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To appear in: Computer Standards & Interfaces Received date: 16 March 2016 Revised date: 24 August 2016 Accepted date: 18 September 2016 Cite this article as: Xiaoqing Tan, Xiaoqian Zhang and Tingting Song, Verifiable delegated quantum computation with χ-type entangled states, Computer Standards & Interfaces, http://dx.doi.org/10.1016/j.csi.2016.09.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Verifiable delegated quantum computation with χ-type entangled states Xiaoqing Tan1 , Xiaoqian Zhang1,∗ , Tingting Song2 2

1 Department of Mathematics, Jinan University, Guangzhou 510632, China Department of Computer Science, Jinan University, Guangzhou 510632, China

Abstract A delegated quantum computation protocol that includes three servers Bob1, Bob2 and Bob3 and a client Alice is proposed, which is with four-particle χ-type entangled states that have a good property is the other two particles can collapse into a certain Bell state if the ﬁrst two particles are measured. Alice only needs to be capable of getting access to quantum channels and perform Pauli operation and also has a memory of two graph states brickwork state and blind topological entangled state, but she does not need to prepare any quantum state in this paper. Our protocol is without a trusted center which will save funds, sources and quantum devices, and can verify the correctness of measurement outcomes by quantum entanglement of |χij 3124 . At last, we also give blind analysis and proof rigorously. Keywords: Delegated quantum computation, χ-type entangled states, Veriﬁability, Blind analysis and proof 2010 MSC: 94A40 PACS: [2010] 03.67.-a, 03.67.Lx

1. Introduction Quantum cryptography has a good development prospect in the eyes of many researchers. Quantum computing [1, 2, 3] is an important theoretical foundation of quantum cryptography. As a branch of quantum computation, blind quantum computation (BQC), which addresses the problem of security for the client who has less quantum 5

technology and quantum devices, becomes a hot topic in recent years. As a new type of quantum computation, BQC [4, 5, 6, 7] enables Alice to delegate her quantum computation over servers when Alice has limited quantum ability and servers cannot learn anything about Alice’s input, output and algorithm. Based on the principle, various blind quantum computation models [8, 9, 10, 11, 12, 13, 14, 15, 16] are proposed. BQC based on the circuit model is ﬁrstly proposed by Childs [4], in which Alice and Bob perform quantum gate operations to complete the task. Alice is only

10

required to own quantum memory and perform Pauli transformation. There are also some measurement-based quantum computation (MBQC) [7] models including matrix-product states, AKLT-type states, and so on. The BFK protocol was the ﬁrst universal scheme proposed in Ref.[16], which is an unconditionally secure universal BQC with the graph state that is created by linear cluster state and horseshoe cluster state. Alice only needs to prepare single qubits ∗ Corresponding

author. E-mail:[email protected]

Preprint submitted to Information Sciences

October 13, 2016

randomly chosen from a ﬁnite set and sends them to servers in their paper. The BFK protocol is the foundation of 15

many MBQC protocols based on blind cluster states. In BFK protocol, the client has some basic quantum abilities, such as producing single-qubit state or with quantum memory. Server doesn’t know any information about Alice’s secret because of the no-signaling principle. The BFK protocol has already been experimentally demonstrated by Barz et al. [17] that performs blind computation on four-qubit blind cluster states which is combined by optical gates. After the BFK protocol was proposed, more practical BQC protocols [18, 19, 20, 21, 22, 23, 24] based on entangled

20

states are proposed. Inspired by the fault tolerant ability in BFK protocol [16], a kind of fault tolerant blind quantum computation are proposed and realized [24]. In addition, double-server scheme [21] and triple-server scheme [22] are also proposed. However, servers are not allowed to communicate with each other in double-server BQC protocol [21]. Li et al. [22] pointed out that it was not realistic for preventing communication of quantum servers from each other and this problem should be avoided. Therefore, Li et al. [22] proposed a triple-server protocol with a trusted center by using

