High key rate continuous-variable quantum key distribution using telecom optical components

Quantum key distribution (QKD) is one quantum technology that can provide secure encryption keys for data transmission. The secret key rate (SKR) is a core performance indicator in QKD, which directly determines the transmission rate of enciphered data. Here, for the first time, we demonstrate a high-key-rate Gaussian-modulated continuous-variable QKD (CV-QKD) using telecom optical components. The framework of CV-QKD over these components is constructed. Specifically, the high-rate low-noise Gaussian modulation of coherent states is realized by a classical optical IQ modulator. High-baud low-intensity quantum signals are received by an integrated coherent receiver under the shot-noise limit. A series of digital signal processing algorithms are proposed to achieve accurate signal recovery and key distillation. The system can yield a high asymptotic SKR of 10.37 Mbps within 20 km standard telecom fiber, and the secure distance can exceed 100 km. This result confirms the feasibility of CV-QKD with state-of-the-art performance using telecom optical components. Besides, due to the ease of integrating these discrete components, it provides a high-performance and miniaturized QKD solution for the metropolitan quantum network.


Introduction
Quantum key distribution (QKD), one quantum information technology, can provide secure keys for legal parties through an untrusted quantum channel, whose theoretical security is guaranteed by quantum mechanics' basic principles [1].Due to its provable security, the QKD protocol can serve applications requiring high-level security, such as financial transactions and political or strategic decisions.Gaussian-modulated coherent-state (GMCS) continuous-variable QKD (CV-QKD) protocol [2,3] is one well-known QKD protocol, in which the initial key information is encoded on the quadrature components of the quantized electromagnetic field, and is detected by coherent detection.In theory, GMCS CV-QKD has been theoretically proved to be secure against arbitrary collective attacks [4][5][6][7] and coherent attacks [8], when even considering the finite-size effects [9][10][11][12][13].In actual implementation, CV-QKD has made great breakthroughs in secure transmission distance, secret key rate (SKR), integration, and field test [14][15][16][17][18], demonstrating its application prospects in the construction of metropolitan quantum networks.
Nevertheless, before QKD is widely adopted, it still faces some crucial challenges, such as SKR, distance, size, cost, and practical security [19].SKR, one core performance indicator of QKD in actual implementation, determines the transmission rate of enciphered data.Specifically, in quantum private communication, information-theoretically security can only be achieved by combining one-time-pad (OTP) encryption protocol with QKD.OTP protocol requires the key to be used only once and be as long as the message, which means that SKR needs to be consistent with the classical data rate.As the communication techniques improve, classical optical communications that provide 100 Gbit s −1 per wavelength channel are deployed [20].A field trial featuring an aggregate data rate of 54.2 Tbit s −1 has been carried out [21].Some alternative protocols, such as quantum stream cipher [22] and physical-layer secret key generation and distribution [23], can offer a higher rate, but their security has not been proven entirely yet.Therefore, it is clear that if we want to encrypt high volumes of classical network traffic using OTP for applications with high-level security requirements, significant developments on the SKR generated by QKD are required [19].
In terms of improving the SKR, much work has been done for discrete-variable QKD, such as improving the repetition rate to gigahertz [24], developing high-speed post-processing modules [25], exploiting high-dimensional quantum states [26], and using heterogeneous multicore fiber [27].By implementing ultrahigh counting rate superconducting detectors, a low error rate encoding transmitter, and high-speed post-processing algorithms, an SKR exceeding 110 Mbps@10 km has been achieved [28].For CV-QKD, by controlling the excess noise of the system and increasing the post-processing performance, the SKR has also been significantly improved [29][30][31][32][33].In 2022, Jain et al achieved an SKR of 0.0471 bits per symbol@20 km under composable security [34].Besides, Roumestan et al applied a series of digital signal processing and a probabilistic constellation shaping quadrature amplitude modulation (QAM) modulation format to realize 24 Mbps@25 km considering the finite-size effect [35].Moreover, due to the encoding scheme, the theoretical SKR generated by CV-QKD in the theoretical asymptotic regime can be close to the ideal SKR performance curve without relay, which scales as 1/2 of the Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound [36].
At present, the SKR of CV-QKD is mainly limited by the system repetition rate.The classical coherent optical communication, which has a similar optical system structure, can reach the repetition rate of even 100 GBaud [37], whereas Gaussian-modulated CV-QKD has been below 100 MBaud for a long time.The restriction of repetition rate mainly comes from three aspects: (1) Signal modulation rate.Gaussian modulation is different from the conventional modulation format, requiring higher modulation accuracy, and otherwise, unacceptable modulation noise will be introduced.In standard Gaussian modulation, the quantum state is usually modulated by an amplitude and a phase modulator.This scheme is susceptible to the nonlinear response of the modulator and gain-bandwidth product of the amplifier when the modulation rate is high.(2) Signal reception rate.The signal reception in CV-QKD is required to reach the shot-noise limit; otherwise, the noise introduced by eavesdropping will not be easily detected.The coherent detection circuit limits the shot-noise clearance achievable at high frequencies and restricts the detection bandwidth [38].(3) Data processing scheme.Since the quantum signal's signal-to-noise ratio (SNR) under the shot-noise limit detection is relatively low, compensating for channel impairment and imperfections in classical components at a high clock rate has not been realized yet.The excess noise due to incomplete compensation will significantly deteriorate the final SKR.
These three aspects hinder the improvement of SKR.Fortunately, considering the compatibility of CV-QKD with optical communication components, it is possible to use high bandwidth optical components with well-designed algorithms to break through the bottleneck.
In this paper, we report a Gaussian-modulated CV-QKD experiment using high bandwidth telecom components that achieves a record-high SKR.Specifically, the high-rate low-noise Gaussian modulation of coherent states is realized by a classical optical IQ modulator (IQM).High-baud low-intensity quantum signals are received by an integrated coherent receiver under the shot-noise limit.A series of digital signal processing algorithms are proposed to achieve accurate signal recovery and key distillation.The system has finally achieved 10.37 Mbps, 1.61 Mbps, 337.82 kbps, and 58.06 kbps asymptotic SKR within standard telecom fiber of 20 km, 50 km, 70 km, and 100 km, which is comparable to state-of-the-art GMCS CV-QKD experiments.Besides, since all the components used can be integrated through silicon photonics technology, it provides a system solution for a high-performance and cost-effective QKD network.

