NNoname manuscript No. (will be inserted by the editor)
Quantum security computation on shared secrets
Hai-Yan Bai · Zhi-Hui Li · Na Hao
Received: date / Accepted: date
Abstract
Ouyang et al. proposed an ( n, n ) threshold quantum secret sharingscheme, where the number of participants is limited to n = 4 k + 1 , k ∈ Z + ,and the security evaluation of the scheme was carried out accordingly. In thispaper, we propose an ( n, n ) threshold quantum secret sharing scheme for thenumber of participants n in any case ( n ∈ Z + ). The scheme is based on aquantum circuit, which consists of Clifford group gates and Toffoli gate. Westudy the properties of the quantum circuit in this paper and use the quantumcircuit to analyze the security of the scheme for dishonest participants. Keywords
Secret sharing scheme · Quantum circuit · Quantum computa-tion · Unitary matrix
Quantum computing is a new type of computing mode that regulates quantuminformation and follows the laws of quantum mechanics. That is, it enablesquantum bits to achieve the purpose of programming in different quantumlogic gates sequentially, and all kinds of quantum algorithms can be realizedby combining different quantum logic gates. From the perspective of computa-tional efficiency, because of the existence of quantum mechanical superposition,some quantum algorithms are faster than the conventional general computerwhen solving problems.In the field of quantum cryptography, aside from the quantum key distri-bution[1,2], the quantum computation of security computing has attracted the
Hai-Yan Bai, Zhi-Hui Li (cid:12) ,Na HaoSchool of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119,ChinaE-mail: [email protected] a r X i v : . [ qu a n t - ph ] A p r Hai-Yan Bai et al. attention of people, such as secure multiparty computation [3], blind compu-tation [4-7] and verifiable delegated computation [8-12]. The earliest quantumalgorithm is proposed by Jozsa and Deutsch. This quantum algorithm showsthe computing power that classical computers do not have [13]. Ouyang et al.give a different way of security computing quantum algorithm [14], and thesecurity evaluation of quantum secret sharing circuit.In 1999, Hillery [15] and others proposed a quantum secret sharing schemefirstly. The emergence of this concept provides an effective way for the secu-rity of quantum state secrets. After that, the quantum secret sharing schemehas received rapid attention and development [16-22]. The so-called ( k, n )-threshold quantum secret sharing scheme will mean the quantum state secretdistributed to n participants such that no group fewer than k participantscan reconstruct the secret quantum state [23-25], and any k participants canreconstruct the secret quantum state.In the quantum circuits, the implementation of the unitary transformationof the quantum state is equivalent to the role of the logic gate in the quan-tum state. In any dimension Hilbert space, if the unitary transformation ofthe corresponding quantum state can be realized through the combination ofelements in a logic gate group, we call such a logic gate group a universallogic gate group (universal gate), hereinafter referred to as universal gate. Ithas been proved that Toffoli gates can realize all unitary transformation [26]in any dimension Hilbert space, but it is a 3 qubits quantum gate, which isdifficult from the perspective of physical realization. However, in any case, theuniversal gates of quantum computing, like the classical logic gate group, havedifferent forms of expression, as the Ref.