25

entanglement swapping of Bell state. They have shown that the security of the proposed triple-server BQC protocol will not be aﬀected when the mutual communication among servers can be realized except for some special cases. In fact, the triple-server protocol has four servers because a trusted center can be considered as a server. In recent years, some veriﬁable BQC protocols are presented [25, 26, 27] to ensure the correctness of measurement outcomes of servers. For example, Hayashi and Morimae [25] proposed a protocol for veriﬁable measurement-only BQC. They showed that

30

Alice can check the correctness of Bob’s state by directly verifying the stabilizers of some copies. In this paper, we suppose a scenario that Alice only has a small company, so she has limited funds and time. In order to save fund and complete the task in limited time, she needs a suitable BQC scheme. After studying the singleserver BQC scheme [16], double-server BQC scheme [20, 21] and triple-server BQC scheme [22], Alice ﬁnds that she must prepare the quantum states if she performing single-server scheme. If she performs the double-server scheme and

35

triple-server scheme, she must look for a trusted center to prepare quantum states, whereas servers can’t communicate with each other in double-server BQC protocol [21] and there are four servers in triple-server scheme [22] because the trusted center can be considered as a server. So these schemes are not suitable for Alice, Alice must propose her new scheme. Alice oﬀers the money to servers who all possess the most advanced quantum computers, so she hopes herself not to do anything as far as possible. In this new scheme, Alice has the almost classical ability as follows. Alice has

40

memory of two graph states, including the blind brickwork state and blind topological entangled states, and helps her to hide her input, output and the procedure. She possesses a database which is static and ﬁxed only known by herself. Alice performed this scheme in this paper, but she only asks Bob1 to create blind brickwork state in the past. At that time, Alice reserved these datum into this database when brickwork state is created in the step of Alice and Bob1 after performing this three-server scheme. The database helps her to inquire whether measurement outcomes match or not

45

when |χij 3124 entangled states are used and this is helpful for veriﬁcation. However, Alice does not use these value of database because they are too old. So the database is used to verify the correctness of measurement outcomes. She assigns sever Bob2 to prepare quantum states rather than a trusted center because all servers have high credibility.

2

She only has her own quantum channels, the ability of quantum storage and can perform Pauli operations on her own particles. She delegates her quantum computation to three servers Bob1, Bob2, Bob3 and keeps her own input, output 50

and the procedure hidden from the servers. Three servers have the most fully ﬂedged quantum computers and help Alice to complete the delegated quantum computation. Inspired by Morimae et al. [25, 26, 27, 28], we introduce a veriﬁable scheme to ensure that three servers all return the correct measurement outcomes. In this paper, the classical channels are all authenticated. The structure of this paper is as follows. In Sect. 2, the preparation work is presented. In Sect. 3, veriﬁable

55

delegated quantum computation protocol is presented. In Sect. 4, blind analysis and proof is presented. In Sect. 5, performance discussion is presented. In the last, we give a conclusion.

2. Preparation work Similar to Refs. [29, 30], one of χ-type maximally entangled states is deﬁned as following |χ00 3124

=

=

where |Ψxy are Bell states as |Ψxy = 60

1 √ [|0000 − |0011 − |0101 + |0110 2 2 + |1001 + |1010 + |1100 + |1111]3124 1 [|1131 |Ψ00 24 + |0031 |Ψ11 24 2 + |1031 |Ψ01 24 − |0131 |Ψ10 24 ],

√1 [(−1)x |x, y 2

(1)

+ |x ⊕ 1, y ⊕ 1](x, y ∈ {0, 1}). All the orthogonal four-particle

χ-type entangled states can be obtained by the following way {|χij 3124 = σ i ⊗ σ j ⊗ I ⊗ I|χ00 3124 | i, j = 0, 1, 2, 3}

(2)

where σi (or σ j ) is one of four Pauli operator, i.e. σ 0 = |00| + |11|,

σ 1 = |01| + |10|,

σ 2 = |01| − |10|,

σ 3 = |00| − |11|.