CV-QKD protocol
CV-QKD with Gaussian modulation can be implemented according to the preparation-and-measurement model, which is schematically depicted in figure 1. Alice uses the laser source to prepare the original coherent states |a 0 ⟩ with constant intensity and phase.Then these states are modulated following the Gaussian distribution.That is, Alice first prepares Gaussian random number vector {x a , p a }, and then load it on the origin coherent state to generate the coherent state |x a + ip a ⟩.Moreover, she needs to adjust the attenuation to set the desired quadrature component modulation variance V A of the output coherent states.After being transmitted through the quantum channel, the encoded coherent state is detected by dual homodyne detectors (also known as the heterodyne detection in the QKD regime).Dual homodyne detectors can access both quadratures, which has been proved secure in theory [13] and is easy to execute phase recovery in practice.Specifically, The encoded coherent state is first divided equally by a 50:50 beamsplitter (BS) and then detected by two homodyne detectors to obtain the quadrature component value of x B and p B , respectively.The theoretical security of GMCS CV-QKD is proved through the entanglement-based model, and the achievable SKR at a specific distance is estimated according to the information theory.
In the asymptotic conditions, the achievable SKR generated by CV-QKD can be calculated as where f e is the effective symbol rate, I AB is the classical mutual information between Alice and Bob, χ BE is the Holevo bound on the information leaked to Eve, β is the reconciliation efficiency, FER is the frame error rate (FER) of the reconciliation, a is the overhead for frame synchronization, ν is the key fraction disclosed for parameter estimation.The detailed derivation of SKR is given in appendix A. In an actual system, the fraction disclosed for parameter estimation ν has an optimal value and can be calculated [39].The overhead a is determined according to the actual channel condition.The FER and reconciliation efficiency are related to the selected decoder-the selected decoder should trade off these two parameters.The amount of mutual information I AB and χ BE are both related to the modulation variance V A , so V A has an optimal value to maximize the difference between the two.χ BE is also related to the critical parameter excess noise ε.The larger ε is, the greater the amount of information leaked to Eve, and even make SKR drop to 0. Therefore, controlling the excess noise is essential.According to the formula, it can intuitively be seen that by increasing the repetition rate, SKR will increase proportionally.However, it is also necessary to ensure the accurate modulation and reception of the coherent states and the high-precision signal recovery after increasing the repetition rate, guaranteeing excess noise suppression and high SKR performance.

