Anonymous Computation in Quantum Networks

1. Introduction

In recent centuries, the vigorous development of science and technology has greatly enhanced humanity’s ability to understand nature and transform the world [1]. However, science can never answer ultimate questions, such as how the primordial energy that gave birth to all things in the universe came into existence, what force truly drives the operations of the universe, and what is the fundamental cause of quantum entanglement [2]. Many such questions can never be satisfactorily answered solely through scientific exploration. After all, human observation of the universe is extremely limited, and the resulting perspectives are undoubtedly very one-sided [2].

Unfortunately, despite the significant limitations of science, some people treat it as a belief. These individuals do not believe in karma and hold the concept of “hedonism”. At the same time, they behave improperly and without restraint in their pursuit of fame and wealth. In such circumstances, science has undoubtedly become a kind of cult, exploited by some unscrupulous individuals for evil purposes. From the current global situation, the damage caused by technology and industry to the ecological environment has already seriously threatened the survival and development of all humanity.

If technology is allowed to continue developing without proper guidance, the world will eventually be destroyed. Fortunately, some wise individuals have already recognized this and are actively taking countermeasures, including seeking solutions. To our knowledge, the only approach is to understand the core principles of the I Ching and put them into practice, with the success rate of practice determined by the depth of the understanding of the I Ching. The I Ching is the origin of all things, and through it, satisfactory answers can be found to the questions like those mentioned above [2].

A few individuals with extraordinary wisdom have discovered that the world in which humans live is like a virtual world created by a supercomputer [3]. Based on current knowledge, it is acceptable to hypothesize this supercomputer as a quantum computer [3]. Building on this idea, in our recent work [3], we have hypothesized that the universe is created by a vast quantum computer network. All natural phenomena in the universe are developed and designed by this network based on the core theories of the I Ching, with quantum mechanical phenomena serving as a typical example. Therefore, exploring quantum mechanical phenomena is of great help in understanding the I Ching. Furthermore, although the quantum network cannot be directly observed, its operating mechanism can be roughly inferred through the various natural phenomena it exhibits.

As one of the most peculiar phenomena in quantum mechanics, entanglement swapping is a key pillar for building quantum networks, and it has wide and important applications in quantum cryptography, quantum communication and quantum computing [4,5,6]. In this paper, based on our previous work [7], we design two quantum anonymous computation (QAC) protocols through entanglement swapping. In a QAC protocol, the identity of data holders is confidential, but there are no requirements regarding whether the data is public. Even if the data is public, it is not disclosed which participant each data comes from. We introduce a third party in both protocols, who is responsible for collecting all data and completing the computation of the final public functions. Both protocols use unitary operations for encoding, and achieve information exchange and confidentiality through entanglement swapping.

The remaining part of this paper is arranged as follows: Section 2 describes the two proposed protocols, including the protocol procedures and the main quantum resources used. Section 3 is a summary of the work presented in this paper.

2. Quantum Anonymous Computation Protocols

Before describing the processes of the two protocols, let us first introduce the main quantum resources adopted in the protocols, including quantum states, unitary operations, and entanglement swapping cases.

In the two protocols, the d-level Bell states are used as the information carriers, which can be expressed as

$$\begin{array}{c}\hfill \left|\varphi (u,v)\right.〉={\displaystyle \frac{1}{\sqrt{d}}}{\displaystyle \sum _{j=0}^{d-1}}{\zeta }^{ju}\left|j\right.〉\left|j\oplus v\right.〉,\end{array}$$

(1)

where $u,v=0,1,\dots ,d-1$, and the symbol ⊕ denotes addition modulo d throughout this paper. It is easy to get

$$\begin{array}{c}\hfill \left|\varphi (0,0)\right.〉={\displaystyle \frac{1}{\sqrt{d}}}{\displaystyle \sum _{j=0}^{d-1}}\left|j\right.〉\left|j\right.〉.\end{array}$$

(2)