(3)

The sixteen χ-type basis states can be deterministically distinguished only by individual detection of related qubits showed in Ref. [31]. The χ-type entangled state has a good character which the ﬁrst two particles are measured by using the basis {|00, |01, 10, |11}, the ﬁnally two particles will collapse into a deterministic Bell state. Three severs 65

help Alice complete her computation according to Alice’s instructions. The lengths and order of particles received by Bob1, Bob2 and Bob3 are diﬀerent. And the value of rsi is randomly chosen and only known by Alice. We choose χ-type entangled states rather than other entangled states to complete this scheme because the advantages of χ-type entangled states attract us. The four-particle χ-type entangled states have some advantages as follows. • The four-particle χ-type states have a broad application such as quantum private comparison [32], quantum secret

70

sharing [33], quantum secure direct communication [34, 35], quantum teleportation [36], and so on. 3

• The four-particle χ-type state |χ00 3124 could be a likely candidate for the genuine four-partite since it cannot be reducible to a pair of Bell states [36]. It has the maximum entanglement between 32 and 14, between 31 and 24. In contrast, for a pair of Bell states |Ψ00 12 ⊗ |Ψ00 34 , the entanglement between 31 and 24 is zero. • It has been shown that a new Bell inequality is optimally violated by |χ00 3124 but not by the other types of 75

four-qubit entangled states [37, 38]. There are many experimental schemes [31, 38, 39, 40] to generate χ-type entangled states. Bob2 loyally prepares a sequence of n χ-type states S = [P31 , P11 , P21 , P41 , P32 , P12 , P22 , P42 , ..., P3n , P1n , P2n , P4n ] in the form of |χ00 3124 as mentioned above that were used in Refs.[31, 38, 39, 40] instead of the trusted center in other BQC schemes, where Pkl is the kth particle of the lth χ-type state and k ∈ {1, 2, 3, 4}, l ∈ {1, 2, 3, ..., n}. If Bob2 prepares

80

fake entangled states, his creditworthiness becomes low. This is inadvisable for him, so he can not do that. Bob2 takes particles 3 and 1 to form a sequence SA in order. The particles 2 form a sequence SB1 and the particles 4 form another sequence SB2 in order too. Here SA = [P31 , P11 , P32 , P12 , ..., P3n , P1n ],

SB1 = [P21 , P22 , ..., P2n ],

SB2 = [P41 , P42 , ..., P4n ].

Firstly, Bob2 sends SA and SB1 to Alice when Bob2 receives Alice’s instruction. Alice randomly and uniformly chooses two of four Pauli operators {σ 0 , σ 1 , σ 2 , σ 3 } on the particles 3 and 1. The |χ00 3124 are randomly transformed 85

into one of sixteen orthogonal χ-type entangled states |χij 3124 . Here, Pauli operations are uniformly chosen to prevent servers from stealing any secret information. The brief process of BQC scheme is shown by Figure 1. QM

SB3

s

s

s

s

[P3 1 ,P1 1 ,",P3 l ,P1 l ]

Bob3 QC

3

2

" "

4

"

1

Bob2

QC

"

SA

SB1

[P31,P11," ,P3n ,P1n ]

PO

[P21,P22,",P2n ]

FAD

Alice

QC

QM

Bob1

c

c

c

c 1 [P2s1 ,P2s2 ,",P2sD ] SB

CZ

Figure 1: The brief process of delegated quantum computation protocol including Alice, Bob1, Bob2 and Bob3, where QC, QM, PO, FAD and CZ respectively represent quantum channel, quantum measurement, Pauli operations, filter and disrupt and controlled-Z gate respectively.

3. Verifiable delegated quantum computation Alice interacts with Bob3, Bob2 and Bob1 respectively by the following steps to complete tasks of delegated quantum computation. Here, Alice only has m useful measurement outcomes, which are reordered as s1 , s2 , ..., sm in this scheme. 4

90

Between Alice and Bob3 (A1) Alice scrambles the order of particles 3 and corresponding particles 1 of the operated sequence SA , and randomly chooses l(l < n) particles 3 and 1 to form the new sequence SB3 that is only known by herself, where SB3 = [P3s1 , P1s1 , · · · , P3sl , P1sl ] and s1 , s2 , · · · , sl represent the positions of chosen particles in SA . Alice transmits SB3 to the sever Bob3.

