proposed quantum error-correcting codes[10],[11].Gottesman and Alderbank proposed the stabilizer system[6],[7],[12],[13]and constructed quantum Hamming-bound satu?rated quantum error-correction codes[14]-[16].With these codes,the error of one quantum state can be corrected using three quantum state or five quantum state coding[6],[7],[14].In 2010,Lv Hongjun proposed a 3,5,7 quantum error-correc?tion circuit that made quantum network communication be?tween sites possible[2].Recently,Zhou Nanrun et al.pro?posed a quantum synchronous communication protocol[15]-[17].Zhou Xiaoqing et al.have also proposed a model of a token-ring quantum transmission network using three-parti?cle entangled state[18].They also calculated the fidelity of this model[19].In[20],the interconnection and routing strate?gy of point quantum teleportation network was described.To minimize transmission errors caused by environmental factors,a five-particle entangled quantum state encoding and er?ror-correction circuit was designed.This circuit can correct an error of up to two quantum states.It was no longer confined to the situation of one quantum error.A more simplified er?ror-correction circuit wasdesigned.

2 Channel Coding

In a classical channel,noise affects transmission.Flip errors sometimes occur in the received bit.The sent bit may be 1,but the received bit is 0.In the transmission of quantum informa?tion,errors are easily caused by noise and quantum incoher?ence.Quantum information errors may be more complicated than those that affect classic information.There are three cate?gories of error that affect quantum information:reverse turn,phase reversal,and reverse turn with phase[5].As in classical information science,quantum information science also uses channel coding.By structuring the state of information that re?peats itself and increasing redundancy,the system can auto?matically correct errors and ensure that information is correct[1].We assume that the input qubit signal of an information channel is│Ψ〉=α│0〉+β│1〉and that classical bit coding is used to reduce the error rate as much as possible.More than 1 bit is encoded to transfer the information.Quantum informa?tion can transmitted by an encoder,and quantum states are│Ψ〉=α│00000〉+β│11111〉,that is,code one qubit with five qubits.Thisalso creates the possibility for quantum correc?tion(Fig.1).

2.1 Five-Particle Bit-Reversal Error Correction

According to the basic principles of quantum mechanics,measurement leads to the collapse of the quantum state,and in?formation carried by quantum state is lost[17].Therefore,it is important to find a corrective measure without quantum bit flip.Here,we only consider an error of no more than two bits,and the error coding state can be corrected without the need for measuring by introducing redundant qubits[20].Collapse of the quantum state due to measurement is avoided.To reduce the quantum auxiliary bits and quantum logic gates,we use three quantum bit error-correction method[2]to design a five-particle entangled state bit inversion error-correction cir?cuit.Fig.2 showstheerror situation of a,b,c,and d.

For example,M1is 0;M2is 1;and M3is a,b are both incor?rect.The variety of error conditionsisshown in Table 1.

No more than two qubit error conditions are corrected,and these error conditions determine the quantum bit inversion op?eration.Then,the five entangled particle error-correction cir?cuit is used.From Fig.2,the error conditions of qubit a,b,c can be corrected,then we anticipate that the qubits of a,b,c are correct.Therefore,these qubits can help correct d and e to concert and assist the qubit.In this way,no more than two quantum bits are error-corrected.The five entangled particle bit-reversal error-correction circuit isshown in Fig.3.

▲Figure2.Measuring theerror condition of a,b,c.

Because of environmental factors,the quantum state and auxiliary bit quantum state of the sender has changed from 00000 000000 to 10010 000000.

Before position 1,a,b and c are control bits that control the NOT operation,the auxiliary bitδ,is︳0〉;the auxiliary bitβis︳1〉;and theoutput is10010 010000.

Before position 2,the auxiliary bitsδ,βare control bits;c is the target bit that operates the Toffoli gate and remains un?changed;and the output is 10010 010000.

Before position 3,the target bit isβand operates the NAND gate;βis︳0〉;and theoutput is10010 000000.

Before position 4,the auxiliary bitsδ,βare control bits;b is the target bit that operates the Toffoli gate and remains un?changed;and the output is 10010 000000.

Before position 5,the target bits areδ,βand operate the NANDgate;δis︳1〉;βis︳1〉;and theoutput is10010 110000.

Before position 6,the auxiliary bitsδ,βare control bits;a is the target bit that operates the Toffoli gate;a is︳0〉by rever?sion;and the output is00010 110000.

▲Figure3.Fiveparticlebit-reversal error-correction circuit.

▲Figure1.Five-particleencoder.

Before position 7,b,c and d are control bits that control the NOToperation;the auxiliary bitγis︳0〉;the auxiliary bitμis︳1〉;and theoutput is00010 111100.

Before position 8,the auxiliary bitsγandμare control bits;d is the target bit that operates the Toffoli gate;d is︳0〉by re?version;and theoutput is00000 111100.

Before position 9,the target bit isμand operates the NAND gate;μis︳0〉;and theoutput is00000 111000.

Before position 10,the auxiliary bitsγandμare control bits;c is the target bit that operates the Toffoli gate and re?mainsunchanged;and the output is 00000 111000.

Before position 11,the target bitsγandμoperate the NANDgate;γis︳0〉;μis︳1〉;and theoutput is00000 110100.

Before position 12,the auxiliary bitsγandμare the control bits;b is the target bit that operates the Toffoli gate and re?mainsunchanged;and theoutput is00000 110100.

Before position 13,c,d and e are control bits that control the NOT operation;the auxiliary bitνis︳0〉;and the output is 00000 110100.