$\left|\varphi (u,v)\right.〉$ can be generated by

$$\begin{array}{c}\hfill (I\otimes \left|U_{(u,v)}\right.〉)\left|\varphi (0,0)\right.〉=\left|\varphi (u,v)\right.〉,\end{array}$$

(3)

where I is the identity matrix, the symbol ⊗ is the tensor product symbol, and

$$\begin{array}{c}\hfill \left|U_{(u,v)}\right.〉={\displaystyle \frac{1}{\sqrt{d}}}{\displaystyle \sum _{j=0}^{d-1}}{\zeta }^{ju}\left|j\oplus v\right.〉\left.〈j\right|.\end{array}$$

(4)

The entanglement swapping used in the protocol is realized by performing a joint measurement on one particle in the d-level cat state and one particle in the d-level Bell state, which can be expressed as [5,8]

$$\begin{array}{cc}\hfill & \left|\varphi \left(u_0^1,u_1^1,\dots ,u_m^1\right)\right.〉\otimes \left|\varphi \left(u_0^2,u_1^2\right)\right.〉\hfill \\ \hfill =& {\displaystyle \frac{1}{d}}{\displaystyle \sum _{v_0,v_1=0}^{d-1}}{\zeta }^{\left(u_k^1\ominus v_1\right)\left(u_0^2\ominus v_0\right)}\left|\varphi \left(u_0^1\oplus u_0^2\ominus v_0,u_1^1,u_2^1,\dots ,u_1^2\oplus u_k^1\ominus v_1,\dots ,u_m^1\right)\right.〉\otimes \left|\varphi \left(v_0,v_1\right)\right.〉,\hfill \end{array}$$

(5)

where the particle $u_k^1$ in the cat state and the particle $u_1^2$ in the Bell state are swapped, the symbol ⊖ denotes subtraction modulo d throughout this paper, and the d-level cat state is given by [5,8]

$$\begin{array}{cc}\hfill & \left|\varphi (u_1,u_2,\dots ,u_m)\right.〉\hfill \\ \hfill =& {\displaystyle \frac{1}{\sqrt{d}}}{\displaystyle \sum _{j=0}^{d-1}}{\zeta }^{j{u}_1}\left|j,j\oplus u_2,j\oplus u_3,\dots ,j\oplus u_m\right.〉,u_1,u_2,\dots ,u_m\in \{0,1,\dots ,d-1\}.\hfill \end{array}$$

(6)

2.1. Quantum Anonymous Computation for Multiple Sets of Data

Suppose that there are n mutually distrustful participants, $P_1,P_2,\dots ,P_n$, and that each party $P_i$ has a set of secret data, denoted as ${\mathbb{S}}_i=\{x_i^1,x_i^2,\dots ,x_i^{l_i}\}$, where

$$\begin{array}{c}\hfill x_i^j\in \{0,1,2,\dots ,\xi \},\xi \in \mathbb{N},\phantom{i}i=1,2,\dots ,n,\phantom{i}j=1,2,\dots ,l_i.\end{array}$$

(7)

Here, the value of $\xi$ is set according to an actual situation. For example, if ${\mathbb{S}}_i$ represents the exam scores of all subjects for a student and the maximum score for each subject is 100 points, then $\xi$ can be set to 100. $P_i$ arranges the values of data in descending order as follows:

$$\begin{array}{c}\hfill y_i^1,y_i^2,\dots ,y_i^{k_i},\phantom{i}{k}_i\in \left\{z|z\in {\mathbb{Z}}^{+},1\le z\le l_i\right\},\end{array}$$

(8)

where $k_i(1\le k_i\le l_i)$. Next, $P_i$ counts the number of the data equal to $y_i^1,y_i^2,\dots ,y_i^{k_i}$, respectively, and records their quantities as

$$\begin{array}{c}\hfill {\eta }_i^1,{\eta }_i^2,\dots ,{\eta }_i^{k_i}.\end{array}$$

(9)

