Clock offset recovery with sublinear complexity enables synchronization on low-level hardware for quantum key distribution

TL;DR

iQSync, a clock-offset recovery method designed for implementation on low-level hardware, such as field-programmable gate arrays or microcontrollers, for quantum key distribution (QKD), is introduced, demonstrating its performance and conformity with analytically derived success probabilities for channel attenuations exceeding 70dB.

Abstract

We introduce iQSync, a clock offset recovery method designed for implementation on low-level hardware, such as FPGAs or microcontrollers, for quantum key distribution (QKD). iQSync requires minimal memory, only a simple instruction set (e.g. no floating-point operations), and can be evaluated with sublinear time complexity, typically involving no more than a few thousand iterations of a simple loop. Furthermore, iQSync allows for a precise clock offset recovery within few seconds, even for large offsets, and is well suited for scenarios with high channel loss and low signal-to-noise ratio, irrespective of the prepare-and-measure QKD protocol used. We implemented the method on our QKD platform, demonstrating its performance and conformity with analytically derived success probabilities for channel attenuations exceeding 70 dB.

Figures (9)

• Figure 1: A typical QKD setup. Qubits, e.g. encoded in coherent quantum states with low mean photon number $\mu \lesssim 1$, are transmitted from Alice to Bob using a quantum channel. Due to channel losses, Bob usually receives states with $\mu \ll 1$. A bi-directional data channel with high (e.g. a few hundred ms) unknown and undeterministic (but bounded) latencies is used for classical communication between Alice and Bob, and for QKD post-processing. An optional clock channel is often used to establish a common clock phase.
• Figure 2: Relation between the total synchronization pattern duration $N_\mathrm{s}$ and the max. recoverable offset $\Delta_\mathrm{max}$ for different degrees of interleaving $d_\mathrm{i}$. Cross-correlation refers to approaches utilizing a single FFT, or the dual-step FFT approach as presented in calderaroFastSimpleQubitbased2020.
• Figure 3: Calculated synchronization failure probability $P_\mathrm{fail}$ for two experimental conditions, depending on the total pattern duration $N_\mathrm{s}$. The thick black lines depict a scenario with high signal and low noise detection probability ($p_\mathrm{sig} = 10^{-3}$, $p_\mathrm{noise} = 10^{-7}$). The thin blue lines represent low signal and high noise detection probability ($p_\mathrm{sig} = p_\mathrm{noise} = 10^{-6}$). Cross-correlation refers to approaches utilizing a single FFT and the dual-step FFT approach as presented in calderaroFastSimpleQubitbased2020.
• Figure 4: Tolerable channel attenuation $\eta$ for iQSync, depending on the maximum recoverable offset $\Delta_\mathrm{max}$, the degree of interleaving $d_\mathrm{i}$, and the probability for a noise detection per symbol $p_\mathrm{noise}$. The solid (dashed) lines represent the settings with 50% success probability for no (maximal) interleaving, with the lower (upper) edge of the shaded areas marking the 90'th (10'th) success percentile. The case of no noise is depicted in (a). The most pessimistic scenario for QKD is shown in (b), where for each configuration a QBER of 11% is assumed. The case of $p_\mathrm{noise}=10^{-7}$ is shown in (c), representative for the detectors used during our experimental validation, cf. Section \ref{['sec:experiment']}. The position of the kink depends on $p_\mathrm{noise}$ and always sits at the point, where the channel transmission equals $p_\mathrm{noise}$. The area with a $\mathrm{QBER} > 11\%$ is shaded red.
• Figure 5: TC of iQSync in terms of the number of iterations $\bar{N}_\mathrm{loop}$ over the inner loop of Algorithm \ref{['alg:pattern-analysis-iqsync']} for different levels of noise, corresponding to the lines in Fig. \ref{['fig:att_over_max_offset_3']}. The solid (dashed) lines represent the settings with 50% success probability for no (maximal) interleaving. The three gray lines depict the TC of the FFT, the dual-step approach by Calderaro et al. calderaroFastSimpleQubitbased2020, and the theoretical limit $\log_2 n$, given by the number of bits of the binary representation of the offset. The light gray lines depict linear TC as a reference.