# Quantum Coding via Quasi-Cyclic Block Matrix
## 1 Introduction  
Quantum communication requires that environmental effects and decoherence should be reduced with reliable quantum information processing. In practice, information reconciliation in quantum key distribution (QKD) is important for the secret key rate and also affects the maximum transmission distance [1; 2; 3; 4]. Error-correcting code generally may be applied in a quantum channel to correct errors caused by channel noise and possible interventions from eavesdroppers. To obtain an acceptable level, QECC is an essential method because of its robustness and efficiency in quantum computation [5; 6; 7; 8; 9; 10]. One of the advantages of quantum computation is that its high efficiency compares favorably to classical computation; it is able to handle large-scale data that classical computation cannot. Previously, most of the construction methods for QECC focused on the generation of stabilizers, and there was little research on the long code type. Furthermore, due to some fields in quantum information becoming gradually more practical, this has prompted researchers to identify a good coding method for quantum error-correcting codes of long length. For example, encoding large-scale data has potential applications in the field of machine learning (ML) with respect to big data [11; 12; 13; 14]. Therefore, how to efficiently express classical massive data with physics-based codes is also an important research field. To obtain a general quantum code, the question is usually converted to a problem of stabilizer generation. One typical method is to obtain the generation matrix of a quantum code based on two classical codes, which are called Calderbank-Shor-Steane (CSS) codes [6; 7]. By resorting to the generalization of cyclic codes, a class of classical codes called quasi-cyclic (QC) codes can build linear codes based on algebraic structure, which has improved some fundamental minimum distances [15; 16; 17; 18]. For instance, it can satisfy the modified version of the Gilbert-Vashamov (GV) bound [19; 20; 21]. It has shown good performance when applied to form quantum codes. In the construction process for long-sequence QECCs, how to design the generators of stabilizers based on the block matrix is key. Methods from QC low density parity check (LDPC) codes have been studied by [PERSON] et al. in terms of a probabilistic method [22; 23]. In 2018, [PERSON] et al. applied two generators to construct a quantumversion with dual-containing QC [24]. Some QC short-length codes were obtained with good parameters. Not long after that, [PERSON] et al., in 2019, used QC codes with large Hermitian hulls to form QECCs over fields \(F_{4}\) and \(F_{9}\)[25], so that a record-breaking binary QECC was obtained. Furthermore, [PERSON] et al. proposed some new binary quantum codes derived from one generator quasi-cyclic codes with a stabilizer [26; 27]. Recently, some researchers have considered the application of this kind of quantum code in quantum key distribution [4]. However, few researchers have paid attention to the construction of long-length quantum QC code involved with massive data. Inspired by the previous work, we presented QC code constructions of long length to generate QECCs with a family of orthogonal jacket matrices, the main property of which are that the inverse matrix can be obtained by its element-wise inverse or block-wise inverse [28; 29]. Therefore, it can be realized relatively easily with a physical circuit. Furthermore, since [PERSON] first proposed LDPC codes in the 1960s [30], this class of classical code has shown good performance approaching the channel capacity [31; 32; 33; 34; 35]. Subsequently, its quantum versions has been investigated [22; 23]. However, the achievements in this field have been explored far less than their classical counterparts. The constructed quantum codes have shown good performance for low density codes based on iterative coding in our proposed construction method.