[27] points out that any of the T gate,the controlled phase gate and the Toffoli gate and Clifford group gates canform a group of universal gate. In this regard, Ouyang et al. considers theuniversal gate composed of discrete Clifford group gates and T gate [14].In this paper, we consider the situation that Clifford group gates and Toffoligate constitute a universal gate, and propose an ( n, n ) threshold quantumsecret sharing scheme for any positive integer n . Then we use the quantumcircuit composed of discrete Clifford group gates and Toffoli gate to evaluatethe security. In this section,we introduce a quantum circuit (See Fig.1, Fig.2) with a proofof security and give corresponding conclusions of the quantum circuit.In Fig.1, the first s qubits of the first column qubits are the quantum statesecrets to be shared, and the later t qubits are the ancilla to help implementthe Toffoli gate. The notation a x,y labels the qubit on the x -th row and the y -th column, and R x labels the qubits in the x -th row, where x ∈ { , , ··· , s } , y ∈{ , , · · · , n + 1 } . uantum security computation on shared secrets 3 Fig. 1: Secret sharing process corresponding to n + 1 participantsIn Fig.2, U represents the unitary matrix on the x -th row of qubits R x inFig.1. When i ∈ { , , ··· , n } , V i represents the corresponding unitary matrix ofthe i column, and we order A = V n V n − ··· V . W i represents the correspondingunitary matrix of the n + i column, and we order B = W n W n − · · · W , then U = BA .Fig. 2: Schematic diagram of the corresponding circuit of n + 1 participantsAccording the quantum circuit given in Fig.1 and Fig.2, we give the fol-lowing two properties. Property 1 when n is an odd number, U ( σ ⊗ I ⊗ n − ) U † = σ ⊗ n , (1)where σ ∈ P = { I, X, Y, Z } . Prove
For σ = I, X, Y, Z, we give a proof of (1) respectively. That is, weprove the following equations are set up U ( I ⊗ I ⊗ n − ) U † = I ⊗ n , (2) Hai-Yan Bai et al. U ( X ⊗ I ⊗ n − ) U † = X ⊗ n , (3) U ( Y ⊗ I ⊗ n − ) U † = Y ⊗ n , (4) U ( Z ⊗ I ⊗ n − ) U † = Z ⊗ n , (5)Firstly, by U = BA , it is easy to know that the above Eq.(2) is clearlyestablished when σ = I . Next, the cases of σ = { X, Y, Z } are proved bymathematical induction.When σ = X , we prove Eq.(3) to be established. First,the following Eq.(6)is established. A ( X ⊗ I ⊗ n − ) A † = X ⊗ n , n ∈ Z + . (6)where A = | (cid:105)(cid:104) | ⊗ I ⊗ n − + | (cid:105)(cid:104) | ⊗ X ⊗ n − A ( X ⊗ I ⊗ n − ) A † = ( | (cid:105)(cid:104) | ⊗ I ⊗ n − + | (cid:105)(cid:104) | ⊗ X (cid:78) n − )( X ⊗ I ⊗ n − )( | (cid:105)(cid:104) | ⊗ I ⊗ n − + | (cid:105)(cid:104) | ⊗ X ⊗ n − )= | (cid:105)(cid:104) | ⊗ I ⊗ n − + | (cid:105)(cid:104) | ⊗ X ⊗ n − = X ⊗ n . It is easy to know that when n = 3 , , Eq.