Experimental set-up
The entire optical layer set-up is based on the local local oscillator (LLO) scheme architecture [40][41][42][43][44][45], shown in figure 2. At Alice, a narrow-linewidth continuous-wave (CW) laser with 100 Hz linewidth (NKT Koheras BASIK X15) generates the optical carrier, followed by an optical isolator and a variable optical attenuator (VOA) to control its power.Then, the optical carrier is sent into the high-speed optical IQM to realize the Gaussian modulation.Specifically, an arbitrary wave generator (AWG, Tektronix, AWG5200) is loaded in Gaussian random number data (which is generated by a quantum random number generator in appendix B) and then outputs two Gaussian distributed electronic analog signals with 500 MHz symbol rate, which are then amplified by a two-channel microwave amplifier (MA) to meet the voltage requirement of the modulator.The AWG can achieve a maximal sampling rate of 5 GSample s −1 and has a sampling depth of 16 bit.The pilot data is interposed between Gaussian data.Its value is relatively large and alternates between positive and negative to form an AC signal.The generated pilot signals are reference symbols with agreed information and location, helping the data processing but not forming part of the secret key generation.The input and output optical fields of the IQM can be expressed as where V π is the half-wave voltage, and ϕ is the phase difference between two arms of IQM.function of the bias controller (BC) is to control the bias voltage to operate in the linear region while ensuring ϕ = π/2, which adopts automatic feedback control to ensure constellation modulation's stability.
In order to reduce the crosstalk between signal pulses, an intensity modulator with a high extinction ratio (EOSPACE) is employed to cut CW light into pulsed light.The AWG also generates the pulsed electrical signal, and the duty cycle is set to 20%.Then it is amplified by another MA and then loaded on the intensity modulator.The bias point of the modulator is controlled by a DC stabilized power supply to ensure stable optical pulse generation.A manual VOA is deployed to control the output power.A deployed 90:10 BS monitors the output power in real-time, followed by several single-mode fiber (SMF) spools for long-distance transmission.
After the signal arrives at the receiver, it is adjusted by the polarization controllers and then interferes with the locally generated local oscillator (LO).VOA3 is used to close the signal path and thereby verify the shot noise.The LO is CW mode and comes from another narrow linewidth laser (NKT Koheras BASIK X15), the center wavelength of which can be controlled by software to align with the signal laser.The LO optical power at the receiver is calibrated to 9.98 dbm, which can make the shot noise 6.51 dB higher than the electrical noise, as shown in figure 8.Both the LO and signal are injected into an integrated coherent detector (ICR, FUJITSU).ICR is a standard component in classical communication that can be integrated into the coherent optical module.The ICR parameter relevant with CV-QKD is tested, where the detectors in ICR have the same quantum efficiencies η = 0.42 and electronic noises v el = 0.18 in shot noise units (see appendix C).The shot noise is calibrated in real-time.After the LO and signal enter the ICR, they are interfered with, detected, and amplified, and finally, four detection outputs are provided.Two detection  3), finishing the quantum communication procedure.To achieve clock synchronization, the AWG signal from Alice is used to trigger the oscilloscope sampling, which can be further solved by another wavelength or clock recovery algorithm.Finally, these raw data will generate the final key through data processing (see appendix D).

Gaussian modulation characterization
Gaussian modulation is a crucial criterion in the theoretical CV-QKD protocol, making it convenient to use Gaussian optimality to realize unconditional security proof.Therefore, to be consistent with the theoretical protocol, the transmitter must also perform the Gaussian modulation format.Specifically, the amplitude of the modulated electrical signal at the transmitter is restricted to the linear interval of the response function of the IQM to ensure the Gaussian property of the modulated optical field, and the coherent detection at the receiver is also within the linear operating interval.In figure 4, the distribution of the modulation data at Alice and the received data at Bob (after digital signal processing) are shown in (a) and (b), respectively.The Gaussian performance is tested by Shapiro and Wilk's W-test [46].Results show that at the significance level of α = 0.05, both the modulation and received data obey the Gaussian distribution (W-test statistic W = 0.995, 0.997, P-value = 0.182, 0.799), proving the consistency of the implementation with the theoretical protocol, but the strict criteria are not unified at present and need relevant standards.Moreover, figure 4(c) shows the linearity of the transmitted data and the received data.The SNR of the quantum signal is 2.58 dB, and the statistical sample size is 10 4 .The slope of the fitting curve is k = 0.996, expressing the consistency of data sent and received.Figure 4(d) is the raw collected data at the receiver, including quantum signal and pilot.The SNR of the pilot is set as 12.53 dB here.Due to the phase drift, the pilot data forms a ring in the phase space, and the signal data maintains a two-dimensional Gaussian distribution.