95

(A2) Alice asks Bob3 to perform two-particle measurement by using two-particle basis {|00, |01, |10, |11} which corresponds to classical bits {00, 01, 10, 11}, when she wants Bob3 to measure two particles P3 and P1 . After measurement, Bob3 sends the measurement outcomes corresponding to two classical bits r3si r1si ∈ {00, 01, 10, 11} to Alice by authenticated classical channel. (A3) According to the ith measurement outcome r3si r1si , Alice can know Bell state of two particles P2si and P4si . For

100

example, if Alice performs σ 1 ⊗ σ 3 ⊗ I ⊗ I on the state |χ00 3124 , then |χ00 3124 is transformed into |χ13 3124

σ 1 ⊗ σ 3 ⊗ I ⊗ I|χ00 3124 1 √ [|1000 − |1011 + |1101 − |1110 2 2 + |0001 + |0010 − |0100 − |0111]3124 1 [−|0131 |Ψ00 24 + |1031 |Ψ11 24 2 + |0031 |Ψ01 24 + |1131 |Ψ10 24 ],

= =

=

(4)

where |Ψxy (x, y ∈ {0, 1}) are four Bell states. Suppose Bob3’s the measurement outcome is |0131 , Alice can immediately infer that Bell state shared between Bob1 and Bob2 is |Ψ00 24 according to Eq.(5). Between Alice and Bob2 (B1) Bob2 reserves n particles 4 and Alice sends n classical messages {ηi }ni=1 to Bob2, where {ηs1 , ηs2 , ..., ηsα } 105

are randomly selected from the set S = { kπ 4 | k = 0, 1, ..., 7)} and ηi is uniformly distributed in the set S when ηi ∈ / {ηs1 , ηs2 , ..., ηsα }, where α < n and {s1 , s2 , ..., sα } ∈ {1, 2, ..., n}. (B2) Bob2 performs measurement on her particle 2 with the basis |±ηi = 7π 4 }.

√1 (|0 ± eiηi |1), 2

where ηi ∈ {0, π4 , 2π 4 ,...,

After measurement, Bob2 transmits the measurement outcomes corresponding to one classical bit {xi }ni=1 (xi ∈

{0, 1}) to Alice and Alice just keeps xs1 , xs2 , ..., xsα , where {|+ηi , |−ηi } corresponds classical bits {0, 1}. The notations 110

α {si }α i=1 represent those useful positions of particles in SB2 . Alice chooses α particles of SB1 in positions {si }i=1 and s

s

s

s

scrambles the order to form a new sequence SB1 = [P2 1 , P2 2 , P2 3 , ..., P2 α ]. Finally, Alice sends SB1 to Bob1. The α α |±ηs +xs π . particles that Bob1 holds now is

Between Alice and Bob1

i=1

i

i

(C1) Alice has the structures of the brickwork state and blind topological entangled states, and only tells Bob1 to 115

perform Controlled-Z (i.e. CZ) gates, where CZ = |0000| + |0101| + |1010| − |1111|. Bob1 does not know the structures of brickwork state and blind topological entangled states. After Bob1 performs CZ operation, Alice knows

5

that the blind brickwork state and topological entangled states are created (Figure 2). The number of particles in blind topological entangled state and blind brickwork state are all t (α = 2t > 2m).

Bob1

CZ

a

CZ

b

Figure 2: Bob1 performs CZ gates to create blind brickwork state and topological entangled state, where (a) a cell of brickwork states (b) a cell of blind topological entangled states.