(3) is held, so we assumed that Eq.(3)is established when n = 2 k + 1, next certificate that Eq.(3) is also establishedwhen n = 2 k + 3. According hypothesis and Eq.(6) we know that B = W k W k − · · · W ,U ( X ⊗ I ⊗ k ) U † = X ⊗ k +1 ,A (cid:48) ( X ⊗ I ⊗ k +2 )( A (cid:48) ) † = X ⊗ k +3 . When n = 2 k + 3 , B (cid:48) = W (cid:48) k +2 W (cid:48) k +1 W (cid:48) k W (cid:48) k − · · · W (cid:48) = W (cid:48) k +2 W (cid:48) k +1 ( W k ⊗ I ⊗ I ) · · · ( W ⊗ I ⊗ I )= W (cid:48) k +2 W (cid:48) k +1 ( W k ⊗ · · · ⊗ W ⊗ I ⊗ I )= W (cid:48) k +2 W (cid:48) k +1 ( B ⊗ I ⊗ I ) , where W (cid:48) k +2 = I ⊗ I ⊗ k ⊗ | (cid:105)(cid:104) | ⊗ I + X ⊗ I ⊗ k ⊗ | (cid:105)(cid:104) | ⊗ I,W (cid:48) k +1 = I ⊗ I ⊗ k ⊗ I ⊗ | (cid:105)(cid:104) | + X ⊗ I ⊗ k ⊗ I ⊗ | (cid:105)(cid:104) | . then U (cid:48) ( X ⊗ I ⊗ k +2 )( U (cid:48) ) † = B (cid:48) A (cid:48) ( X ⊗ I ⊗ k +2 )( A (cid:48) ) † ( B (cid:48) ) † = B (cid:48) ( X ⊗ k +3 )( B (cid:48) ) † = W (cid:48) k +2 W (cid:48) k +1 ( X ⊗ k +3 )( W (cid:48) k +1 ) † ( W (cid:48) k +2 ) † = X ⊗ k +1 ⊗ ( | (cid:105)(cid:104) | ⊗ X + | (cid:105)(cid:104) | ⊗ X )= X ⊗ k +3 . uantum security computation on shared secrets 5 It is known that the Eq.(3) holds according the inductive hypothesis.When σ = Y , we prove Eq.(4) to be established. Firstly, we suppose Y = | (cid:105)(cid:104) | − | (cid:105)(cid:104) | , and it is clear from Y = i | (cid:105)(cid:104) | − i | (cid:105)(cid:104) | that it is also true. Thesame reason has the next form A ( Y ⊗ I ⊗ n − ) A † = Y ⊗ X ⊗ n − , n ∈ Z + . (7)It is easy to know that when n = 3 , , Eq.(4) is held, so we assumed that Eq.(4)is established when n = 2 k + 1, next certificate that Eq.(4) is also establishedwhen n = 2 k + 3. According hypothesis and Eq.(7) wo know that B = W k W k − · · · W ,U ( Y ⊗ I ⊗ k ) U † = Y ⊗ k +1 ,A (cid:48) ( Y ⊗ I ⊗ k +2 )( A (cid:48) ) † = Y ⊗ X ⊗ k +2 . When n = 2 k + 3 ,U (cid:48) ( Y ⊗ I ⊗ k +2 )( U (cid:48) ) † = B (cid:48) ( Y ⊗ X ⊗ k +2 )( B (cid:48) ) † = W (cid:48) k +2 W (cid:48) k +1 ( Y ⊗ k +1 ⊗ X ⊗ X )( W (cid:48) k +1 ) † ( W (cid:48) k +2 ) † = Y ⊗ k +1 ⊗ ( −| (cid:105)(cid:104) | ⊗ Y + | (cid:105)(cid:104) | ⊗ Y )= Y ⊗ k +3 . It is known that the Eq.(4) holds according the inductive hypothesis.When σ = Z, we prove Eq.(5) to be established. The same reason has thenext form A ( Z ⊗ I ⊗ n − ) A † = Z ⊗ X ⊗ n − , n ∈ Z + . (8)It is easy to know that when n = 3 , , Eq.(5) is held, so assumed that Eq.(5)is established when n = 2 k + 1, next certificate that Eq.(5) is also establishedwhen n = 2 k + 3. According hypothesis and Eq.(8) we know that B = W k W k − · · · W ,U ( Z ⊗ I ⊗ k ) U † = Z ⊗ k +1 ,A (cid:48) ( Z ⊗ I ⊗ k +2 )( A (cid:48) ) † = Z ⊗ I ⊗ k +2 . When n = 2 k + 3 ,U (cid:48) ( Z ⊗ I ⊗ k +2 )( U (cid:48) ) † = B (cid:48) ( Z ⊗ I ⊗ k +2 )( B (cid:48) ) † = W (cid:48) k +2 W (cid:48) k +1 ( Z ⊗ k +1 ⊗ I ⊗ I )( W (cid:48) k +1 ) † ( W (cid:48) k +2 ) † = Z ⊗ k +1 ⊗ ( | (cid:105)(cid:104) | ⊗ Z − | (cid:105)(cid:104) | ⊗ Z )= Z ⊗ k +3 . Hai-Yan Bai et al.
It is known that the Eq.(5) holds according the inductive hypothesis.To sum up: the Property 1 has to be proved.