A semi-honest third party (named TP throughout this paper), who executes the protocol honestly but is curious about the identity of the data owners, help the participants calculate the public function

$$f\left(x_1^1,x_1^2,\dots ,x_1^{l_1};x_2^1,x_2^2,\dots ,x_2^{l_2};\dots ;x_n^1,x_n^2,\dots ,x_n^{l_n}\right).$$

(10)

The protocol steps are as follows:

Step 1. $P_i$ prepares $\xi +1$ copies of d-level Bell state $\left|\varphi (0,0)\right.〉$, where

$$d\in \left\{z|z\in {\mathbb{Z}}^{+},z>max\left\{l_1,l_2,\dots ,l_n\right\}\right\},$$

(11)

and marks them by

$${\left|\varphi (0,0)\right.〉}_r,r=0,1,\dots ,\xi .$$

(12)

TP prepares $\xi +1$ copies of the d-level ($n+1$)-particle cat state $\left|\varphi (u_0,u_1,\dots ,u_n)\right.〉$, and marks them by

$$\left|\varphi \left(u_0^r,u_1^r,\dots ,u_n^r\right)\right.〉.$$

(13)

Then, TP takes the particles with the marks $u_1^r,u_2^r,\dots ,u_n^r$ out from $\left|\varphi \left(u_0^r,u_1^r,\dots ,u_n^r\right)\right.〉$, and arrange them as follows:

$$\left(u_1^0,u_1^1,\dots ,u_1^{\xi }\right),\left(u_2^0,u_2^1,\dots ,u_2^{\xi }\right),\dots ,\left(u_n^0,u_n^1,\dots ,u_n^{\xi }\right),$$

(14)

and marks them by ${\mathcal{S}}_1,{\mathcal{S}}_2,\dots ,{\mathcal{S}}_n$, respectively.

Step 2. TP sends ${\mathcal{S}}_1,{\mathcal{S}}_2,\dots ,{\mathcal{S}}_n$ to $P_1,P_2,\dots ,P_n$ through quantum channels, respectively. To verify the security of quantum channels, TP can use decoy photons or the entanglement correlation of entangled states for eavesdropping detection. These two eavesdropping detection methods are detailed in Ref. [10] and will not be elaborated here.

Step 3. If there are no eavesdroppers in quantum channels, $P_i$ encodes his secret data which equal to the value $y_i^{t_i}\left(t_i\in \{1,2,\dots ,k_i\}\right)$. Specifically, $P_i$ chooses the d-level Bell state with the subscript which equals to $y_i^{t_i}$, and establishes a variable $w_i^{t_i}$ and sets $w_i^{t_i}={\eta }_i^{t_i}$. For the rest of d-level Bell states with the subscripts $h_i\in \left\{0,1,\dots ,\xi \right\}\setminus \left\{y_i^{t_i}\right\}$, $P_i$ sets $w_i^{h_i}=0$. Then, $P_i$ uses the unitary operation shown in Eq. 3 to generate the d-level Bell states

$$\begin{array}{c}\hfill \left|\varphi (v_i^r,w_i^r)\right.〉={\displaystyle \frac{1}{\sqrt{d}}}{\displaystyle \sum _{l=0}^{d-1}}{\zeta }^{l{v}_i^r}\left|l\right.〉\left|l\oplus w_i^r\right.〉.\end{array}$$

(15)

Step 4. $P_i$ performs d-level Bell measurements on the particle with the marks $u_i^r$ in the cat state and the particle with the mark $v_i^r$ in $\left|\varphi (v_i^r,w_i^r)\right.〉$, respectively. The measurement results are marked by

$${\left\{\left|\varphi ({\lambda }_i^r,{\gamma }_i^r)\right.〉\right\}}_{r=0}^{\xi }.$$

(16)

Then, $P_i$ sends all unmeasured particles in d-level Bell states to TP. To ensure the security of transmission, the same method as Step 2 is used for eavesdropping detection.