(C2) Alice randomly chooses {rsi }α i=1 and computes δsi = (ηsi + xsi π + φsi + rsi π) mod 2π, (i = 1, 2, ..., α), where 120

φsi is target rotation and rsi ∈ {0, 1} is random binary which is hidden from Bob1. Then {δsi }α i=1 is sent to Bob1. (C3) Bob1 performs measurement on the si th (i = 1, 2, ..., α) qubit with the basis {|±δs = |0 ± e

iδs

i

i

|1} and sends

the measurement outcomes corresponding to one classical bit bsi ∈ {0, 1} to Alice, where {|+δs , |−δs } corresponds i

i

to classical bits {0, 1}. Alice only needs m measurement outcomes s1 , s2 , ..., sm and the others are used to verify the correctness of measurement outcomes, where {s1 , s2 , ..., sm } ∈ {s1 , s2 , · · · , sα }. If rsi = 1, Alice ﬂips bsi . Otherwise, 125

she does nothing. Verifiability of delegated quantum computation Morimae and Fujii proposed a single-server blind computation scheme based on blind topological entangled states [23]. Here we use the idea of blind topological entangled states inspired by the paper of Morimae and Fujii [23] in veriﬁability. Alice has a memory of two graph states with brickwork state or blind topological state. Next, we give the

130

veriﬁable process. Alice has all measurement outcomes of Bob1, Bob2 and Bob3 already. The order and length of particles owned by Bob1, Bob2 and Bob3 are diﬀerent. Only Alice knows which particle is brickwork state and which is blind topological entangled state. Alice randomly chooses some particles of brickwork state to verify the correctness and honesty of three servers by comparing the correlation of |χij 3124 entangled states. Brickwork states are used to verify, and m

135

measurement outcomes of blind topological entangled states are used for Alice’s computation. Alice has already adjusted the correct orders of measurement outcomes about SB1 , SB2 and SB3 after measurement and accesses to the databases

of her classical computer. If there exist matching outcomes, the veriﬁcation success by quantum entanglement. Alice knows that three servers return correct measurement outcomes.(see Figure 3) So far, Alice delegates quantum computation to three servers successfully, and keeps her input, output and procedure 140

privately. Whether Bob1, Bob2 and Bob3 cooperate or not, they cannot get any private information because Alice scrambles the order and reduces the length of these particles. The geometry structures of blind brickwork state and

6

4 3

1

2

Alice

S B2

S B3

c 1 SB

Figure 3: Alice compares the measurement outcomes satisfying the entanglement to realize verification. These particles connected by line in circle is from blind brickwork state. reconstruct a |χij 31254 entangled state, where particles of SB1

topological entangled states are also secret for severs.

4. Blind analysis and proof Three servers cannot implement general attacks including intercept-resend attack, measurement-resend attack, an145

cillary particles attack and joint measurement attack to obtain other servers’ information because they don’t know who are the other servers except himself and they also maintain their own reputation. They only can try to ﬁnd leaked information from Alice by communicating with each other. But we will show that our scheme is secure whether three servers cooperate with each other or not. 4.1. Three servers cooperate with each other

150

Bob3 helps Alice to implement measurement with basis {|00, |01, |10, |11} in order to let Bob1 and Bob2 share Bell states. However, Alice randomly performs two of four Pauli operators σ i ⊗ σ j on state |χ00 3124 , so Bob3 does not know the corresponding classical value r2si r4si of Bell state. Three servers belong to diﬀerent company, so they can’t cooperate with each other in principle. Obviously, three servers without cooperation or two servers with cooperation would obtain fewer secret information than three servers cooperate with each other. Therefore, we just need to discuss

155

the case of three servers cooperate with each other. Three servers wants to steal some useful information of Alice by cooperating secretly. Bob1 knows and reveals the values of δsi and bsi (i = 1, ..., α). Bob2 knows and reveals the value of {ηi }ni=1 and {xi }ni=1 . Bob3 knows and reveals the value of {(r3si r1si )}li=1 , where α < l < n. They can’t get any information of Alice because Alice scrambles the order and changes the length of particles. The value of rsi ∈ {0, 1} is randomly chosen by Alice and only known by Alice.

160

Three severs cannot recover the original |χij (i, j ∈ {0, 1, 2, 3}) entangled states by performing joint measurement, so they cannot steal information about ηsi , xsi , φsi and rsi .

