Polar-code-based information reconciliation scheme with the frozen-bit erasure strategy for quantum key distribution

Information reconciliation (IR) ensures the correctness of quantum key distribution systems by correcting the error bits that existed in the sifted keys. In this article, we propose a polar-code-based IR scheme with the frozen-bit erasure strategy, where an equivalent transmission of sifted keys is conducted so that each frozen bit in the decoding procedure is erased to zero. Thus, our IR scheme is no longer limited by the assumption of true random numbers and can be implemented simply and efficiently. Furthermore, we implement the proposed IR scheme with the fast simplified successive-cancellation list decoder and its throughput reaches $0.68\mathrm{Mbps}$ with the yield of 0.8333; when the decoder list size is 16, the block size is $1\mathrm{Mb}$ and the quantum bit error rate is 0.02.

Figure 1

The diagram of polar-code-based IR schemes with (a) DD strategy and (b) BFD strategy. The block size is $n$ and the length of information bits is $k$. $V$ is the frozen vector precalculated according to QBER $E$. TRN means true random number. $\mathrm{Encode}\left(U\right)$ represents encoding $U$ to $U{G}_n$. $U=\mathrm{Decode}(X,V,Y)$ represents decoding $X$ to $U$ with the frozen vector of $V$ and the frozen bits value of $Y$. In (a), $W=\mathrm{Ext}(U,V)$ represents extracting a vector $W$ from $U, W$ is composed of the elements $u_i$ when $v_i=1, U=[u_0,u_1,\cdots ,u_{n-1}]$, and $V=[v_0,v_1,\cdots ,v_{n-1}]$. In (b), $W$ is composed of the $k$ random bits. $\mathrm{Fill}(U,V,W)$ means filling $W$ into the information bits of $U$ when $V$ is the frozen vector.

Figure 2

The diagram of the polar-code-based IR scheme with FBE strategy. $Y=\mathrm{Ext}(U,V)$ represents extracting the vector $Y$, which is composed of the elements $u_i$ when $v_i=1, U=[u_0,u_1,\cdots ,u_{n-1}]$, and $V=[v_0,v_1,\cdots ,v_{n-1}]$. $T=\mathrm{CRC}\left(X\right)$ represents calculating the CRC value $T$ of vector $X$. $U,\sigma ={\mathrm{Decode}}^{\mathrm{CA}}(X,V,0,T)$ represents conducting the CRC-aided decoding procedure on the codeword $X$ with the CRC value of $T$, the frozen vector of $V$, and the frozen bits of zero, and output the decoded vector $U$ and the decoding flag $\sigma$, which represents whether $X$ is decoded successfully.

Figure 3

The yield of the polar-code-based IR schemes with FBE strategy against the failure probability, when the decoder is the FSSCL decoder, QBER equals 0.02, the list size is in $\{1,2,8,16,32,64\}$, and the block size is $1\mathrm{Mb}$. The data of Tang's scheme are from the forward reconciliation phase [20].

Figure 4

The yield of the polar-code-based IR schemes with FBE strategy against the corresponding efficiency. The data of Tang's scheme are from the forward reconciliation [20].

Figure 5

Throughput comparison of the polar-code-based IR schemes with FSSCL and SCL decoders. The block size is $1\mathrm{Mb}$ and the QBER is 0.02.