A Phase Estimation Algorithm for Quantum Speed-Up Multi-Party Computing

Security and privacy issues have attracted the attention of researchers in the field of IoT as the information processing scale grows in sensor networks. Quantum computing, theoretically known as an absolutely secure way to store and transmit information as well as a speed-up way to accelerate local or distributed classical algorithms that are hard to solve with polynomial complexity in computation or communication. In this paper, we focus on the phase estimation method that is crucial to the realization of a general multi-party computing model, which is able to be accelerated by quantum algorithms. A novel multi-party phase estimation algorithm and the related quantum circuit are proposed by using a distributed Oracle operator with iterations. The proved theoretical communication complexity of this algorithm shows it can give the phase estimation before applying multi-party computing efficiently without increasing any additional complexity. Moreover, a practical problem of multi-party dating investigated shows it can make a successful estimation of the number of solution in advance with zero communication complexity by utilizing its special statistic feature. Sufficient simulations present the correctness, validity and efficiency of the proposed estimation method.


Introduction
In recent years, security and privacy in edge computing have become a major challenge to the increased scale of information processing in sensor networks. Some people say that the data created by Internet of Things (IoT) sensors must be better protected. With more and more data protection regulations, new public awareness of tracking and the explosion of devices, simple device password solutions are no longer enough.
A new architecture using Security Agents should be constructed actively in local routers and networks to deal with Internet of Things security and computing, rather than offloading digital processing to data centers or clouds, or actually trying to execute it on IoT devices with limited resources. From the perspective of [1], Internet of Things security should be handled at the network level, not at the device level, in order to achieve the best results. Different from the security framework based on edge computing proposed in [1], we focus on another way to solve this problem, which is known as quantum computing that can accomplish both security and speed-up. Information stored and transmitted in a quantum state cannot be identified by eavesdroppers thanks to the measurement principle in quantum mechanics [2]. It means that the quantum method is able to be naturally secure as the message can be coded physically at the quantum level. What's more, quantum speed-ups are fascinating discoveries in recent years. The key technique to gain such an advantage is so-called quantum computing which is applicable to solve a series problems in such a wide range of quantum information processing. It can reduce the algorithm complexity in multimedia processing [3][4][5] and benefit the information security as well through protecting communication from eavesdropping [6][7][8]. Some papers focus on the states preparation for quantum communication [9,10]. And other field like quantum machine learning has gain much attention of computer scientists [11]. This special quantum parallelism totally different from conventional information techniques also can be applied in communication complexity which is a standard that can determine how much communication cost spent in a distributed computing task [12][13][14]. The first big progress proposed by Shor make the decoding RSA secret keys easy, which is definitely hard to implement in the classical case [15]. This class of quantum algorithms such as Shor's can downgrade the computation complexity from NP to P, but not applicable to the most of the classical algorithms. More generally, the database search algorithm called the Grover's quantum search algorithm has a larger range application to its classical counterpart than Shor's; however, the speed-up efficiency is mostly quadric level only [16][17][18][19][20][21][22]. The implementation of this type of quantum distributed (QD) algorithms for multi-party computing (MPC) definitely requires Grover's iteration [23][24][25][26].
One of the prerequisites for applying Grover's iteration is to know the number of solutions in order to determine the optimal number of iterations [16][17][18]. However, for many practical problems, it is difficult to know how many solutions the problem itself exactly has before it is answered. To solve this problem, a key algorithm called quantum phase estimation is used to estimate a possible number of solutions of one specific problem that uses Grover's iteration in the solving process [16].
We propose a distributed algorithm of phase estimation, which is used to estimate the number of solutions of MPC problem, and study how much quantum communication complexity is added to QD algorithm by using this method.
Additionally, considering the constraints of the statistical characteristics of the MPD (multi-party dating) problem, which can schedule appointments among different users and is seen as one of the practical applications of MPC, we develop the distributed estimation algorithm for estimating the number of solutions. This algorithm can be implemented without any multi-party communication through some simple local calculations. Finally, given many simulation results about this estimation algorithm prove the efficiency and correctness of the proposed method.