Phase noise elimination
In the experiment, equally spaced pilot signals are inserted between the quantum signals to compensate for the frequency offset (FO) and phase drift caused by the independent LO laser.Specifically, the intensity of the generated pilot signal is proportional to the amplitude of the modulated electrical signal, so it is convenient to set different power intensities.At the receiver, a phase compensation method based on the pilot signal is proposed, which can eliminate different phase drift rates.We set up pilot signals of different powers and then statistics on the remaining phase noise after phase compensation, shown in figure 5(a).One can see that as the pilot power increases, the residual phase noise gradually decreases.When the SNR is higher than 15 dB, the phase noise can be less than 0.01 rad 2 , and when the SNR is greater than 25 dB, the phase noise can be less than 0.001 rad 2 .This result determines the set of pilot signal power in the key distribution procedure.
In addition, we also analyzed the FO of the two lasers through the pilot, which is shown in figure 5(b).The red color represents the FO of the two NKT narrow-linewidth lasers at different times.Due to the high stability of the laser wavelength (±0.04 pm within 60 min), the FO changes very slightly.The FO of the two commercial integrable tunable laser assembly (ITLA) lasers from classical optical modules represented in blue are used for comparison.Since its stability is only ±1 pm within 15 min, it will jitter at MHz level in a short period and finally exceed the detector bandwidth after a while.Therefore, for ITLA lasers, the center wavelength needs to be controlled through real-time feedback algorithms to ensure long-time system operation.

Excess noise measurement
In the actual key distribution procedure, a data frame is composed of the frame header, pilot, and raw key data.The length of one frame is 5 × 10 5 symbols, where the frame header occupies 1 × 10 3 symbols, data occupies 2.49 × 10 5 symbols, and the interleaved pilot is 2.5 × 10 5 symbols.After receiving, the signal will be processed through a series of digital signal processing algorithms, finally recovering the original key information.The remaining excess noise of the system, composed of amplitude and phase noise, determines the maximum amount of information Eve can acquire, and therefore must be estimated accurately.The traditional parameter estimation scheme is adopted [47].The excess noise refers to the output of Alice's side.Figure 6 shows the excess noise measured by the system under a 20 km standard telecommunication fiber.By testing 20 data frame, the average value of the excess noise is 0.062, which can be below the 10 Mbps threshold bound and therefore has 10 Mbps SKR capability.In the experiment, the remaining excess noise of the system mainly comes from the limited phase compensation accuracy, while the excess noise fluctuation comes from the device's instability.For example, the laser intensity fluctuation will cause signal modulation variance and pilot power changes.In addition, the limited data length is also an important factor leading to bias in parameter estimation.

SKR analysis
According to the measured excess noise and the key experimental parameters (shown in table 1, the SKR that can be achieved in the experiment is shown in table 2. The length of the telecom fiber spools used for test is  The achievable SKR at different distances in the experiment is shown in figure 7. The red line represents the SKR bound estimated according to the results of the communication experiment, in which the reconciliation efficiency and FER are constant at different distances.The red five-pointed star is the achieved SKR after post-processing.Since the reconciliation efficiency and FER are changed at different distances, and the excess noise and the modulation variance fluctuate with time, there will be a slight drop compared with the achievable bound.To illustrate the performance, the previous CV-QKD experimental results, whether using an LLO scheme or a transmitted local oscillator (TLO) scheme, are also marked and compared in the figure .It can be seen that our system is comparable to state-of-the-art CV-QKD systems in terms of SKR.One can see that the theoretical SKR per symbol in CV-QKD can be close to the PLOB bound [36].Considering the finite-size effect, we conducted SKR simulations for different block sizes [9].It was found that when the block size exceeds 10 11 , the finite-size SKR remains consistent with asymptotic conditions at 50 km.However, when the block size is less than 10 8 .For 100 km, SKR can only be generated when the block size reaches 10 11 .If we further consider that, we also need to ensure that the excess noise in a long block is small, which has a high requirement for the stability of the system.

