Greenberger-Horne-Zeilinger states-based blind quantum computation with entanglement concentration

In blind quantum computation (BQC) protocol, the quantum computability of servers are complicated and powerful, while the clients are not. It is still a challenge for clients to delegate quantum computation to servers and keep the clients’ inputs, outputs and algorithms private. Unfortunately, quantum channel noise is unavoidable in the practical transmission. In this paper, a novel BQC protocol based on maximally entangled Greenberger-Horne-Zeilinger (GHZ) states is proposed which doesn’t need a trusted center. The protocol includes a client and two servers, where the client only needs to own quantum channels with two servers who have full-advantage quantum computers. Two servers perform entanglement concentration used to remove the noise, where the success probability can almost reach 100% in theory. But they learn nothing in the process of concentration because of the no-signaling principle, so this BQC protocol is secure and feasible.

Blind quantum computation (i.e. BQC) 1-7 is still a challenging research field, where a client has not enough quantum computability, and delegates her quantum computing to the servers who have full-advanced quantum computers. In long-distance BQC, quantum entanglement plays an important role and three mainly blind entangled states have already been studied which are blind brickwork state 1 , blind topological state 2 and Affleck-Kennedy-LiebTasaki (i.e. AKLT) state 3 . Some BQC protocols 1,[4][5][6] are based on the blind brickwork state which is proposed by Broadbent et al. 1 . Later, Barz et al. 7 demonstrated the blindness of the brickwork state. Broadbent et al. 1 in 2009 proposed a single-server BQC protocol based on single-qubit states and double-server BQC protocol based on the entanglement swapping of Bell states. However, the quantum entanglement of Bell states in double-server BQC protocol 1 will suffer quantum channel loss due to the influence of noisy channel. To solve this problem, Morimae and Fujii 4 proposed a method of entanglement distillation to extract high-fidelity Bell states, meanwhile its security can also be guaranteed. Li et al. 5 proposed a triple-server BQC protocol based on Bell states. Sheng and Zhou 6 proposed a double-server BQC protocol based on Bell states, where the deterministic entanglement distillation can remove the noise that transforms pure entangled states into mixed entangled states. As we can see that the aims of BQC protocols 1, 4-6 are all to obtain the single-qubit states ± θ i with θ ∈ … π π π Results BQC protocol based on maximally GHZ entangled states. Photons are the best physical systems for the long-distance transmission of entangled states, thus entangled photons states are used as quantum information carriers in BQC. In this BQC protocol, we use |0〉 and |1〉 to express photons. In entanglement concentration, we use |H〉 and |V〉 to express photons, where |H〉 is equal to |0〉 and |V〉 is equal to |1〉. In this section, we propose the BQC protocol based on maximally GHZ photons entangled states = + + + GHZ ( 001 010 100 111 ) A B C 1 2 j j j ( = … j n 1, 2, , ) (Fig. 1). The cross-Kerr nonlinear can be used to construct a CNOT gate in ref. 19. There are also many other methods to realize it [19][20][21][22][23] . In the BQC protocol, we suppose that these quantum devices are all ideal. The client owns quantum channels with two servers and quantum disturbing device.
• Bob generates enough maximally GHZ entangled states GHZ A B C j j j , where the subscripts A j , B j and C j represents photons A j , B j and C j . Bob keeps photons sequences S B = [B 1 , B 2 , …, B n ] and sends photons sequences S A = [A 1 , A 2 , …, A n ] and S C = [C 1 , C 2 , …, C n ] to Alice successively. After receiving photons sequences, Alice disturbs the order of photons sequences S A and S C . The reordered photons sequences are rewritten as . , where |α| 2 + |β| 2 + |δ| 2 + |η| 2 = 1. In order to get states , Bob and Charlie firstly perform entanglement concentration. • Bob performs one of four operations {I, σ x , iσ y , σ z } randomly chosen by Alice on photons B j and states evolve into one of four states | ′ ′ The remaining steps are the same as steps (2-3) of the BFK protocol 1 or steps (2-5) of blind topological BQC protocol 2 . The blindness of graph states and the correctness of quantum computation have already been exhibited in refs 1 and 2 in detail.
In the step 1 of this BQC, entanglement concentration is used to remove the noise. In the following, the process of entanglement concentration is showed with optical system. Entanglement concentration of pure maximally GHZ entangled state. In a practical transmission, there exist two kinds of quantum channel noises, i.e. pure maximally entangled states evolve into mixed states or less-entangled states. Entanglement purification 24-28 is applied to extract high-fidelity maximally entangled states from mixed entangled states. Entanglement concentration [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45] is often used to distill less-entangled states into pure maximally entangled states by local operations and classical communication (i.e. LOCC). Bennett et al. 29 firstly proposed an entanglement concentration protocol by using Schmidt projection. In 2003, Zhao et al. 42 not only demonstrated the entanglement concentration scheme in ref. 30 but also verified a quantum repeater in experiment. Li et al. 39 proposed two protocols to concentrate hyper-entangled GHZ states by using a single-photon state of two freedoms and two less-entangled states respectively. Sheng et al. 32 proposed to concentrate arbitrary W states by using two steps. Afterwards, a universal concentration scheme of an arbitrary less-entangled N-photon W state is proposed in ref. 43. Here, we consider a special quantum channel noise, i.e. pure maximally entangled states evolve into less-entangled states, which can be distilled by entanglement concentration. In the following, we give the entanglement concentration of GHZ states that were experimentally prepared in refs 46-48.
The first round of entanglement concentration. In the BQC, the maximally GHZ states can be rewritten in the form of where we define |H〉 = |0〉 and |V〉 = |1〉. The subscripts a 1 , b 1 and c 1 represent the spatial-mode of photons ′ A t 2 , B j and ′ C t 1 . We consider the noisy model that pure maximally entangled states evolve pure less entangled states. Suppose less-entangled pure photons states are where four real numbers α, β, δ, η satisfy |α| 2 + |β| 2 + |δ| 2 + |η| 2 = 1. Two identical less-entangled states, which the parameters are all unknown, can distill a maximally entangled state in Eq. (2). The schematic of entanglement concentration is shown in Fig. 2. Here, only Alice knows whether entanglement concentration is successful and the correct orders of ′ A t 2 , B j and where polarization photons a 1 , b 1 and c 1 are flipped and relabeled as a 2 , b 2 and c 2 . The entanglement concentration is divided into two steps. In the first step, the system composed of six photons is     If the detectors  (1) 1   The probabilities of getting quantum states ϕ a b c