Estimation of the Number of Solutions of MPC Problem Based on Universal Discriminant Function
The quantum speed-up model of two-party computing is first discussed in paper [23]. For a reasonable extension of two-user model proposed in [23], we introduce the MPC model based on the same composite Boolean-valued function. Supposing that there are K users, marked as user 1 to user K, provided with the function in turn. user1 : user2 : . . .
The kth user wants to compute the function g k x ð Þ, 1 k K. The function g k x ð Þj1 k K f g could be arbitrary function in the application, but only satisfying that they share the same function domain and codomain. Besides, denote the function F y 1 ;y 2 ; Á Á Á ;y K ð Þas an arbitrary K-ary Boolean-valued function, only to satisfy that F y 1 ;y 2 ; Á Á Á ;y K ð Þ2 0; 1 f g and y k ¼g k x ð Þ, 1 k K.
Without loss of generality, denote the domain of function g k x ð Þ as X ¼ xj0 x N À 1; x 2 Z f g , the length of X is N. For the convenience let N be an integer that satisfies N ¼ 2 n (As for the case of 2 n < N <2 nþ1 , simply have N ¼ 2 nþ1 , the expanded part of X does not influence on the solving of the problem), therefore it's feasible to use the length of n ¼ logN bits information to describe the function domain. And similarly, supposing that the codomain of g k x ð Þ is Y ¼ yj0 y H À 1; y 2 Z f g , and the length to store the codomain information is h ¼ logH bits. So we have In summary, the goal of our research on the multi-party computing task is to find a solution x to equation F g 1 x ð Þ;g 2 x ð Þ; Á Á Á ;g K x ð Þ ð Þ ¼ 1 by comparing the results of K arbitrary functions g k x ð Þj1 k K f g calculation held by the multi-users. And the total communication complexity of quantum distributed algorithms that are designed to solve this MPC problem is determined as O K logNþK 2 logH À Á ffiffiffi ffi N p À Á .
Any quantum distributed algorithm for MPC needs to know the number of solutions of a specific problem. When the number of solutions is unknown, the number of solutions must be estimated.
Grover's database search problem is iterated by phase estimation algorithm, and the rotation angle of Grover problem is estimated by success rate and m bit accuracy [2].
For the MPC problem studied, we need to design a multi-party phase estimation algorithm to estimate the number of solutions of the MPC problem. The communication cost of the new phase estimation algorithm is deduced and evaluated.
According to the result of phase estimation algorithm of Grover's search problem in paper [2], the input of our phase estimation algorithm for MPC problem is two quantum registers, the first is t-qubit, the initial value is n + 1 qubit, and the second is n + 1 qubit. And we can conclude the whole steps of phase estimation algorithm in Tab. 1. Step 1: Quantum states are transformed into uniform superposition states via Hardmand gates.
Step 2: Quantum states are iterated by controlled Grover's gates.
Step 3: The quantum states in the first register are inversely transformed by quantum Fourier transform.
Step 4: The estimated rotation angle of Grover's iteration is obtained by measuring quantum states in the first register.
Assuming that the estimation error probability of the algorithm is 0, the probability of obtaining the rotation angle of M bit accuracy is obtained after measurement. The estimation error of rotation angle at this time. And the error of the number of solutions at this time Since the accuracy and success rate of the estimates depend on the magnitude of M and e, the setting of both will ultimately affect the upper limit. Here, we note that if the accuracy is assumed, the substitution Eq.
The estimation algorithm needs to call Grover's iteration 2 t times altogether. Therefore, the algorithm totally requires Â ffiffiffi ffi N p À Á times of Grover's iterations.
For MPC problems, Oracle operations are distributed, which means that estimation algorithms are also distributed. Each Grover's iteration in Fig. 1 consists of distributed quantum computing. The DOO quantum circuit of the phase estimation algorithm is shown in Fig. 1.
Theorem 1: Communication complexity required by phase estimation algorithms for MPC problem is O K logNþK 2 logH À Á ffiffiffi ffi N p À Á .
Proof: The communication complexity of the estimation algorithm can be deduced from the above results.