Discussions
We have demonstrated a complete GMCS CV-QKD system using mature telecom components.In particular, we use a high-speed IQM to realize accurate Gaussian modulation, employ ICR to detect the encoded coherent states, and propose a series of algorithms to eliminate the impairment in the signals.Finally, the excess noise in CV-QKD is suppressed to a lower level, reaching 0.062, 0.050, 0.054, and 0.037 under the standard single-mode fiber of 20 km, 50 km, 70 km, and 100 km.The corresponding SKR achieved with our post-processing scheme is 10.37 Mbps, 1.61 Mbps, 337.82 kbps, and 58.06 kbps, respectively.
It is anticipated that with the improvement of experiment details, including automatic polarization control and laser power stability, the excess noise in the system can be further controlled stable and low.Besides, the finite-size effect can be mitigated by employing high-speed analog-to-digital converters (ADCs) and high-volume storage to acquire a larger block.Further, by developing hardware and introducing field-programmable gate array cores, the real-time random number modulation and real-time data processing can be realized, supporting the long-term secret key generation of the system [54].
As for integrated lasers, we need to consider two important parameters: frequency stability and linewidth.Integrated lasers have poor frequency stability, so a feedback algorithm is needed to achieve calibration of the laser center frequency.For the linewidth, since the symbol rate increases, the interval between the pilot signal and the quantum signal becomes smaller, and the phase noise is further reduced.Therefore, the linewidth of tens of kHz of conventional commercial lasers can also meet the requirements of CV-QKD.
It is worth noting that the internet architecture that comprises communication segments relayed by optical amplifiers, so there is still a certain distance for our system to use in the existing network.At this stage, our system-level solution can be directly deployed on fiber links without optical amplifier relays, such as access networks, 5G mobile Fronthaul links [55].For long-distance transmission, trusted relays need to be placed at the optical amplifier, and signal separation, conversion and recombination can be carried out by wavelength division multiplexing technology to realize the forwarding of the key.
In addition to the system's feasibility, the system's practical security is also worthy of further study.Monitoring the conditions of crucial components will further ensure the practical security of QKD and ultimately ensure the integrity of the data.In [56], the author mentioned that using optical single sideband modulation will generate mirrored sideband, due to the finite sideband suppression ability of IQMs, leading to side channels.In our work, we have used optical baseband modulation instead of sideband modulation, thus avoiding security issues.Besides, although CV-QKD based on discrete modulation still needs to complete its theoretical security, the consistency of its modulation format with classical optical communication gives it a competitive advantage in terms of the high SKR [57,58].
In conclusion, our work has verified the feasibility of a high-performance GMCS CV-QKD using high bandwidth optical components.The system is comparable to state-of-the-art QKD systems in SKR and is promising to be realized by standard optical modules since all the components can be integrated.Considering this scheme's high performance and ease of integration, it provides a high-speed and cost-effective QKD solution to build the quantum secure communication network.
where β ∈ (0, 1) is the efficiency of reverse reconciliation, I het AB is the mutual information between Alice and Bob, and χ het BE is the maximum information available to Eve on Bob's key bounded by the Holevo quantity.Specifically, I het AB can be identified as where V = V A + 1, and χ tot representing the total noise referred to the channel input can be calculated as χ tot = χ line + χ het /T, in which χ line = 1/T − 1 + ε, and Above essential parameters are: V A is the modulation variance, T is the channel transmittance, ε is the excess noise, η is the quantum efficiency, and v el is the electronic noise.Besides, χ het BE is identified as follows where G(x) = (x + 1) log 2 (x + 1) − x log 2 x. λ i are symplectic eigenvalues derived from the covariance matrices and can be expressed as where According to the above calculation formula, the SKR per second can be evaluated as [47] where f e is the effective symbol rate, FER is the frame error rate of the reconciliation, a is the overhead for frame synchronization, ν is the key fraction disclosed for parameter estimation.Once the above parameters are determined, the SKR of the system can be determined.
Figure 8. Noise power spectrum of the integrated coherent receiver.The curves from top to bottom represent the noise power at 9.98 dBm, 7.28 dBm, 5.32 dBm, 2.07 dBm LO and without LO.When the LO is 9.98 dBm, the total noise power is 6.51 dB higher than the electronic noise power at 500 MHz.

