A Scheme for Controlled Cyclic Asymmetric Remote State Preparation in Noisy Environment

During the past few years, quantum communication theory and technology have been further extended. Quantum entanglement is a crucial resource in quantum information processing tasks, and has been widely applied in various fields such as quantum teleportation (QT), remote state preparation (RSP), quantum key distribution (QKD) and quantum secret sharing (QSS). In 1982, Aspect successfully confirmed the phenomenon of quantum entanglement through experiments [1]. In 1993, Bennett proposed the concept of QT [2], which has great potential for protecting sensitive information and accelerating classical computation. Then, Lo [3] first introduced remote state preparation (RSP) at the beginning of the 21st century. In RSP [4,5,6], Alice can remotely prepare an arbitrary single-qubit quantum state for Bob by using a classical bit communication and a shared entangled state. Besides, Alice knows all the information of the single-qubit quantum state to be prepared by Bob, and Alice does not have to hold the single-qubit quantum state. Since then, the novel type of RSP has drawn considerable attention, and various related schemes [7,8,9,10,11], like CRSP (controlled RSP) [12,13], CJRSP (controlled joint) [14] and CBRSP (controlled bidirectional RSP), have been presented. In addition, with the development of entanglement source preparation, the scheme of RSP has been expanded from single qubit to multiple qubits [15]. In 2016, Chen et al. pointed out an efficient scheme for the RSP of an arbitrary five-qubit Brown-type state [16]. Binayak and Ding introduced the joint remote preparation of an arbitrary six-qubit cluster-type state [17,18]. In 2018, Wu and Fang discussed the bidirectional and hybrid quantum information transmission schemes [19,20]. A JRSP scheme of the arbitrary eight-qubit cluster-type state was put forward in 2020 [21]. However, we note that almost all the protocols mentioned above are only considered in one-way directional or bidirectional preparations. In fact, the information network is made up of two or more nodes, that is, a large information system usually is composed by some subsystems. Naturally, some cyclic RSP schemes [22,23] are investigated, information switching is not located in two-party system. In 2019, Li [24] et al. discussed the scheme of the controlled cyclic quantum teleportation of an arbitrary two-qubit entangled state by employing a ten-qubit entangled state as a quantum channel. Sang [25] et al. explored a scheme of four-party cyclic RSP of an arbitrary single-qubit state by employing a ten-qubit maximally entangled state as the quantum channel. In quantum communication, the remote preparation of multi-particle quantum states between multi-party is inevitable. So we put forward the cyclic asymmetric remote preparation scheme with qubits sequentially increasing, which is a novel transmission mode of cyclic RSP. Furthermore, we generalize the scheme to send multiple qubits quantum information among multiple players in an asymmetric manner.

The influence of decoherence caused by the actual environment has always been the major factor affecting the effect of quantum state transmission; it is meaningful to investigate the cyclic controlled RSP in a noisy environment. In recent years, some RSP schemes that operated in noisy environments have been proposed [26,27,28,29]. In 2018, Sun [30] put forward an asymmetrically controlled two-way joint RSP in a noisy environment for the preparation of single and three equatorial states. The deterministic hierarchical remote state preparation of a two-qubit entangled state employing a Brown state in a noisy environment was proposed in 2020 [31].

Since 2016, IBM Quantum Experience has allowed players to design quantum circuits employing an interactive graphical user interface, test and simulate these circuits on classic computers and actual quantum processors. The actual relevance of an optimized scheme lies in the experimental realization of the scheme. The different kinds of theoretical protocols for quantum communication and computation [32,33,34] have already been verified in the IBM quantum experience. IBM Quantum Experience has been utilized to perform many practical experiments on quantum chips, including quantum simulation, the development of quantum algorithms [35], testing quantum information theory tasks, quantum cryptography, quantum error correction [36], quantum applications [37], and so forth.

In this paper, we put forward a novel (N + 1)-party cyclic RSP protocol for preparing sequentially increasing qubits among arbitrary number of players with a controller. To be specific, the i-th party prepare an i-qubit state for the (i + 1)-th party in a loop of N(1, 2, … i, i + 1, … n)-party, and the n-th party prepares the n-qubit state for the first party. Moreover, we demonstrate the special case of our scheme on the IBM quantum computer by designing appropriate quantum circuits using single-qubit and two-qubit quantum gates. For verifying the practicality of our scheme, we also discuss the variation of the fidelity of the scheme affected by different noises.

This paper is structured as follows—in Section 2, we specifically introduce a controlled cyclic asymmetric RSP scheme in which Alice remotely prepares an arbitrary single-qubit state for Bob, Bob prepares an arbitrary two-qubit state to Charlie, and Charlie prepares an arbitrary three-qubit state for Alice. Then, we construct the quantum circuit simulation of this scheme in IBM’s quantum simulator and analyse the mean probability of the output results. Furthermore, we generalize the scheme to (N + 1)-party controlled cyclic RSP protocol for sequentially increasing qubits in Section 3. In Section 4, we analyze and compare the fidelity of output state in four noisy environments (depolarization noise, amplitude damping noise, phase damping noise, and bit-phase flip noise). Finally, we give a brief comparison and conclusion of the scheme in Section 5.

