# Blind quantum computation with noise environment

###### Abstract

Blind quantum computation (BQC) is a new type of quantum computation model. BQC allows a client (Alice) who does not have enough sophisticated technology and knowledge to perform universal quantum computation and resorts a remote quantum computation server (Bob) to delegate universal quantum computation. During the computation, Bob cannot know Alice’s inputs, algorithm and outputs. In single-server BQC protocol, it requires Alice to prepare and distribute single-photon states to Bob. Unfortunately, the distributed single photons will suffer from noise, which not only makes the single-photon state decoherence, but also makes it loss. In this protocol, we describe an anti-noise BQC protocol, which combined the ideas of faithful distribution of single-photon state in collective noise, the feasible quantum nondemolition measurement and Broadbent-Fitzsimons-Kashefi (BFK) protocol. This protocol has several advantages. First, Alice does not require any auxiliary resources, which reduces the client’s economic cost. Second, this protocol not only can protect the state from the collective noise, but also can distill the single photon from photon loss. Third, the noise setup in Bob is based on the linear optics, and it is also feasible in experiment. This anti-noise BQC may show that it is possible to perform the BQC protocol in a noisy environment.

###### pacs:

03.67.Ac, 03.65.Ud, 03.67.Lx## I Introduction

Quantum computation has attracted much interest for its ultrafast computation ability. Shor’s algorithm for integer factorization shor (), Grover’s algorithm and the optimal Long’s algorithm for unsorted database search grover (); long (), all have displayed the great computing power of quantum computers. Small-scale quantum computers in ions ion (), superconduction superconduct (), photons photon (), and some other important quantum systems have been widely investigated other1 (). It is not a dream to successfully product a quantum computer in the foreseeable future. Like current supercomputers, the first generation of quantum computers must be very expensive and owned by very few governments or big companies. As an ordinary quantum computer client, he or she has poor quantum ability and are insufficient to realize universal quantum computation. Certainly, he or she can resort the quantum computation server’s help to realize the computation. Moreover, he or she should protect his data without leakage. Blind quantum computation (BQC) is a new type of quantum computation model that the client who does not have enough quantum knowledge and sophisticated technology and resorts the quantum computation servers to perform the universal quantum computation. During the computation, the client’s inputs, algorithms and outputs should be absolutely security.

In 2005, Childs proposed the first BQC model blind1 (). It is the standard quantum circuit model. Bob needs to perform the quantum gates and Alice requires the quantum memory. In 2009, Broadbent, Fitzsimons, and Kashefi (BFK) proposed a BQC protocol based on the one-way quantum computation model blind2 (). In their protocol, Alice only requires to generate the single-qubit quantum state and a classical computer. The most advantage of this protocol is that Alice does not need the quantum memory. There are also some other important BQC protocols blind3 (); blind4 (); blind5 (); blind6 (); blind7 (); blind8 (); blind9 (); blind10 (); blind11 (); blind12 (); blind13 (); blind14 (); blind15 (); blind16 (); blind17 (); blind18 (); blind19 (); blind20 (); blind21 (). For example, Morimae et al. proposed two BQC protocols based on the Affleck-Kennedy-Lieb-Tasaki state blind3 (). Fitzsimons and Kashefi constructed a new verifiable BQC protocol blind5 (). The experiment of the BFK protocol based on the optical system was also reported blind10 (). Generally, these kinds of BQC protocols can be divided into three groups. The first group is the single-server BQC model blind1 (); blind2 (); blind3 (); blind5 (); blind6 (); blind7 (); blind8 (); blind9 (); blind10 (); blind12 (); blind13 (); blind14 (); blind15 (); blind16 (); blind17 (); blind18 (); blind21 (). The second gourp is double-server BQC model blind2 (); blind11 (); blind20 () and the third gourp is triple-server BQC model blind19 (). In single-server BQC model, the client Alice is required to has the quantum ability of generating and distributing the single quantum states. In double-server BQC model and triple-server BQC model, the client Alice can be completely classical.

In single-server BQC protocol, the client Alice should distribute the single-photon states to the server Bob. In previous single-server protocols, the quantum channel is assumed to be ideal. However, the ideal quantum channel does not exist, and all the quantum states will suffer from noise. The environment noise will make the quantum state become error, and it will also make the distributed photons loss. In quantum communication, various error correction and error rejection
methods are proposed collective0 (); collective1 (); collective2 (); collective3 (); collective4 (); collective5 (). For instance, Walton et al.
proposed a scheme for rejecting the errors introduced by
noise with decoherence-free subspaces collective0 (). In 2005, Kalamidas proposed two interesting linear-optical single photon
schemes to reject and correct arbitrary qubit errors
without additional particles collective1 (). By adding one extra photon with a fixed polarization, a distribution scheme of polarization states of a single photon over a collective-noise
channel was proposed collective2 (). In 2007, Li and Deng described a faithful qubit transmission scheme with linear optics against collective noise without ancillary qubits collective3 ().
On the other hand, quantum state amplification is an efficient tools to resist the photon loss amplification1 (); amplification2 (); amplification3 (); amplification4 (); amplification5 (); amplification6 (); amplification7 (); amplification8 ().
The quantum state amplification can be increase the probability of single photon and decrease the probability of photon loss.