7

4.2. Blind proof In our protocol, three servers use Bell measurement and single-particle measurement, and the client Alice only uses quantum technology with unitary operation and generating states |χ00 , like Refs.[31, 38, 39, 40]. We have already 165

analyzed the blindness of our protocol at previous section. Now we will give concrete mathematical proof according to Baye’s theorem. If the quantum computation scheme satisﬁes the following two conditions, it is blind [18]. 1) The conditional probability distribution of Alice’s computational angles, given all the classical information Bob1, Bob2 and Bob3 can obtain during the protocol, and given the measurement results of any POVMs which Bob1, Bob2 and Bob3 may perform on their systems at any stage of the protocol, is equal to the priori probability

170

distribution of Alice’s computational angles, and 2) The conditional probability distribution of those measurement angles from Alice, given all the classical information Bob1, Bob2 and Bob3 can obtain during the protocol, and given the measurement results of any POVMs which Bob1, Bob2 and Bob3 may perform on their systems at any stage of the protocol, is equal to the priori probability distribution of these measurement angles.

175

Proof: We ﬁrstly analyse the eﬀect of Alice’s computational angles information Φ = {φsi }m i=1 on Alice’s privacy. Suppose X = {xsi }m }m }m }m }m i=1 , Ω = {ηs i=1 , Φ = {φs i=1 , R = {rs i=1 and ℵ = {δs i=1 , where X is Bob2’s measurement i i i i outcomes, R is the random variables chosen by Alice and {ηsi , δsi , φsi } ∈ S = { kπ 4 | k = 0, 1, 2, · · · , 7} . Let Πsi be

Bob2’s POVM measurement and A = {Asi }α i=1 be Bob2’s measurement outcomes. Bob2’s knowledge about Alice’s = j and ℵ = {δsi }m secret angles is given by the conditional probability distribution of Φ = {φsi }m i=1 given by As i=1 . i 180

Based on Baye’s theorem, we have p(Φ = {|φsi }m = j, ℵ = {δsi }m i=1 | As i=1 ) i = =

}m }m }m p(Asi = j | Φ = {|φsi }m i=1 , ℵ = {δs i=1 )p(Φ = {|φs i=1 , ℵ = {δs i=1 ) i i i m p(Asi = j, ℵ = {δsi }i=1 )

}m }m }m p(Asi = j | Φ = {|φsi }m i=1 , ℵ = {δs i=1 )p(Φ = {|φs i=1 )p(ℵ = {δs i=1 ) i i i p(Asi = j | ℵ = {δsi }m }m i=1 )P (ℵ = {δs i=1 ) i

T r[Πj 1 = m 8

T r[Πj

m i=1

m i=1 11 82

1 2

1 rs =0

|δsi − rsi π − φsi δsi − rsi π − φsi |]

i

1

φs ∈S rs =0 i

|δsi − rsi π − φsi δsi − rsi π − φsi |]

i

1 = m 8 Since p(Φ = {|φsi }m i=1 ) =

1 8m .

It shows that the conditional probability distribution of Alice’s computational angles is

equal to a priori probability distribution of Alice’s computational angles. So the scheme satisﬁes the ﬁrst condition 1). In our scheme, Alice changes the length of the particles owned by Bob1, Bob2 and Bob3. And we discuss the conditional probability distribution of measurement angles of Bob2 obtained from Alice. Bob2 doesn’t know the values

8

185

n of {ηsi }m i=1 since she doesn’t know the positions of s1 , s2 , · · · , sm even though she knows the values {ηi }i=1 . We can

get the conditional probability = j, ℵ = {δsi }m p(Ω = {ηsi }m i=1 | As i=1 ) i = =

m }m }m p(Asi = j | Ω = {ηsi }m i=1 , ℵ = {δs i=1 )p(Ω = {ηsi }i=1 , ℵ = {δs i=1 ) i i m p(Asi = j, X = {xsi }i=1 )

m }m }m p(Asi = j | Ω = {ηsi }m i=1 , ℵ = {δs i=1 )p(ℵ = {δs i=1 )p(Ω = {ηi }i=1 ) i i

p(Asi = j | ℵ = {δsi }m }m i=1 )p(ℵ = {δs i=1 ) i

T r[Πj 1 = m 8

T r[Πj

m i=1

m i=1 11 82

1 8

1 φs ∈S

|δsi − φsi − rsi πδsi − φsi − rsi π|]

i

1

1

rs =0 φs ∈S i

|δsi − rsi π − φsi δsi π − rsi π − φsi |]

i

1 = m 8 At the same time, p(Ω = {ηsi }m i=1 ) =

1 8m .