C.1. Quantum efficiency calibration
The quantum efficiency represents the overall signal loss at the receiver, including path loss and photoelectric conversion loss.Considering that the responsivity of the ICR detector is R = 0.60 A W −1 , the photoelectric conversion loss can be converted as η ICR = R × q/hν = R × 0.80 = 0.48, where q is the energy of an electron, and hν is the energy of a photon.The path loss, including the manual polarization controller, the attenuator, and the connector loss, can be calibrated using the optical power meter.The final attenuation is 0.53 dB, which is converted into η Loss = 0.88.So the receiver's quantum efficiency is η = η ICR × η Loss = 0.42.Besides, the quantum efficiency can also be measured by testing the input optical power and the output power of the trans-impedance amplifier, which is basically consistent with the above value.It is worth noting that due to the limited accuracy of the equipment, there will be some deviation in the value.This may lead us to overestimate quantum efficiency and thus overestimate the actual SKR.Therefore, by using the monitoring output of ICR and conducting a global evaluation of receiver efficiency, its practical security can be effectively guaranteed.

C.2. Electronic noise calibration
Reaching the shot-noise limit detection is an important criterion to ensure the security of the CV-QKD implementation.First, we analyze the shot-noise limitation in the frequency domain.The noise power spectrum of the output signal of the ICR with different intensities of the LO is tested by a spectrum analyzer (Agilent Technologies, N9020A), the result of which is shown in figure 8.It can be seen that the greater the intensity of the LO, the higher the noise power.It is because the shot noise power is proportional to the intensity of the local oscillator in theory.When the optical input power of the LO reaches 9.98 dbm, the noise power will increase by 6.51 dB compared to the electronic noise power at 500 MHz, indicating that the detection of the shot-noise limit can be achieved under the local LO intensity.Besides, one can find that the electronic noise will increase slightly at higher frequencies, which means that the proportion of shot noise will decrease slightly under the same LO intensity.Moreover, the noise at low frequencies rises because the amplification gain at low frequencies is the largest, and parasitic capacitance will affect the gain at higher frequencies, so the noise at low frequencies is higher.The similar situation also appears in [38].In this case, the low-frequency electronic noise is larger, so the overall total noise will be larger in the low-frequency region.
In order to more accurately calibrate the electronic noise in shot noise units, we have conducted an experimental test in the time domain.We measure the total noise of each output port of the ICR (with LO input and without signal input) and the electronic noise (without LO input).The sampling rate and the filter used are consistent with the signal acquisition.The result is shown in figure 9.The noise power is different in the eightoutput ports of the ICR, but the fluctuation over time can be ignored according to the error bar.Considering the balance of detecting x and p components, port 1 and port 3 are used to detect the signal's pilot signal is set to a fixed value, the variance of pilot component data x pilot and p pilot obtained by the receiver will be equal theoretically due to the phase drift.However, the unbalanced response makes the variance inconsistent.This inconsistent variance is exploited to determine the scaling ratio.The method is as follows: step 1. Calculate the variance of the component data of the collected pilot signal, i.e var{x pilot } and var{p pilot }.Step 2. Get conversion ratio r = √ var{x pilot }/var{p pilot }.
Step 3.All the p component data p sig , p pilot is scaled by p pilot1 = p pilot × r, p sig1 = p sig × r.After scaling, quadrature component data is balanced.