Firstly, in the DOO part, in the forward communication process from user 1 to user K, the communication cost is as follows: The upper bound of t in Eq. (1) is substituted, and the upper bound is expanded as Similarly, the reverse communication process has the same communication overhead as the forward process. Therefore, the communication overhead of a complete DOO is: According to the conclusion that the MPC phase estimation algorithm needs Â ffiffiffi ffi N p À Á times of Grover's iterations, the communication complexity of the phase estimation algorithm can be expressed as By given e, log 2 þ 1=2e ð Þ d eis a constant value. So after simplifying the above equation, we can get O K logNþK 2 h À Á ffiffiffi ffi N p À Á .
By replacing h which is the information length of function range of MPC with logH, the expression of communication complexity to be proved is obtained as O K logNþK 2 logH À Á ffiffiffi ffi N p À Á .
It can be seen that the phase estimation algorithm has the same communication complexity as the MPC quantum distributed algorithm. This means that using phase estimation algorithm to estimate the number of solutions in advance will not increase the communication complexity of MPC quantum distributed algorithm.

Estimation of the Solutions Amount of MPD Problem
An MPD problem is related to the appointment scheduling issue [24][25][26]. Paper [23] shows us a quantum computing result for two-party appointment. For the MPD case it belongs to MPC problems and actually is a Boolean-valued model as well.
Suppose T 1 ;T 2 ; Á Á Á T K are the calendars of K users respectively. Where And f k i ð Þ denotes whether the kth user is idle or not on day i. The discriminant function of the corresponding MPD problem iŝ Obviously, for any i, if^K i ð Þ¼ 1 then the K users succeeded in dating; Otherwise, it means the K users do not have any same free day in their calendar, the appointment task will fail.
Assuming that K users want to make a successful appointment with a certain probability P, the length of the schedules prepared by users will be constrained. Obviously, the more users participate in the joint appointment, the longer the schedule of the appointment will be on the premise of guaranteeing a certain success rate. Therefore, given the expected probability P of successful appointment, under the premise of K users' appointment, to meet P the probability of successful appointment, the relationship between the length of the schedule N and P is For MPD problems, the probability distribution of M which denotes the number of solutions satisfies the following equation Let P b ¼2 ÀK . It is known that M obeys Bernoulli distribution, that is Mb M; N;P b ð Þ. Therefore, it is easy to obtain the statistical average (mathematical expectation) of M.
And the standard deviation of M is: The above formula shows that if M the mathematical expectation of M is chosen as the estimated value of M; the average error of the estimated value is O ffiffiffiffi ffi M p .
Comparing with the phase estimation algorithm discussed in the previous section, we find that: On the one hand, the estimation error obtained by using the phase estimation algorithm is O ffiffiffiffi ffi M p À Á after calculating the statistical average, the error is O ffiffiffiffi ffi M p . This has the same complexity as the case of using M as the estimation.
On the other hand, it can be inferred that the communication complexity O K 2 ffiffiffiffiffi ffi 2 K p and the cost of other local quantum computation are needed if the phase estimation algorithm is used to estimate the number of solutions of MPD problem. However, it is much easier to get the numerical value of M without any multi-party communication. At most, it is necessary to use classical computers locally to get a mathematical expectation of Bernoulli distribution. The cost is negligible compared with MPD problem.
For the MPD problem, we choose the estimation method of using M to improve the performance of the algorithm. Next, we will analyze the impact of the difference between the estimated value and the real value on the performance of the quantum distributed algorithm in a given MPD problem.
According to paper [2], for a search problem whose number of solutions is M and the total search length is N, the Grover's search algorithm is equivalent to the process of gradually rotating the quantum state ci j to the vicinity at each angle. Here, Suppose we perform a specific MPD task, and its actual number of solutions is M opt . It is known that the angle between the initial state c 0 i j and the orthogonal ground state ai j is The corresponding optimal number of iterations is where CI (x) means that if the fractional part of X is greater than 0.5, it is rounded up; if the fractional part of X is less than or equal to 0.5, it is rounded down. In addition, the error probability of QD algorithm measurement results is P opt e ¼ M opt N .