Practical BQC protocol should also works under the noise environment. In double-server BQC, Morimae and Fujii first described an efficient secure entanglement distillation for double-Server BQC blind11 (). They showed that it is possible to perform entanglement distillation in the double-server scheme without degrading the security of blind quantum computing. In 2015, we proposed the deterministic entanglement distillation for secure double-server BQC blind20 (). In single-server BQC, recently, Takeuchi et al. first considered the model of single-server BQC over a collective-noise channel, which is called DFS-BQC blind21 (). They described three variations of DFS-BQC protocols, combined the ideas based on the DFS and the BFK protocol. In this paper, we describe another anti-noise BQC protocol, based on the original BFK protocol blind2 (). This protocol has some advantages. First, Alice does not require to generate the Bell pair or coherent light, and only to distribute and operate the single photon with linear optics. which reduces the client’s economic cost. Second, this protocol not only can protect the state from the collective noise, but also can distill the single photon from the photon loss. Third, the noise setup in Bob is based on the feasible linear optics.

## Ii Basic model of anti-noise BQC protocol

Before we explain this protocol, we first briefly describe the original BFK protocol. It runs as follows blind2 (): a) Client Alice first prepares rotated qubits and distributes to the server Bob. Here and is the set of edges of and CZ is the CZ gate between the th and th qubits. b) Bob prepares the Graph state , which Alice tells her. Here . c) Bob performs the measurement on the th qubit according to measurement angle , which Alice tells her. Here is a random number and . is the modified version of according to the previous measurement results. d) Bob sends the measurement results to Alice and Alice completes the computation with classical computer.

The basic model of this anti-noise BQC protocol is shown in Fig. 1. In the side of Alice, Alice first prepare rotated qubits . In an optical system, we denote the horizontal polarization photon as and vertical polarization photon as , respectively. In traditional BFK protocol, the single-photon state is sent to Bob directly. In this protocol, Alice first encodes the state as shown in Fig. 2.

(1) |

The photon will suffer from the noise, which will make

(2) |

and

(3) |

Here , and . and are the spatial modes as shown in Fig. 2. is the short arm and is the long arm. The noise model is also the collective noise model blind21 (). Therefore, after transmission, if the photon does not loss, the state becomes

(4) | |||||

Certainly, the single-photon state may also suffer from the photon loss and become a vacuum state . Generally, Bob will receive a mixed state , which can be written as

(5) |

Here denotes the transmission efficiency of the photon. From Eq. (5), before Bob starting the BQC protocol, he should distill the mixed state and obtain the original state deterministically. The Noise Processer showed in Fig. 1 is to complete the task. The Noise Processor is detailed in Fig. 3. From Eq. (4), if the single photon does not lose, it will be in the spatial modes or with equal probability of 50%. The noise processor in Fig. 3 shows the distillation in spatial mode . If the photon is in spatial mode , Bob can distill it with the same setup. We take the photon in spatial mode for example. As shown in Fig. 3, the quantum state can be evolved as

(6) | |||||

From Eq. (6), if the photon does not lose, it will be in the different arriving time, i. e., , , or , respectively. Therefore, Bob can get the uncorrupted states in spatial mode , or in spatial mode in the determinate time corresponding to and . Here . One can perform a bit-flip operation to convert to . From Eq. (4), if the single photon is in the spatial mode , Bob can deal with the same principle, using BS, HWP and PBS. Therefore, if the photon does not lose, combining with the single photon being in the spatial mode with the same probability, Bob will obtain a single-photon entangled state in the time bin as

(7) | |||||

Here the spatial modes and are shown in Fig. 4. It means that if the photon is in the spatial mode as shown in Fig. 2, Bob can deal with the collective noise with the setup of Fig. 4.