Step 5. $P_1,P_2,\dots ,P_n$ cooperate together to calculate

$${\mathcal{A}}^r={\displaystyle \sum _{i=1}^n}{\gamma }_i^r,$$

(17)

and then announce ${\mathcal{A}}^r$ to TP publicly. Here, if all participants gather together, they can directly complete the calculation, otherwise they can use public quantum channels to complete the calculation.

Step 6. After receiving all cat states (note here that all unmeasured particles in the d-level Bell states and cat states collapse into new cat states), TP measures each cat state and obtains the marks

$$\left(u_0^r+{\displaystyle \sum _{i=1}^n}{k}_i^r-{\displaystyle \sum _{i=1}^n}{\lambda }_i^r\right)modd,u_1^r\oplus w_1^r\ominus {\gamma }_1^r,u_2^r\oplus w_2^r\ominus {\gamma }_2^r,\dots ,u_n^r\oplus w_n^r\ominus {\gamma }_n^r.$$

(18)

Then he calculates

$${\mathcal{R}}^r={\displaystyle \sum _{i=1}^n}\left(u_i^r\oplus w_i^r\ominus {\gamma }_i^r\right)-{\displaystyle \sum _{i=1}^n}{u}_i^r+{\mathcal{A}}^r,$$

(19)

which means the number of all participants’ data that equal to r is ${\mathcal{R}}^r$. By adding subscripts, the data of all participants can be marked as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{0}_1,0_2,\dots ,0_{{\mathcal{R}}^0},\hfill \\ 1_1,1_2,\dots ,1_{{\mathcal{R}}^1},\hfill \\ ⋮\hfill \\ {\xi }_1,{\xi }_2,\dots ,{\xi }_{{\mathcal{R}}^{\xi }},\hfill \end{array}\right.\end{array}$$

(20)

such that TP can calculate the public function

$$f\left(0_1,0_2,\dots ,0_{{\mathcal{R}}^0};1_1,1_2,\dots ,1_{{\mathcal{R}}^1};\dots ;{\xi }_1,{\xi }_2,\dots ,{\xi }_{{\mathcal{R}}^{\xi }}\right).$$

(21)

2.2. Quantum Anonymous Computation for a Set of Data

Suppose that there are n mutually distrustful participants, $P_1,P_2,\dots ,P_n$, who have the secret data $x_1,x_2,\dots ,x_n$, respectively, where

$$\begin{array}{c}\hfill x_i\in \{0,1,2,\dots ,\xi \},\phantom{i}i=1,2,\dots ,n.\end{array}$$

(22)

TP helps the participants complete the calculation of the public function

$$f(x_1,x_2,\dots ,x_n).$$

(23)

The steps of the protocol are as follows:

Step 1. TP prepares a n-particle n-level singlet state, which is given by [9]

$$\begin{array}{c}\hfill \left|\Psi \right.〉={\displaystyle \frac{1}{\sqrt{n!}}}\sum _{l_1,l_2,\dots ,l_n\in \mathbb{S}}{(-1)}^{{\omega }_{\mathbb{S}}}\left|l_1{l}_2\dots l_n\right.〉,\end{array}$$

(24)

where $\mathbb{S}$ denotes the permutation of $\left\{0,1,\dots ,n-1\right\}$, and ${\omega }_{\mathbb{S}}$ the inverse number of $\mathbb{S}$ (i.e., the number of transpositions of pairs of elements in $\mathbb{S}$ which are used to place the elements in ascending order [9].) TP takes out the i-th particle in each state in turn, and then sands them to $P_i$. As before, in order to ensure the security of particle transmission, eavesdropping detection is required, which will not be repeated here.

Step 2. After confirming that there are no eavesdroppers in quantum channels, $P_i$ performs single-particle measurements on the singlet state $\left|\Psi \right.〉$, and marks measurement result by $m_i$.

