## 2 Information Reconciliation in QKD Post-Processing
### Rate-Adaptive LDPC Codes and Efficiency Metric of Information Reconciliation
Therefore, the efficiency of rate-adaptive reconciliation can be determined as a function of the a priori QBER estimation \(q_{est}\), as follows:  
\[\eta_{IR}=\frac{1-R_{C}^{(opt)}}{H(q_{est})}=\frac{M-n_{p}}{(N-n_{p}-n_{S}) \cdot H(q_{est})}. \tag{6}\]Furthermore, the secret key throughput (\(\tau_{\text{SK}}\)) can be used to evaluate the performance of QKD post-processing by considering the inherent parameters of a BB84 QKD system [33; 34; 35]. It is defined as follows:  
\[\tau_{\text{SK}}=p_{exp}\cdot\epsilon_{\text{BB84}}\cdot f_{clk}\cdot(1-\text{ FER})\cdot r_{real}, \tag{7}\]  
where:  
* \(p_{exp}\) is the total detection rate for events where photons are transmitted from _[PERSON]_ to _[PERSON]_. * \(\epsilon_{\text{BB84}}\) is the theoretical efficiency of the BB84 protocol. * \(f_{clk}\) is the operational clock rate of QKD devices. * FER is the frame error rate, indicating the failure probability of decoding, which affects the likelihood of non-identical reconciled keys for _[PERSON]_ and _[PERSON]_. * \(r_{real}\) is the actual secret key rate, depending on \(\eta_{IR}\) from Equation (6), which can be defined as \(r_{real}=H(X|Z)\)\(\eta_{IR}\cdot H(X|Y)\). Rate-Adaptive LDPC Codes for Information Reconciliation: Integrating Syndrome-Based Error Estimation and Subblock Confirmation  
In this section, the proposed rate-adaptive LDPC codes are introduced for an information reconciliation protocol and its integration with effective algorithms in channel error estimation and confirmation steps. In the context of QKD post-processing, the proposed scheme commences after the quantum information distribution and sifting steps of the discrete-variable quantum key distribution (DV-QKD) protocol. At this stage, _Alice_ and _Bob_ obtain correlated sifted keys of the same length, which then proceed to the subsequent steps of the proposed syndrome-based error estimation, rate-adaptive reconciliation, and polynomial-based hash subblock confirmation. Firstly, the syndrome-based error estimation provides the value of the estimated QBER by utilizing the maximum likelihood estimator, which is based on the syndrome encoding of sifted keys between _Alice_ and _Bob_ using the maximum code rate \(R_{\text{C}}^{\text{ (}max\text{)}}\) of irregular LDPC codes. The estimated QBER is subsequently used to adjust the optimal channel coding rate \(R_{\text{C}}^{\text{ (}opt\text{)}}\) of irregular LDPC codes. The aim of this adjustment is to optimize \(R_{\text{C}}^{\text{ (}opt\text{)}}\) by determining the number of puncturing bits \(n_{p}\), and the number of shortening bits \(n_{s}\) for the rate-adaptive reconciliation scheme. After completing the information reconciliation steps, the reconciled keys are segmented into subblocks. A specific polynomial-based hash function is then randomly generated and subsequently employed for key identity verification within each subblock. If certain subblocks fail verification, the keys from these unverified subblocks are subjected to additional rounds of syndrome-based error estimation and rate-adaptive reconciliation by using irregular LDPC codes with block lengths that match the sizes of the subblocks. This process is repeated until all subblocks achieve successful verification during the confirmation step, which effectively prevents key discarding in QKD post-processing. Figure 3 illustrates the flowchart of the proposed scheme, which includes syndrome-based error estimation, rate-adaptive reconciliation, and polynomial-based hash subblock confirmation. This flowchart is described by the following four main steps:Figure 3: Flowchart of the unified procedure for syndrome-based error estimation, rate-adaptive information reconciliation, and subblock confirmation using polynomial hashing. **(1) Initialization of LDPC code parameters:**_[PERSON]_ and _[PERSON]_ mutually agree on two sets of irregular LDPC codes with block lengths \(N\) and \(N_{sb}\).