Before position 14,the auxiliary bitsνandοare control bits;e is the target bit that operates the Toffoli gate,and the output is00000 110100.

Before position 15,the target bitοoperates the NAND gate;οis︳1〉;and the output is 00000 110101.

Before position 16,the auxiliary bitsνandοare control bits;d is the target bit that operates the Toffoli gate and re?mainsunchanged;and theoutput is00000 110101.

Before position 17,the target bitsνandοoperate the NANDgate;νis︳1〉;οis︳0〉;and the output is 00000 110110.

Before position 18,the auxiliary bitsνandοare control bits;c is the target bit that operates the Toffoli gate and re?mainsunchanged;and theoutput is00000 110110.

Error-correction resultsareshown in Table2.

The five quantum entanglement error-correction code can correct errors of no more than two qubits without completely destroying the encoded state.In the received︳abcde〉,a repre?sents 0 or 1.We measure the number of 1 in a,b,c,d,e and decode︳abcde〉into︳00000〉.In contrast,︳abcde〉is decoded to︳11111〉.This means that︳D determines the means of de?coding[18]:D{︳00000〉︳00001〉︳00010〉︳00100〉︳01000〉︳10000〉︳11000〉︳10100〉︳10010〉︳10001〉︳01100〉︳01010〉︳01001〉︳00110〉︳00101〉︳00011〉}=︳00000〉D{︳11111〉︳11110〉︳111101〉︳11011〉︳10111〉︳01111〉︳00111〉︳01011〉︳01101〉︳01110〉︳10011〉︳10101〉︳10110〉︳11001〉︳11010〉︳11100〉}=︳11111〉

2.2 Five Particle Phase-Flip Error Correction

Assuming︳00000〉+︳11111〉flipsthrough channels,the received information is︳00000〉-︳11111〉because of the phase reversal of the first qubit.In this case,the previous method cannot be used for correction.Thus,we use the phase-inverted channel error-correction method.The Hadamard transformisgiven by[1]

▼Table2.A variety of input and output

Wedefine:

Let︳+〉flip through channels which hasa phase flip For example,α︳0〉+β︳1〉turns intoα︳+〉+β︳-〉after a Hadamard transform,and theoccurr phaseflip isequal tothe operation ofwith theresultα︳-〉+β︳+〉.The error of the phase flip was converted intobit-flip errors Fig.4.The result of phase flip│+〉and│-〉is similar to the preced?ing phase-flip error.This provides the basis for correcting the phase flip.We only need to flip through the phase error of the H gate logic circuit,which can be converted into bit-flip errors for processing.The phase-inversion error-correction circuit in Fig.5 can be used to correct the reversal error.A phase rever?sal is turned into a phase inversion.The results can be referred to the inversion error correction.Phase inversion error is shown in Fig.5.

3 Efficiency of Five Particle Error Correction

We assume that the qubit is︳φ〉,the decoded qubit is︳φ′〉,and fidelity is

After using 3 quantum states︳000〉+︳111〉to encode,the probability of the decoding results isα︳0〉+β︳1〉,which is equal to the probability of a qubit inversion,i.e.(1-p)3+3p(1-p)2.The probability of a decoding error is equal to the prob?ability of more than 2 qubit reversal,i.e.3p2(1-p)+p3.The fi?delity F isgiven by

We obtain min F3≥[(1-p)3+3p(1-p)2]=1-3p2+2p3with the normalization condition│α│2+│β│2=1 and(α*β+β*α)2≥0.Then,F＞1-p because0＜p＜i.e.theloyalty of thechannel is increased.Similarly,an error rate of no more than two quan?tum states can be corrected after five entangled particle encod?ing.Theloyalty F isgiven by min F5≥(1-p)5+5p(1-p)4+10p2(1-p)3.We can see that min F5≥F3by 0＜p＜,and the phase-flip error can be corrected by turning into error.There?fore,its fidelity algorithm is also similar to the fidelity algo?rithmsof abit-flip.

▲Figure4.Schematic diagram of phaseflip converted intobit-flip by H transform.

▲Figure5.Fiveparticlephase-inversion error-correction circuit.

In the process of quantum error correction,the error proba?bility of a quantum bit is T/N,where T is the time of error cor?rection and N is the number of error corrections.The larger N is,thesmaller theinterval and bit error rateis.Theerror proba?bility of theremainder of the quantumbit isproportional to T2/N 2 after thefirst operation.

As a consequence of this,the error probability of quantum qubit is proportional to N(T 2/N 2)[1].It illustrates that the er?ror rate of system is reduced while the number of operation is larger.For example,If p=0.1,the probability of the condition which there is no more than one qubit inversion error occurs in threequbitsis

The probability when there are no more than two qubit er?rorsin afiveentangled particlesis

The comparison of error correction efficiency and fidelity be?tween five and three quantum states can illustrate that the effi?ciency and fidelity have been improved while use five quantum statesencodingfor transmittinginformation.

4 Conclusion

In this paper,we proposed an encoding and decoding scheme for channel transmission with five-particle entangled state.With this scheme,the quantum state is not collapsed by measurement,and errors of no more than two quantum bits are automatically corrected.In this paper,the quantum channel coding and error correction of five-particle entangled state was described.Error correction is no longer limited to only one quantum error.Using our scheme,errors can be corrected for two quantum states.We compared the transmission efficiency and error-correction capability of five qubits and three qubits in the channel.Transmission efficiency and fidelity of the channel were improved.To minimize the error rate,the trans?mission information must be encoded into more quantum states before transmission.This paper provides new ideas for design?ing and optimizing an error-correction circuit for multipartite quantumstates.