Property 2 when n is an even number, U ( σ ⊗ I ⊗ n − ) U † = (cid:26) I ⊗ σ ⊗ n − , σ ∈ { I, X } Z ⊗ σ ⊗ n − , σ ∈ { Y, Z } . (9) Prove
For σ = I, X, Y, Z, we give a proof of (9) respectively. That is, weprove the following equations are set up U ( I ⊗ I ⊗ n − ) U † = I ⊗ n . (10) U ( X ⊗ I ⊗ n − ) U † = I ⊗ X ⊗ n − . (11) U ( Y ⊗ I ⊗ n − ) U † = Z ⊗ Y ⊗ n − . (12) U ( Z ⊗ I ⊗ n − ) U † = Z ⊗ Z ⊗ n . (13)Firstly, by U = BA , it is easy to know that the above Eq.(10) is clearlyestablished when σ = I . Next, the cases of σ = { X, Y, Z } are proved bymathematical induction.When σ = X , it is easy to know that when n = 2 , , Eq.(11) is held, so weassumed that Eq.(11) is established when n = 2 k , next certificate that Eq.(11)is also established when n = 2 k + 2. According hypothesis and Eq.(6) we knowthat B = W k − W k − · · · W ,U ( X ⊗ I ⊗ k − ) U † = I ⊗ X ⊗ k − ,A (cid:48) ( X ⊗ I ⊗ k +1 )( A (cid:48) ) † = X ⊗ k +2 , when n = 2 k + 2 , B (cid:48) = W (cid:48) k +1 W (cid:48) k W (cid:48) k − W (cid:48) k − · · · W (cid:48) = W (cid:48) k +1 W (cid:48) k ( W k − ⊗ I ⊗ I ) · · · ( W ⊗ I ⊗ I )= W (cid:48) k +1 W (cid:48) k ( W k − ⊗ · · · ⊗ W ⊗ I ⊗ I )= W (cid:48) k +1 W (cid:48) k ( B ⊗ I ⊗ I ) , where W (cid:48) k = I ⊗ k ⊗ | (cid:105)(cid:104) | ⊗ I + X ⊗ I ⊗ k − ⊗ | (cid:105)(cid:104) | ⊗ I,W (cid:48) k +1 = I ⊗ k ⊗ I ⊗ | (cid:105)(cid:104) | + X ⊗ I ⊗ k − ⊗ I ⊗ | (cid:105)(cid:104) | , then U (cid:48) ( X ⊗ I ⊗ k +1 )( U (cid:48) ) † = B (cid:48) A (cid:48) ( X ⊗ I ⊗ k +1 )( A (cid:48) ) † ( B (cid:48) ) † = B (cid:48) ( X ⊗ k ⊗ X ⊗ )( B (cid:48) ) † = W (cid:48) k +1 W (cid:48) k ( I ⊗ X ⊗ k +1 )( W (cid:48) k ) † ( W (cid:48) k +1 ) † = I ⊗ X ⊗ k +1 . uantum security computation on shared secrets 7 It is known that the Eq.(11) holds according the inductive hypothesis.When σ = Y , it is easy to know that when n = 2 , , Eq.(12) is held, so weassumed that Eq.(12) is established when n = 2 k , next certificate that Eq.(12)is also established when n = 2 k +2 . According hypothesis and Eq.(7) we knowthat B = W k − W k − · · · W ,U ( Y ⊗ I ⊗ k − ) U † = Z ⊗ Y ⊗ k − ,A (cid:48) ( Y ⊗ I ⊗ k +1 )( A (cid:48) ) † = Y ⊗ X ⊗ k +1 , when n = 2 k + 2 , B (cid:48) = W (cid:48) k +1 W (cid:48) k ( B ⊗ I ⊗ I ) , then U (cid:48) ( Y ⊗ I ⊗ k +1 )( U (cid:48) ) † = B (cid:48) A (cid:48) ( X ⊗ I ⊗ k +1 )( A (cid:48) ) † ( B (cid:48) ) † = B (cid:48) ( Y ⊗ X ⊗ k +1 )( B (cid:48) ) † = W (cid:48) k +1 W (cid:48) k ( Z ⊗ Y ⊗ k − ⊗ X ⊗ )( W (cid:48) k ) † ( W (cid:48) k +1 ) † = Z ⊗ Y ⊗ k +1 . It is known that the Eq.(12) holds according the inductive hypothesis.When σ = Z , it is easy to know that when n = 2 , , Eq.(12) is held, so weassumed that Eq.(13) is established when n = 2 k , next certificate that Eq.(13)is also established when n = 2 k +2 . According hypothesis and Eq.(8) we knowthat B = W k − W k − · · · W ,U ( Z ⊗ I ⊗ k − ) U † = Z ⊗ k ,A (cid:48) ( Z ⊗ I ⊗ k +1 )( A (cid:48) ) † = Z ⊗ I ⊗ k +1 , when n = 2 k + 2 , B (cid:48) = W (cid:48) k +1 W (cid:48) k ( B ⊗ I ⊗ I ) , then U (cid:48) ( Z ⊗ I ⊗ k +1 )( U (cid:48) ) † = B (cid:48) A (cid:48) ( Z ⊗ I ⊗ k +1 )( A (cid:48) ) † ( B (cid:48) ) † = B (cid:48) ( Z ⊗ I ⊗ k − ⊗ I ⊗ )( B (cid:48) ) † = W (cid:48) k +1 W (cid:48) k ( Z ⊗ k ⊗ I ⊗ )( W (cid:48) k ) † ( W (cid:48) k +1 ) † = Z ⊗ k +2 . It is known that the Eq.(13) holds according the inductive hypothesis.To sum up, the Property 2 is proved.
