Simple Efficient Decoders for Quantum Key Distribution Over Quantum Repeaters with Encoding

We study the implementation of quantum-key-distribution (QKD) systems over quantum-repeater infrastructures. We particularly consider quantum repeaters with encoding and compare them with probabilistic quantum repeaters. To that end, we propose two decoder structures for encoded repeaters that not only aid system performance but also make the implementation aspects easier by removing two-qubit gates from the QKD decoder. By developing several scalable numerical and analytical techniques, we then identify the resilience of the setup to various sources of error in gates, measurement modules, and initialization of the setup. We apply our techniques to three- and five-qubit repetition codes and obtain the normalized secret key generation rate per memory per second for encoded and probabilistic quantum repeaters. We quantify the regimes of operation, where one class of repeater outperforms the other, and find that there are feasible regimes of operation where encoded repeaters—based on simple three-qubit repetition codes—could offer practical advantages.

Figure 1

Schematic of the QKD setup on a repeater chain based on the three-qubit repetition code. The small oval pairs represent bipartite entangled states prepared in advance. Using remote cnot gates, an encoded entangled state is generated across elementary links, and stored in memories represented by large ovals. The encoded entanglement is then extended across the entire link by performing entanglement-swapping (ES) operations on the middle nodes. The two users will then apply decoding operation on this state to generate their raw key.

Figure 2

Schematic of different decoder structures considered in this paper: (a) the original decoder proposed in Ref. [2], where a decoding circuit is used to generate a qubit, on which a QKD measurement, either in $Z$ or $X$ basis, is performed. The decoding circuit would generate syndrome data $d$, which is used for state classification. (b) A modified version of the decoder proposed in Ref. [9], which is very similar to (a) except that the decoder measurement outcome $d$ is not used for classification. They still use the information in $m$ in their key-rate extraction. Note that, in both (a),(b), Eve can control the decoder module, but has to pass some measurement data to users. (c) First alternative decoder, proposed in this work, where users directly measure the three qubits either all in $Z$ basis or in $X$ basis. They use majority (parity) rules, in $Z$ ($X$) basis, to decode the key bit. (d) Our second alternative decoder, which is very similar to (c), except that in the $Z$ basis a perfect match 111 (000) is mapped to bit 1 (0). In (c),(d), we assume Alice and Bob have control over the final set of memories in their secure box.

Figure 3

Secret fraction for different decoder (Dec) structures versus (a) gate error probability $\beta$ at $F_0=1$ and $\delta =0$, and (b) measurement error probability $\delta$ at $F_0=1$ and $\beta =0$. In the curves corresponding to perfect decoders, all error parameters assume their ideal values just in the decoder module; the corresponding value in the rest of the system is as the graph shows.

Figure 4

Secret fraction $r_{\mathrm{\infty }}^{\mathrm{opt}}$ versus (a) $\beta$, at $\delta =1-F_0=0$; (b) $1-F_0$, at $\delta =\beta =0$; (c) $\delta$, at $\beta =1-F_0=0$; and (d) $\beta$, at $\delta =0.01$ and $1-F_0=0$. The curves labeled by good correspond to the output states where no error is detected at the ES stage, whereas the bad curves are for the output states where some errors are detected at the ES stage. The curves labeled by total are the weighted sum of good and bad terms as given by Eqs. (12) and (15).

Figure 5

Secret fraction $r_{\mathrm{\infty }}^{\mathrm{opt}}$ versus gate error probability $\beta$ for the first three nesting levels under three different approximation methods, with initial fidelity $F_0=1$ and measurement error probability $\delta =0$.

Figure 6

Secret fraction $r_{\mathrm{\infty }}^{\mathrm{opt}}$ for three-qubit repetition code, as a function of gate error probability $\beta$ for different nesting levels, using our numerical approximation technique at $N_{\mathrm{top}}=20$, with initial fidelity $F_0=1$ and measurement error probability $\delta =0$. Here, the errors in the encoding and decoding circuits are included. The exact simulation results for the first three nesting levels are shown (solid yellow lines) as comparison.

Figure 7

Secret fraction $r_{\mathrm{\infty }}^{\mathrm{opt}}$ of QRs encoded with three-qubit repetition code (solid lines) and five-qubit repetition code (dashed lines) for the first three nesting levels as a function of gate error probability $\beta$, with initial fidelity $F_0=1$ and measurement error probability $\delta =0$. We use our numerical approximation method at $N_{\mathrm{top}}=20$.

Figure 8

Normalized secret key rates for the encoded QRs with and without multiplexing for the first three nesting levels as a function of the total distance, with initial fidelity $F_0=0.99$, gate error probability $\beta =0.01$, and measurement error probability $\delta =0.005$. The secret key rate is calculated for the better of decoders 3 and 4 at $p=0.5$, ${\eta }_d=0.9$, and $L_{\mathrm{att}}=22$ km using our numerical approximation technique at $N_{\mathrm{top}}=20$.

Figure 9

Normalized secret key rates for QRs with encoding (solid, three-qubit code) and probabilistic QRs (dashed) in the absence of multiplexing for up to six nesting levels as a function of the total distance, with different error parameters: (a) $F_0=0.999$, $\beta =0.0005$, and $\delta =0.0001$; (b) $F_0=0.99$, $\beta =0.005$, and $\delta =0.001$; (c) $F_0=0.98$, $\beta =0.02$, and $\delta =0.01$. Other parameters are as in Fig. 8. In the encoded repeater case, the secret key rate is calculated for the better of decoders 3 and 4 using our numerical approximation method at $N_{\mathrm{top}}=20$.

Figure 10

Region plots showing the distribution of the optimal QR protocol in a three-dimensional parameter space at $L_{\mathrm{tot}}=1000\mathrm{km}$ for (a) three-qubit repetition code and (b) five-qubit repetition code. Other parameters are as in Fig. 8. In the encoded repeater case, the secret key rate is calculated for the better of decoders 3 and 4 using our numerical approximation method at $N_{\mathrm{top}}=20$.