In this section, we aim at a situation where the amount of information transmitted between multiple participants is sequentially increasing. Suppose that there are four participants, employing ten quantum entangled states as quantum channels. Alice prepares any single-qubit state $\left|{\xi }_A\right.〉=a_0{e}^{i{\theta }_0}\left|0\right.〉+a_1{e}^{i{\theta }_1}\left|1\right.〉$ for Bob, Bob prepares any two-qubit state $\left|{\xi }_B\right.〉=b_0{e}^{i{\varphi }_0}\left|00\right.〉+b_1{e}^{i{\varphi }_1}\left|11\right.〉$ for Charlie, Charlie prepares any three-qubit state $\left|{\xi }_C\right.〉=c_0{e}^{i{\gamma }_0}\left|000\right.〉+c_1{e}^{i{\gamma }_1}\left|111\right.〉$ for Alice, and David is the controller, where $a_0,a_1,b_0,b_1,c_0,c_1\in R$, $a_0^2+a_1^2=1$, $b_0^2+b_1^2=1$, $c_0^2+c_1^2=1$, ${\theta }_0,{\theta }_1,{\varphi }_0,{\varphi }_1,{\gamma }_0,{\gamma }_1\in \left[0,2\pi \right]$. The scenario of the scheme is shown in Figure 1.

The quantum channel linking the Alice, Bob, Charlie and David has the form where Alice holds qubits (2,3,4,9), Bob holds qubits (5,8), Charlie holds qubits (1,6,7), and David holds qubit 10.

The controlled cyclic asymmetrical RSP protocol of sequentially increasing qubits states can be extended to multiparty scenario easily. Suppose there are n players $Alic{e}_1$, $Alic{e}_2$, … and $Alic{e}_n$, and controller Charlie. $Alic{e}_i$ can prepare an i-qubit state to his neighbor $Alic{e}_{i+1}$ under the control of Charlie. The entangled channel share among participants is $\left|\phi ({\displaystyle \sum _{i=1}^n}(i+1)+1)\right.〉$. where $A_1^n$ represents the nth qubit owned by $Alic{e}_1$, $A_n^n$ represents the nth qubit owned by $Alic{e}_n$, and $A_{i+1}^i$ represents the i-th qubit owned by $Alic{e}_{i+1}$, and $i\in \{1,2,\cdots ,n\}$. Charlie holds control qubit labeled by ${\displaystyle \sum _{i=1}^n}(i+1)+1$. For example, if n = 3, the entangled channel shared should have the form as Equation (1).

If we want to change the direction of the controlled cyclic asymmetrical RSP protocol, we need to change the distribution of qubits, and the quantum channel will be changed to the following expression. where $A_2^1$ represents the 1th qubit owned by $Alic{e}_2$, $A_i^{i+1}$ represents the (i + 1)th qubit owned by $Alic{e}_i$, Charlie holds control qubit labeled by ${\displaystyle \sum _{i=1}^n}(i+1)+1$. Specifically, there is ${(\left|\underset{n+1}{\underbrace{0\cdots 0}}\right.〉+\left|\underset{n+1}{\underbrace{1\cdots 1}}\right.〉)}_{A_2^1,A_1^1,\cdots ,A_1^n}$. When n is even, there is ${(\left|\underset{\left[\frac{n+1}{2}\right]}{\underbrace{0\cdots }}\underset{\left[\frac{n+1}{2}\right]+1}{\underbrace{\cdots 1}}\right.〉-\left|\underset{\left[\frac{n+1}{2}\right]}{\underbrace{1\cdots }}\underset{\left[\frac{n+1}{2}\right]+1}{\underbrace{\cdots 0}}\right.〉)}_{A_2^1,A_1^1,\cdots ,A_1^n}$. When n is odd, there is ${(\left|\underset{\frac{n+1}{2}}{\underbrace{0\cdots }}\underset{\frac{n+1}{2}}{\underbrace{\cdots 1}}\right.〉-\left|\underset{\frac{n+1}{2}}{\underbrace{1\cdots }}\underset{\frac{n+1}{2}}{\underbrace{\cdots 0}}\right.〉)}_{A_2^1,A_1^1,\cdots ,A_1^n}$.

In this section, we discuss our scheme in four noisy environments, including amplitude-damping, phase-damping noise, bit-flip noise, and phase-flip noise.

The content shared by the participants in advance is constructed by an organization called the QDC (Qubits Distribution Center). After preparing the quantum channel, QDC allocates the qubits $A_1,A_2,A_3,A_4$ to Alice, the qubits $B_1,B_2$ to Bob, and sends the qubits $C_1,C_2,C_3$ to Charlie, sends the qubits $D_1$ to David. If the quantum channel is in an ideal environment without noise influence, it is a pure state, and its density matrix can be expressed as $\rho =\left|\zeta \right.〉\left.〈\zeta \right|$. In practical applications, when affected by a noisy environment [38], the pure state may be converted to a mixed state. The corresponding density matrix can be written as follows: where j is a label, and different noise channels have different values; the superscript indicates that the operator E acts on different noise environment and $q\in \left\{D,A,P,W,B,S\right\}$. If $q=D$, for depolarized noise, then $j=0,1,2,3$; if $q=A$, for amplitude damping noise, then $j=0,1$; if $q=P$, for phase damping noise, then $j=0,1$; if $q=S$, for bit-phase flip noise, then $j=0,1$.