Hai-Yan Bai et al.
In the Ref.[14], an ( n, n ) threshold quantum secret sharing scheme is proposed,in which the number of participants is limited to n = 4 k + 1 , k ∈ Z + , and thescheme use quantum circuits to evaluate security after the participants gettheir own quantum states. In this section, we discuss the number of partici-pants n in any case ( n ∈ Z + ). The specific scheme is as follows: Input procedure
The first s bits in the first column are initialized as a quantum secret, andthe later t bits are auxiliary state: | φ + (cid:105) = U T ( H ⊗ H ) | (cid:105) = ( | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) ) , let s = 3 k, t = 3 k (cid:48) , k (cid:48) /k ∈ Z + . The rest of the column isinitialized to the maximum mixed state I , where I, X, Y, Z are common Paulioperator, and U T is a Toffoli gate. Encoding procedure
When x ∈ { , , · · · , s } , U acts on x -th qubit R x , so U ⊗ s act on s ( n + 1)bit quantum state and encodes the quantum state secret into a highly mixedstate. Sharing procedure
Prepared U encrypted n + 1 column mixed quantum state, Alice holdingfirst column quantum state, y -th column quantum state send to the y − y ∈ { , , · · · , n + 1 } . Decoding procedure (a) Collecting the shared state of n + 1 participants;(b) The unitary matrix U † acts on the mixed quantum state shared by the n + 1 column, discards the rest of the columns, and the remaining first column s bits are quantum state secrets. In order to analyze quantum circuits acting on shared secrets, each participantperforms quantum computation on his own share only, and its computation iscarried out between sharing and decryption procedure. In Ref. [27], it is shownthat any gate in C \ C and Clifford group gates can form a set of universalgate. In this paper, we consider a set of discrete gate sets consisting of Cliffordgroup gates and Toffoli gate. where U T = ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) ⊗ I + | (cid:105)(cid:104) | ⊗ X. Quantum circuit consists of any multiple Clifford gates and aconstant k (cid:48) /k Toffoli gates. Now we consider a sequence gate U = ( U , · · · , U L )that acts on the s bit quantum state secret, where U , · · · , U L gates are allunitary and are known to the participants.When U i is a Clifford gate, each participant implements U i on a list of itsown qubit subsets, so n +1 party participants jointly implement U ⊗ n +1 i . When U i is a single-qubit Clifford gates, as shown in Fig.3, when U i is a double-qubitCNOT gate, as shown in Fig.4.When U i is a Toffoli gate, each participant can perform a constant k (cid:48) /k Toffoli gates on its own quantum secret. To implement the j -th Toffoli gate uantum security computation on shared secrets 9 Fig. 3: Schematic diagramof Clifford gate action Fig. 4: Schematic diagram of CONTgate actionat the m, n, l bit qubit, each participant operates like Fig.5, where m, n, l ∈{ , · · · , s } , j ∈ { , · · · , k (cid:48) /k } . Fig. 5: Toffoli gate diagramHere the M frame represents the measuring device, when the result isclassic bit 1, the operation in the above box is implemented, and the Toffoligate is finally realized. Here | x (cid:105) , | y (cid:105) , | z (cid:105) is the quantum state of the m, n, l bit, | φ + (cid:105) = U T ( H ⊗ H ) | (cid:105) = ( | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) ) is an auxiliaryparticle. From Fig.5, it can be seen that each implementation of a Toffoli gateconsumes an auxiliary particle | φ + (cid:105) . In order to better understand the above scheme and analyze the implemen-tation of quantum gates on the shared quantum secret, we give two examplesof n = 4 , . Example