Therefore, when estimating M opt by M, it can be obtained M ¼ 2 ÀK log 1ÀP 1À2 ÀK AE Ç . When the number of users K is large enough, it can be calculated through M ¼ Àln 1 À P ð Þ. Referring to the solving process of Eqs. (5) and (6), it is easy to know that the number of iterations using estimates is as follows.
As depicted in Fig. 2, this is equivalent to regard the initial state c 0 i j as c 0 0 i , and rotating it to the area near to bi j by iterations. Therefore, according to Eq. (5), it is easy to figure that in the process of solving the MPD problem the actual rotation angle isRÂu opt ¼R Â arcsin As a result, the correct probability of the measurement results using the estimated QD algorithm is

Numerical Simulation for Estimating the Number of Solutions
In this section, we simulate the MPD problem using the effect of evaluation. The number of experiments is C, and each experiment simulation produces a specific MPD task. The real value of the corresponding number of solutions is calculated by experiment, and the correct probability of measurement results is calculated.
For an MPD problem, we need to specify the probability of a successful appointment and the number of users in advance when we don't know which time of the parties is free. And we generally set the success rate as a lower value of 50% and a higher value of 90%. Similarly, due to the exponential growth of resources consumed in MPD simulation, the number of users should be set to consider the value more suitable for the simulation environment. For the number of experiments, a smaller value and a larger value are selected to facilitate comparison. So the simulation involves the following settings: (1) the success rate of appointment P is set to be 50% or 90%; (2) the number of users K is set to 6 or 12; (3) the number of experiments C is set to 10 or 1000.
Figs. 3 to 5 show the actual number of solutions produced in each experiment, and the distribution of probability of getting the correct results after every measurement using the estimation method. It can be seen from the experimental results that, except for the MPD problem without solution, the estimation method can always give the answer with a certain probability as long as the solution can be found. By observing their distribution, most of them can achieve the correct results of measurement with a probability greater than 0.5. Further, using the above results for calculation, we can get more intuitive results in Tabs. 2 and 3 (here, we consider the case where the correct probability is greater than 0.6 and 0.8 respectively). Figure 3: Estimation results when params change as K = 6 with P = 50%, K = 6 with P = 90%, K = 12 with P = 50% and K = 12 with P = 90% (four columns from left to right), C = 10 for all When the number of experiments is equal to 10, we can clearly see the distribution of the number of solutions from Figs. 3 and 4. However, in the comparison of Tabs. 2 and 3, we can find that the stability of the results of 10 experiments is significantly less than that of 1000 experiments. In the distribution of more than 0.6, the results of 1000 experiments were basically stable at about 90%, but 10 experiments did not get the similar result.
As is shown in Tabs. 2 and 3, except that the actual number of solutions is zero, the correct probability obtained in each experiment can basically fall in a higher probability interval. This shows that except for the actual number of solutions is zero, the correct measurement results can be obtained by solving MPD problem with high probability by using estimation method. According to the characteristics of Grover's search algorithm, for an example with a certain correct probability (a fixed value with a correct probability greater than 0), the probability of getting the correct result will be close to 1 if the same algorithm process is repeated several times [2]. For the case that the actual number of solutions is zero, no matter how repetitive the operation is, the measurement output of solving MPD problem by using the estimation method will be solvable, which is consistent with the actual number of solutions being 0.

Conclusion
Even if the quantum parallelism can speed many algorithms of multi-party computing up the realization of a quantum distributed algorithm still has been limited by solution's amount estimation. It must be assured that the total communication complexity should not increase after apply solution's amount estimation. Based on our study on the phase estimation algorithm for MPC problem the result shows it is possible to make a phase estimation without any complexity increasing. Theorem 1 presents that the communication complexity of phase estimation algorithm is O KlogNþK 2 logH À Á ffiffiffi ffi N p À Á , which means the communication cost is as the same level as MPC itself. Therefore, no more communication complexity induced by apply proposed estimation algorithm. Additionally, by utilizing the special attributes of MPD problem we able to estimate the number of solutions in advance without any communication. Because that the MPD has a known probabilistic distribution of Bernoulli. It only cost a few local iterations to get the result and dramatically degrades the communication complexity to 0. Amount of simulations for MPD example show the efficiency when the quantum algorithm applied.