12
(1) 1 These are all failed cases, but they can be used as the initial states in the second round. For quantum state its process of concentration is the same as ϕ a b c 12 1 1 1 1 and we can get This is the maximally GHZ entangled state. The success and failure probabilities of ϕ a b c 12 2

Discussion
Blindness and correctness analysis of the proposed BQC protocol. In the following, we will show that the proposed BQC protocol is secure by analyzing the blindness and correctness. First, we show the blindness of the proposed BQC protocol.
(1) Bob performs one of four Pauli operations randomly chosen by Alice on his photons and the initial states are correspondingly changed into one of SCientifiC REPORTs | 7: 11104 | DOI:10.1038/s41598-017-06777-w . Whether Bob colludes with Charlie or not, they guess the correct Bell state with the probability of 1 4 . When this BQC protocol is repeated n times, the probability of obtaining correct quantum states is and disturbs the order of photons A j , B j , C j .
Bob and Charlie know nothing about the states | ± 〉 θ i because of the no-signaling principle. After repeating n times, the probability of guessing correct θ i is = →∞ ( ) lim 0 n n 1 8 . In the process of entanglement concentration, Bob and Charlie cannot eavesdropping any useful information by exchanging their results because of difference of orders of three photons.
(3) The structures of blind brickwork states and blind topological states are private for servers. Therefore, Bob and Charlie can't obtain anything about Alice's private information whether they communicate with each other or not. The blindness of BFK single-server protocol and blind topological single-server protocol are showed in refs 1 and 2 in detail respectively.
Second, the correctness of quantum computation in BFK single-server protocol and blind topological single-server protocol are presented in refs 1 and 2 in detail.
So this BQC protocol is blind and correct.

Analysis of the success probabilities in iteration.
In the above discussion, we have already elaborated the first round of the entanglement concentration with cross-Kerr nonlinearity in detail. QND provides a strong tool for us to perform a quantum nondemolition measurement that does not destroy entanglement of photons, which ensures that each step can be operated independently. Here, we analyse the second round and the k-th round of entanglement concentration. In the second round, for the quantum states In the k-th (k > 1) round, the success and failure probabilities are   In the second step of the k-th round, for the quantum states  The probabilities of obtaining four quantum states in the first step or the second step of the k-th round are  The total probability is = ∑ = P P total k n k 1 , which depends on the number of iterations and parameters of the initial states. The relationship of the total success probability, parameters and the number of iterations is shown in Fig. 4. It can be seen that the total success probability has kept increasing with the parameters β and δ in the range of       0, 3 2 . When n = 4, the success probability has already reached 0.9196. When n = 9, the success probability has already reached 0.9975. Therefore, the entanglement concentration is successful in theory.
In this paper, we only consider the ideal CNOT gate [19][20][21][22][23] . In experiment, there exist many nonideal factors such as the double effect of parameter conversion, the imperfect matching of the crystal lattice and phases, and so on. The probabilities of intrinsic error of experimental methods are unavoidable, such as QND measurements and CNOT operations. Thus optimizing the experimental system is a very meaningful research direction. In the BQC protocol, we only give the concrete quantum channel noise model but not universal. So, we will further study entanglement purification of GHZ states.

Methods
The optical devices are used to complete the entanglement concentration, where the parity check devices are based on cross-Kerr nonlinearity that can construct QND 38,39,41 to improve the successful probability. The cross-Kerr nonlinearity medium is described by the Hamiltonian, where † a s and † a p are the creation operators, a s and a p are the annihilation operators, a Fock state |n〉 and a coherent state |α c 〉 interact. The whole system evolves into  1, 2, 3),where real numbers μ 0 , μ 1 , λ 0 and λ 1 satisfy the normalization condition |μ 0 | 2 + |μ 1 | 2 = 1, |λ 0 | 2 + |λ 1 | 2 = 1, respectively. Then the composite quantum system τ τ α ϒ = ⊗ ⊗ k k c 1 1 2 evolves to