Combined with the case of photon loss, by selecting the time bin , state in Eq. (5) can be rewritten as

(8) |

From Eq. (8), the next step of Bob is to distill from the mixed state deterministically. Here we exploit the linear noiseless amplification (NLA) to complete the task. Here we introduces a pair of ancillary polarized photons of the form

(9) |

As shown in Fig. 3, the partially polarization beam splitter (PPBS) can reflect the vertically polarized photon totally, while reflect the horizontally polarized photon with the coefficient of , and transmit it with the coefficient of . Another PPBS can transmit the horizontally polarized photon totally, while reflect the vertically polarized photon with the coefficient , and transmit it with the coefficient of . For instance, the PPBS can make amplification5 ()

(10) |

Here is the creation operator. The subscripts , , and are the spatial modes as shown in Fig. 3. Therefore, by using PPBS and PPBS, we can obtain the relationship

The mixed state combined with the polarization Bell state can be described as follows. With the probability of , it is in the state and with the probability of , it is in the state . We first discuss the item . It evolves as

(12) | |||||

The first item evolves as

(13) | |||||

The second item can evolve as

The third item can evolve as

The forth item can evolve as

Interestingly, from Eq. (13) to (LABEL:collapse4), if they pick up the case that the single-photon detectors DD or DD register one photon respectively, they will obtain the state deterministically in the output mode, for the cases in Eq. (LABEL:collapse2) to (LABEL:collapse4) can not lead both single-photon detectors DD or DD register one photon, and only the item in Eq. (13) can satisfy the selection condition.

Once Bob obtain the single-photon state deterministically, he can start to perform the BQC protocol. In this way, the whole BQC protocol can be modified as follows: 1) Alice prepares rotated qubits . 2) Alice encodes the photon with linear optics, as shown in Fig. 2. 3) Alice distributes the photon to Bob, which will suffer from collective noise and photon loss. 4) Bob distill the polluted single-photon states with Noise Processer setup, as shown in Fig. 3. 5) Bob prepares the Graph state . 6) Bob performs the measurement on the th qubit. 7) Bob sends the measurement results to Alice and Alice completes the computation with classical computer.

## Iii Discussion and conclusion

So far, we have completely explained the anti-noise BQC protocol. In original BQC protocol, Alice prepares and distributes the state directly. Bob also receives the and perform the BQC subsequently, for they do not consider the noise environment. Similar to the pioneer work of Ref.blind21 () in collective-noise BQC protocol, Alice and Bob should perform the pretreatment for noise, before starting the BQC protocol, following the approaches suggested in Refs.collective3 (); amplification5 (). In BQC protocol, two essential properties are correctness and blindness. The correctness means that the output of the protocol in Alice’s desired one as long as Alice and Bob follow the procedure of the protocol is faithfully. On the other hand, the blindness means that Bob cannot know any information about Alice’s inputs, algorithm, and outputs, whenever Alice follows the procedure of the protocol. Obviously, this protocol is correctness for Bob can obtain the faithful qubits after the Noise Processer. On the other hand, this protocol is also blindness. The information sent from Alice to Bob is , which is decided by . Bob does not know the exact information of , which ensures that the protocol is blindness.

We can calculate the total success probability of this protocol. From Fig. 2, we explain this protocol by selecting the case that the photon being in the spatial mode with the probability of 50%. Actually, if the photon is in the spatial mode , they can perform the protocol with the same principle. That is to say, if the photon does not lose and only suffer from the collective noise, by picking up the suitable arriving time as shown in Fig. 4, the total success probability is 50%. On the other hand, as shown in Fig. 3, such setup essentially distill the photons in the spatial modes , i. e., the first item in Eq. (7). Actually, in Eq. (7), other three items which contains the single photon that can also be verified by adding three auxiliary polarized Bell states. The final success probability of this protocol can be calculated as

(17) |

In the previous work of Ref.blind21 (), they presented three important BQC protocols over a collective-noise channel. The first protocol is entanglement-based protocol. The second is single-photon-based protocol and the third is the coherent-light-assisted protocol. The common characteristics of three protocol is that they require auxiliary resources, such as entanglement, single photon or coherent light. Moreover, these protocols suppose that Bob has powerful QND measurement, which can distinguish the arriving photons deterministically. However, they do not explain how to realize such QND measurement. In current technology, QND measurement, such as exploiting cross-Kerr nonlinearity has widely discussed in quantum information processing kerr1 (); kerr2 (); kerr3 (); kerr4 (); kerr5 (), but it is still a big challenge in experiment kerr6 () and it will greatly increase the computation cost.

In conclusion, we have described an efficient anti-noise BQC protocol. Different from previous work, this protocol has several advantages. First, this protocol does not require any auxiliary resources, which makes the client is economic. Second, this protocol not only can protect the state from the collective noise, but also from the photon loss. Third, the Noise Processer for Bob is based on the linear optics, and it is also feasible in experiment.

## Acknowledgements

This work was supported by the National Natural Science Foundation
of China under Grant Nos. 11474168 and 61401222, the Qing Lan Project in Jiangsu Province, and a Project
Funded by the Priority Academic Program Development of Jiangsu
Higher Education Institutions.

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