Suppose that the shared quantum state secret is σ = X ⊗ Y ⊗ Z, t = 6 . After the action of the unitary matrix U in the scheme, the finalstate is as follows ˜ ρ sec = ( X ⊗ Y ⊗ Z ) ⊗ ⊗ Φ,Φ = U ( | φ + (cid:105)(cid:104) φ + | ⊗ ( I ⊗ ) ⊗ ) U † is the result state of the auxiliary particle afterthe action of the unitary matrix U . Assuming that the first three participants (excluding Alice) are dishonest, and the fourth participant are honest, theytrace the quantum state held by fourth participant, namely, find the reduceddensity matrix corresponding to the four dimensional subsystems includingAlice tr (˜ ρ sec ) = tr (( X ⊗ Y ⊗ Z ) ⊗ ⊗ Φ )= ( X ⊗ Y ⊗ Z ) ⊗ ⊗ Φ (cid:48) · tr( X ⊗ Y ⊗ Z ⊗ Φ )= ( X ⊗ Y ⊗ Z ) ⊗ ⊗ Φ (cid:48) · tr( X ⊗ Y ⊗ Z ) · tr( Φ )= ( X ⊗ Y ⊗ Z ) ⊗ ⊗ Φ (cid:48) · · tr( Φ )= 0 ,Φ (cid:48) is the first four column entangled states of Φ , and Φ is the fifth columnquantum state of Φ . The rest of the three participants and the Alice did notget any information about the quantum secret. We consider a sequence gates U = ( C , C , U T , C , U T , C ) on the s = 3 bits quantum state secret, where C , C , C , C are Clifford gates, U T and U T are Toffoli gates, frame U T and U T in the diagram as in Fig.5; Each unitary gate is known to the participants.The operations performed by each participant are shown in Fig.6.Fig. 6: Quantum diagram of a sequence gates Example
Suppose that the shared quantum state secret is σ = X ⊗ Y ⊗ Z, t = 6 . After the action of the unitary matrix U in the scheme, the finalstate is as follows ˜ ρ sec = ( I ⊗ Z ⊗ Z ) ⊗ σ ⊗ ⊗ Φ,Φ = U ( | φ + (cid:105)(cid:104) φ + | ⊗ ( I ⊗ ) ⊗ ) U † is the result state of the auxiliary particle afterthe action of the unitary matrix. Assuming that the first four participants(excluding Alice) try to get secret information, then they trace the quantum uantum security computation on shared secrets 11 state held by fourth participant, namely, find the reduced density matrix cor-responding to the five dimensional subsystems including Alicetr (˜ ρ sec ) = tr (( I ⊗ Z ⊗ Z ) ⊗ σ ⊗ ⊗ Φ )= ( I ⊗ Z ⊗ Z ) ⊗ σ ⊗ ⊗ Φ (cid:48)(cid:48) · tr( X ⊗ Y ⊗ Z ⊗ Φ (cid:48) )= ( I ⊗ Z ⊗ Z ) ⊗ σ ⊗ ⊗ Φ (cid:48)(cid:48) · · tr( Φ )= 0 .Φ (cid:48)(cid:48) is the first four column entangled states of Φ , and Φ (cid:48) is the sixth columnquantum state of Φ . The rest of the four participants and the Alice did notget any information about the quantum secret. In the same way, we considera sequence gates U = ( C , C , U T , C , U T , C ) on the s = 3 bits quantumstate secret, each unitary gate is known to the participants. The operationsperformed by each participant are shown in Fig.6. A ( k, n )-threshold quantum secretsharing scheme satisfies two properties: (1) any k or more parties can per-fectly reconstruct the secret quantum state, (2) any k − k = n the first property is clearly hold, it can be seen thatthe coding program is completely reversible in the encryption phase. Next, wediscuss the second property. The secret quantum state before encrypting is