Step 3. TP prepares n copies of d-level ($n+1$)-particle cat states, marked by

$$\begin{array}{c}\hfill {\left\{\left|\varphi \left(u_0^r,u_1^r,\dots ,u_n^r\right)\right.〉\right\}}_{r=1}^n,\end{array}$$

(25)

where

$$\begin{array}{c}\hfill d\in \left\{z|z\in {\mathbb{Z}}^{+},z>max\left\{x_1,x_2,\dots ,x_n\right\}\right\}.\end{array}$$

(26)

Then, TP takes the particles $u_1^r,u_2^r,\dots ,u_n^r$ out from $\left|\varphi \left(u_0^r,u_1^r,\dots ,u_n^r\right)\right.〉$ to constructs the particle groups

$$\begin{array}{c}\hfill \left(u_1^1,u_1^2,\dots ,u_1^n\right),\left(u_2^1,u_2^2,\dots ,u_2^n\right),\dots ,\left(u_n^1,u_n^2,\dots ,u_n^n\right),\end{array}$$

(27)

and marks them by ${\mathcal{S}}_1,{\mathcal{S}}_2,\dots ,{\mathcal{S}}_n$, respectively. Finally, TP sends ${\mathcal{S}}_i$ to $P_i$.

Step 4. Confirming that there are no eavesdroppers, $P_i$ prepares n copies of the d-level Bell state $\left|\varphi (0,0)\right.〉$, denoted as ${\left\{{\left|\varphi (0,0)\right.〉}_r\right\}}_{r=1}^n$. Then, $P_i$ chooses the $m_i$-th d-level Bell state, and establishes a variable $w_i^{m_i}$ and sets $w_i^{m_i}=x_i$. For the rest of d-level Bell states with the subscripts $h_i\in \left\{1,2,\dots ,n\right\}\setminus \left\{m_i\right\}$, $P_i$ sets $w_i^{h_i}=0$. Finally, $P_i$ uses the unitary operation shown in Eq. 3 to generate

$$\left|\varphi \left(v_i^r,w_i^r\right)\right.〉={\displaystyle \frac{1}{\sqrt{d}}}{\displaystyle \sum _{l=0}^{d-1}}{\zeta }^{l{v}_i^r}\left|l\right.〉\left|l\oplus w_i^r\right.〉.$$

(28)

Step 5. $P_i$ performs d-level Bell measurements on the particle with the marks $u_i^r$ in $\left|\varphi \left(u_0^r,u_1^r,\dots ,u_n^r\right)\right.〉$ and the particles with the marks $v_i^r$ in $\left|\varphi \left(v_i^r,w_i^r\right)\right.〉$. The measurement results are marked by

$${\left\{\left|\varphi ({\lambda }_i^r,{\gamma }_i^r)\right.〉\right\}}_{r=1}^n.$$

(29)

Then, $P_i$ sends the unmeasured particles in all d-level Bell states to TP.

Step 6. $P_1,P_2,\dots ,P_n$ cooperate together to calculate

$${\mathcal{A}}^r={\displaystyle \sum _{i=1}^n}{\gamma }_i^r,$$

(30)

and then announce ${\mathcal{A}}^r$ to TP.

Step 7. Similar to the sixth step in the previous protocol, TP performs measurements and calculations, and then marks the calculation results by ${\mathcal{R}}^r$. Finally, TP calculates the function

$$\begin{array}{c}\hfill f\left({\mathcal{R}}^1,{\mathcal{R}}^2,\dots ,{\mathcal{R}}^n\right),\end{array}$$

(31)

which is equal to $f(x_1,x_2,\dots ,x_n)$.

2.3. Discussion

The security analysis and the calculation of the success probability of the two protocols proposed above can refer to Refs. [7,10], which will not be elaborated here. The two protocols can be applied to all anonymous computation scenarios, such as quantum anonymous voting [11], quantum anonymous survey [12], and quantum anonymous ranking [13].

3. Conclusions

We have presented two quantum protocols for anonymous computation, both of which allow multiple mutually distrustful users to compute a public function for any non-negative integers. Since both of the protocols are implemented through entanglement swapping, they can adapt to quantum network environments and have significant advantages in communication distance. In our future work, we will consider a more detailed characterization of the success probability and security analysis for the two proposed protocols.