The result shows that the conditional probability distribution of

measurement angles of Bob2 from Alice is equal to a priori probability distribution of measurement angles of Bob2 from Alice. The analysis of other two servers Bob1 and Bob3 are same as Bob2. So the scheme satisﬁes the second condition 190

2). Therefore, our scheme is blind. 2

5. Performance discussion The capacity of Alice is a key factor in blind quantum computation, and she should be classical as far as possible. Considering BFK single-server protocol [16], the client Alice must prepare single-particle state by herself and can directly 195

interact with the sever Bob. In double-server protocol [21], a trusted center should prepare Bell states with replacing the server in order to the security. In fact, the trusted center can also be considered as another server, so it is a three-server protocol [21] actually. In their scheme, Alice becomes more classical, but servers cannot communicate with each other and it is not realistic. In triple-server protocol [22], a trusted center is also used to prepare two sequences of Bell states and those severs can communicate with each other. Actually there are four servers in the triple-server protocol [22] if

200

the trusted center is considered as a server.In our paper, only three servers perform computation and the client Alice uses two graph states to puzzle servers for blind computation. Alice doesn’t need to prepare any source states and servers can communication with each other. Our scheme is the practical three server delegated quantum computation scheme. If servers are not dishonest, they cannot obtain any information of the client Alice and their reputation is ruined. If they are honest, Alice can obtain the correct computation outcomes. In [25], Alice must perform single-qubit

205

measurement to verify the correctness of measurement outcomes. Alice is almost classical and does not need to perform any measurement scheme including the veriﬁable process in the whole delegated quantum computation of this paper.

9

We compare our scheme with some other schemes [25, 26, 27]. In [25], the Raussendorf-Harrington-Goyal (i.e. RHG) states or any bipartite graphs were used to complete BQC protocol. Alice needs to perform measurement of single-qubit states and Bob has the ability to generate and store entangled many-qubit states in their paper. It can 210

check the correctness and honestness of Bob by testing the stabilizers. In [26], the RHG states were used to complete two BQC protocols which the ﬁrst one uses many trap qubits and the other one does not use any trap qubits. It can check the correctness of the computing. Alice also needs to perform measurements and Bob needs to generate resource states in their paper. In [27], device-independent veriﬁable blind quantum computation is proposed. The scheme combines veriﬁed blind quantum computation with Bell state self-testing to solve the problem of verifying the

215

correct operation of such devices. Bell states are prepared for verifying the correctness of computing. However, our aim is to verify the correctness of measurement outcomes of three servers in our scheme. Alice owns quantum channels, and performs Pauli operation and has memory of two graph states for blind computation. Bob2 prepares resource states and performs measurement. We propose a simple method to verify the correctness of three servers Bob1, Bob2 and Bob3 by matching these measurement outcomes of three servers with data in database. Here, we take advantage of the

220

entanglement of |χ states. Compared with Refs. [25, 26], our scheme also can verify the correctness of the computing but Alice does not need to perform measurement. Our protocol is simpler than Refs. [25, 26, 27] too.

6. Conclusion In this paper, we propose a veriﬁable delegated quantum computation protocol with three severs based on χ-type entangled states, which Alice only needs to perform Pauli operators with her own quantum channels and memory of 225

two graph states for blindness. We also analyze and give the proof of blindness about protocol. Alice can obtain the correct outcomes since the veriﬁability is successfully accomplished.

Acknowledgments The research is funded by National Natural Science Foundation of China, under Grant Nos.

61672014 and

61502200, and Natural Science Foundation of Guangdong Province, China, under Grant Nos. 2016A030313090 and 230

2014A030310245, and Science and Technology Planning Project of Guangdong Province, China, under Grant No. 2013B010401018.

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