# Surface-Codes-Based Quantum Communication Networks
## 1 Introduction  
Quantum information processing (QIP) opens up new avenues for reliable communications, high-precision sensing, and high-performance computing [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. Entanglement represents a unique resource for QIP, which allows quantum computers to solve classically intractable problems [7], provides certifiable security [2] for data transmissions, and enables sensors to achieve measurement sensitivities beyond the classical limit [8]. The quantum communication is the key cornerstone to fully exploit the properties of entanglement. The modern classical communications tend to use heterogeneous networks capable of simultaneous data transmission between nodes connected via different types of channels, such as free-space optical (FSO) and fiber-optics links. Nodes in existing quantum communication networks (QCNs), however, have been limited to a single optical medium. Moreover, trusted node assumption [4] is required to operate the current QCNs. As a result, one compromised node in a QCN can undermine the security of the entire QCN. Several quantum key distribution (QKD) testbeds have been reported so far, such as the DARPA QKD network [5], Tokyo QKD network [6], and the secure communication based on quantum cryptography (SECOQC) network [7]. Unfortunately, these different QKD networks employ the dark fiber infrastructure. In this paper, we propose the multipartite heterogenous QCN employing the surface codes, which does not require the trusted node assumption. The research on multipartite entanglement is getting momentum with numerous experimental demonstrations, such as [8]. The surface codes, typically defined on a 2-D lattice, are closely related to the quantum topological codes on the boundary [1], introduced by [PERSON] and [PERSON] [11; 12]. This class of codes is highly popular in quantum computing [13; 14; 15] because only local qubits are involved in stabilizers. In _[PERSON]'s framework_[14], the surface code for quantum computing is represented as a game, played on a board partitioned in a certain number of tiles. On each tile we can place a logical qubit, represented as a _patch_. The edges of qubits represent the logical Pauli operators [1]. The logical qubits correspond to the surface code (SC) patches. By placing the SC patches in nodes of a communication network, and connecting the neighbouring patches by \(d\) wavelength channels, corresponding to the distance of the underlying surface code, wecan create the quantum communication network. The SC patches placed in intermediate nodes can be operated as the SC-based quantum repeaters, thus extending significantly the transmission distance. When the patch edges in the tiles of neighbouring network nodes are different, we can perform the product measurements to entangle them. For instance, the product \(Z\otimes Z\) between adjacent nodes' patches can be simultaneously measured to introduce the entanglement between two adjacent quantum nodes. Namely, we start with the state \(|++\rangle=0.5(|00\rangle+|11\rangle+|01\rangle+|10\rangle)\) and perform the measurement on \(Z\otimes Z\) operator. If the result of the measurement is \(+1\), the qubits end up in state \(2^{-1/2}(|00\rangle+|11\rangle)\); otherwise, they end up in state \(2^{-1/2}\left(|01\rangle+|10\rangle\right)\). In either case, the qubits are maximally entangled. This indicates that the proposed SC-based QCN is highly flexible and have numerous applications, including: (i) to teleport quantum states between any two nodes in the network, (ii) to develop the information infrastructure with unprecedented security level, (iii) to enable distributed quantum computing, and (iv) to enable ultra-high precision for quantum sensing applications. To operate such a quantum network, we propose to employ the software-defined networking (SDN) concepts. The paper is organized as follows. In Section 2, we introduce the surface codes and describe briefly the _Litinski_'s formalism needed in incoming sections.