Concurrent multipath quantum entanglement routing based on segment routing in quantum hybrid networks

Abstract

Introduction

In the future, quantum networks will be an important development direction. Quantum networks should be studied considering quantum's unique physical properties. In quantum networks with limited resources, decoherence will lead to qubits loss, which restricts the time that data qubits can be stored in memory before use. To realize the network function, it is very important to transmit data qubits as quickly as possible. In order to ensure timeliness and make use of entanglement resources as much as possible, this paper proposes a concurrent multipath entanglement routing scheme based on segment routing for quantum hybrid networks.

Methods

In this scheme, centralized routing is used to calculate the “segments” of quantum relay path, and distributed routing is used to transmit control messages between the “segments”. The “concurrent” mechanism is fully adopted in classical messages passing, path nodes BSM (Bell State Measurement) execution and multipath shunt transmission.

Conclusion

This scheme has obvious advantages in multipath concurrent transmission, traffic diversion, reducing transmission time and providing redundancy.

Appendix A

1.1 Quantum entanglement multipath system

Figure 

Fig. 11

Schematic diagram of quantum entanglement multipath system. There are several quantum entanglement paths from the quantum source node A to the quantum destination node B. No matter how many quantum relay nodes are passed on the p-th path, the quantum path can always be changed into node A–X–B model at last due to entanglement swapping. Then, nodes A and X share a Bell pair of ${A}_{(p)}$,${A}_{(p)}^{\mathrm{^{\prime}}}$, nodes B and X share a Bell pair of ${B}_{(p)}$, ${B}_{(p)}^{\mathrm{^{\prime}}}$. By performing BSM on node X, a remote end-to-end quantum entanglement path can be established between node A and node B. The other n–1 paths and so on

11 shows a schematic diagram of a quantum entanglement multipath system. There are several quantum entanglement paths from the quantum source node A to the quantum destination node B. The path in the middle is the p-th path. No matter how many quantum relay nodes are passed on the p-th path, the quantum path can always be changed into node A–X–B model at last due to entanglement swapping. Suppose the number of quantum paths from the source to the destination is n. According to the principle of quantum teleportation [18], on the P-th (p = 1,2… n) path, the system composed of Bell pairs ${A}_{(p)}$,${A}_{(p)}^{^{\prime}}$ and ${B}_{(p)}$,${B}_{(p)}^{^{\prime}}$ is expressed by formula (5).

$$ \begin{aligned}& |\left. \varphi \right\rangle_{{A_{\left( p \right)} A_{\left( p \right)}^{^{\prime}} }} \otimes |\left. \varphi \right\rangle_{{B_{\left( p \right)} B_{\left( p \right)}^{^{\prime}} }} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}\left( {|\left. {01} \right\rangle_{{A_{\left( p \right)} A_{\left( p \right)}^{^{\prime}} }} + |\left. {10} \right\rangle_{{A_{\left( p \right)} A_{\left( p \right)}^{^{\prime}} }} } \right)\left( {|\left. {10} \right\rangle_{{B_{\left( p \right)} B_{\left( p \right)}^{^{\prime}} }} - |\left. {01} \right\rangle_{{B_{\left( p \right)} B_{\left( p \right)}^{^{\prime}} }} } \right) \\ & \quad = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}\left( {|\left. {0110} \right\rangle_{{A_{\left( p \right)} A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)} B_{\left( p \right)}^{^{\prime}} }} - |\left. {0101} \right\rangle_{{A_{\left( p \right)} A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)} B_{\left( p \right)}^{^{\prime}} }} + |\left. {1010} \right\rangle_{{A_{\left( p \right)} A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)} B_{\left( p \right)}^{^{\prime}} }} - |\left. {1001} \right\rangle_{{A_{\left( p \right)} A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)} B_{\left( p \right)}^{^{\prime}} }} } \right) \\ \end{aligned} $$

(5)

Quantum relay node X performs BSM on $A_{\left( p \right)}^{^{\prime}}$ and $B_{\left( p \right)}^{^{\prime}}$, and the expression of the Bell bases is (6).