ρ sec = 2 − s (cid:88) σ ∈ P ⊗ s ω σ σ, (14)where σ = σ ⊗ σ ⊗ · · · ⊗ σ s , P = { I, X, Y, Z } ; it is coefficient ω σ for thenon-trivial Pauli oprators σ in P ⊗ s , and ω σ = 1 when σ is the trivial Paulioprator.When n + 1 is odd, it is known by the Property 1 that the final state ofthe encrypted post is ˜ ρ sec = 2 − s ( (cid:88) σ ∈ P ⊗ s ω σ σ ⊗ n +1 ) ⊗ Φ. (15) Φ = U ( | φ + (cid:105)(cid:104) φ + | ⊗ ( I ⊗ ) ⊗ n ) U † . Assuming that the y -th participant is honest,the other n − n dimensionsubsystem by tracing the quantum state held by the y -th participant˜ ρ sec = 2 − s ( (cid:88) σ ∈ P ⊗ s ω σ σ ⊗ n +1 ) ⊗ Φ = 2 − s (( I ⊗ s ) ⊗ n +1 + (cid:88) σ ∈ P ⊗ s \ I ⊗ s ω σ σ ⊗ n +1 ) ⊗ Φ, where tr y +1 ( (cid:88) σ ∈ P ⊗ s \ I ⊗ s ω σ σ ⊗ n +1 ) = (cid:88) σ ∈ P ⊗ s \ I ⊗ s ω σ tr y +1 ( σ ⊗ n +1 ) = 0 , then tr y +1 { − s (( I ⊗ s ) ⊗ n +1 + (cid:88) σ ∈ P ⊗ s \ I ⊗ s ω σ σ ⊗ n +1 ) } = 2 − s { tr y +1 ( I ⊗ s ) ⊗ n +1 + 0 } = ( I ⊗ s ) ⊗ n . When n + 1 is even, it is known by the Property 2 that the final state ofthe encrypted is ˜ ρ sec = 2 − s ( (cid:88) σ ∈ P ⊗ s ω σ θ ⊗ σ ⊗ n ) ⊗ Φ.Φ = U ( | φ + (cid:105)(cid:104) φ + |⊗ ( I ⊗ ) ⊗ n ) U † . Where θ is the product state of the first columnof quantum bits held by Alice. Similarly, assuming that the y -th participantis honest, the other n − n dimension subsystem by tracing the quantum state held by the y -th participant˜ ρ sec = 2 − s ( (cid:88) σ ∈ P ⊗ s ω σ θ ⊗ σ ⊗ n ) ⊗ Φ = 2 − s ( θ ⊗ ( I ⊗ s ) ⊗ n + (cid:88) σ ∈ P ⊗ s \ I ⊗ s ω σ θ ⊗ σ ⊗ n ) ⊗ Φ. Where tr y +1 ( (cid:88) σ ∈ P ⊗ s \ I ⊗ s ω σ θ ⊗ σ ⊗ n ) = (cid:88) σ ∈ P ⊗ s \ I ⊗ s ω σ θ ⊗ tr y ( σ ⊗ n ) = 0 , then tr y +1 { − s ( (cid:88) σ ∈ P ⊗ s ω σ θ ⊗ σ ⊗ n ) } = tr y +1 { − s ( θ ⊗ ( I ⊗ s ) ⊗ n + (cid:88) σ ∈ P ⊗ s \ I ⊗ s ω σ θ ⊗ σ ⊗ n ) } = 2 − s { θ ⊗ tr y ( I ⊗ s ) ⊗ n + (cid:88) σ ∈ P ⊗ s \ I ⊗ s ω σ θ ⊗ tr y ( σ ⊗ n ) } = θ ⊗ ( I ⊗ s ) ⊗ n − . In the two cases, the dishonest participant can only get a highly mixed state,so there is no information about the quantum secret.Second property is also satisfied when the set of discrete gates composedof Clifford group gates and Toffoli gate act on shared quantum state secret. uantum security computation on shared secrets 13
Suppose that only one of the n participants is honest, while the remaining n − i -th gate the state of the system has theform ρ i = (cid:88) b i ψ Alice ⊗ ( σ ⊗ γ N ) ⊗ χ i . (16)where σ ∈ P ⊗ s , γ is an auxiliary particle that is not destroyed, b i is a set ofccalars, ψ Alice is a column of quantum states held by Alice, and χ i is a set ofoperators on the dishonest parties system. Honesty and other systems exist inthe form of the product state, therefore, the results of the honest measure donot convey any useful information . In this paper, we have constructed an ( n, n ) threshold quantum secret sharingscheme for n with arbitrary number ( n ∈ Z + ) of participants, which is based ona quantum circuit composed of Clifford group gates and Toffoli gate. Becauseof the universal quantum logic gate has different forms, looking for generalquantum gate model set and consider other types of universal quantum circuitgate construction, and more secure quantum secret sharing scheme based onthese quantum circuits is a problem to be studied in the future. References
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