$$ \begin{gathered} |\left. \varphi ^{ \pm }\right\rangle = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sqrt 2 }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sqrt 2 }$}}\left( {\left| {\left. {00} \right\rangle \pm } \right|\left. {11} \right\rangle } \right), \hfill \\ |\left. \psi ^{ \pm }\right\rangle = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sqrt 2 }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sqrt 2 }$}}\left( {\left| {\left. {01} \right\rangle \pm } \right|\left. {10} \right\rangle } \right). \hfill \\ \end{gathered} $$

(6)

Therefore, formula (5) can be rewritten as formula (7).

$$ \begin{aligned}& |\left. \varphi \right\rangle_{{A_{\left( p \right)} A_{\left( p \right)}^{^{\prime}} }} \otimes |\left. \varphi \right\rangle_{{B_{\left( p \right)} B_{\left( p \right)}^{^{\prime}} }} \quad \quad \\ &\quad = \frac{1}{2}\left[ {\left( {|\left. {01} \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }} - |\left. {10} \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }} } \right)\left( {|\left. {01} \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }} + |\left. {10} \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }} } \right)} \right. \\ &\qquad - \left( {|\left. {01} \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }} + |\left. {10} \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }} } \right)\left( {|\left. {01} \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }} - |\left. {10} \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }} } \right) \\ &\qquad + \left( {|\left. {00} \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }} + |\left. {11} \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }} } \right)\left( {|\left. {00} \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }} + |\left. {11} \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }} } \right) \\ &\qquad - \left. {\left( {|\left. {00} \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }} - |\left. {11} \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }} } \right)\left( {|\left. {00} \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }} + |\left. {11} \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }} } \right)} \right] \\ &\quad = \left[ {|\left. \psi \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }}^{ - } |\left. \psi \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }}^{ + } - |\left. \psi \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }}^{ + } |\left. \psi \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }}^{ - } } \right. \\ &\qquad \left. { + |\left. \varphi \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }}^{ + } |\left. \varphi \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }}^{ - } - |\left. \varphi \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }}^{ - } |\left. \varphi \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }}^{ + } } \right] \\ \end{aligned} $$

(7)

For example, if ${|\psi \rangle }_{{A}_{\left(p\right)}^{^{\prime}}{B}_{\left(p\right)}^{^{\prime}}}^{-}$ is chosen, the state of particles ${A}_{(p)}$ and ${B}_{(p)}$ is ${|\psi \rangle }_{{A}_{\left(p\right)}{B}_{\left(p\right)}}^{+}$ after BSM, and entanglement is realized. An entangled quantum channel is successfully established between nodes A and B. Formula (5)–(7) are the derivation of the p-th path assuming that the quantum link bandwidth is 1 (only 1 data qubit can be transmitted). If each path quantum link bandwidth is k (k = 1,2… m), then the system composed of n paths is expressed by formula (8).

$$\begin{aligned}& \mathop \sum \limits_{p = 1}^{n} \mathop \sum \limits_{k = 1}^{m} |\left. \varphi \right\rangle_{{A_{\left( p \right)} A_{\left( p \right)}^{^{\prime}} }} \otimes |\left. \varphi \right\rangle_{{B_{\left( p \right)} B_{\left( p \right)}^{^{\prime}} }}\\&\quad = \mathop \sum \limits_{p = 1}^{n} \mathop \sum \limits_{k = 1}^{m} \left[ {|\left. \psi \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }}^{ - } |\left. \psi \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }}^{ + } - |\left. \psi \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }}^{ + } |\left. \psi \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }}^{ - } + |\left. \varphi \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }}^{ + } |\left. \varphi \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }}^{ - } - |\left. \varphi \right\rangle_{{A_{\left( p \right)} B_{\left( p \right)} }}^{ - } |\left. \varphi \right\rangle_{{A_{\left( p \right)}^{^{\prime}} B_{\left( p \right)}^{^{\prime}} }}^{ + } } \right]\end{aligned}$$

(8)