D.3. Frequency offset estimation
After quadrature balance calibration, we need to estimate the frequency offset.The frequency offset comes from the inconsistency of the center wavelength of the two lasers.The optical spectrums of the signal laser and the LO laser are tested by an optical spectrum analyzer (Finisar WaveAnalyzer 1500S), and the result is shown in figure 10.Here the wavelength of the LO laser and the signal laser are both set at 1550.20 nm, which is compensated by temperature feedback.The spectrum analyzer can see that the signal laser's and LO laser's center wavelengths are basically coincident, implying the coarse alignment.
The remaining center wavelength deviation is processed by the frequency offset (FO) estimation.We Fourier transform on the pilot signal to find the maximum value f off , which corresponds to the frequency offset.However, the direction of frequency offset cannot be judged.According to the traditional method in classical optical communication [62], the pre-compensation of frequency offset is executed on the pilot signal.Firstly, the power spectrum of the experimental pilot data is calculated, and the maximum value of the power spectrum determines the center frequency of the FO as shown in figure 11.We can see the FO this time is f off = 10.56 MHz.Then, the direction of the FO also needs to be determined, that is, whether the phase rotation direction is clockwise or counterclockwise.We adopt the method that we first rotate the current pilot to estimate the next pilot in two directions, respectively.The operation can be expressed as where f off is the FO, t int is the time interval between two pilot, j is the index of the pilot, and A j+1 cw , A j+1 ccw represent the estimation of the next pilot data through clockwise rotation or counterclockwise rotation respectively.Then, we calculate the phase difference between the estimation and the real next pilot data: where δ j+1 cw , δ j+1 ccw represent the phase difference after clockwise or counterclockwise rotation respectively.If the direction is correct, the phase difference will converge to 0; if the direction is wrong, the phase difference will converge to the 4π f off t int .Figure 12 shows the experimental results of this method.After clockwise  It can be seen that the phase angle of the complex amplitude obtained in (a) converges to 0, while the phase angle of the complex amplitude in (b) converges to a non-zero value, so it can be judged that the direction of the FO at this time is clockwise.compensation, we can see that the phase difference converges to 0, implying the direction is clockwise.The residual phase deviation originates from the phase noise, which is relevant to the laser's linewidth.

D.4. Phase compensation
After determining the frequency offset value and direction, the phase drift of the quantum signal can be compensated.Here we use the previous pilot pulse and the subsequent pilot pulse to recover the current quantum signal.In order to eliminate the influence of the amplitude of the pilot signal, we directly use the phase angle of the two pilot signals to rotate the signal, that is, recover the signal with the following formula: where x pprv , p pprv is the x and p component of previous pilot, x psub , p psub is the x and p component of subsequent pilot, x qorg , p qorg is the x and p component of origin quantum signal, and x qrot , p qrot is the x and p component of the quantum signal after rotation.A qrot is the rotation parameter.This calculation is consistent with the method in the [40].But due to the existence of additional noise and the ambiguity of the phase.We need to improve the robustness of the above algorithm further.For this reason, we make a classification discussion according to the direction of FO and the transmitting frequency of the pilot f pilot .The schematic diagram of this scheme is shown in figure 13, which is described as follows: (1) For the counterclockwise FO, it can be divided into the following two situations: (a) The FO is less than f pilot /4.Since the FO is minor, It is necessary to consider the error of the positive and negative phase angle induced by the additional noise.Therefore, we divide it into three sub-cases for consideration.(1) When the previous pilot angle is greater than 0 and the subsequent pilot angle is less than 0, we need to reverse the after rotation (that is, multiply the rotation result −1, because the two angles span the x-axis, there will be positive and negative deviations).( 2) When the angle of the current pilot pulse is less than 0 and the angle of the next pilot pulse is greater than 0, it indicates that positive and negative ambiguity exists.It also needs to be reversed.(3) In other cases, direct rotation is sufficient.(b) If the FO is greater than f pilot /4.Since the FO is large, there is no need to consider the positive and negative errors caused by noise.Therefore, we divide it into two sub-cases for consideration.(1) When the current pilot pulse angle is greater than 0, and the subsequent pilot pulse angle is less than 0, we need to reverse the result after the rotation (multiply the rotation result by −1).(2) In other cases, direct rotation is sufficient.(2) For the clockwise FO, it can be divided into the following two situations: (a) The FO is less than f pilot /4.Since the FO is minor, the positive and negative errors caused by the noise need to be considered.Therefore, we divide it into three cases for consideration.(1) When the previous pilot pulse angle is less than 0 and the subsequent pilot pulse angle is greater than 0, we need to reverse it after the rotation (that is, multiply the rotation result by −1, because the two pulse angles span the x-axis, there will be positive and negative deviations).( 2) When the angle of the current pilot pulse is greater than 0, and the angle of the subsequent pilot pulse is less than 0, it indicates that positive and negative ambiguity exists.It also needs to be reversed.(3) In other cases, direct rotation is sufficient.(b) If the FO is greater than f pilot /4, the FO is large at this time, and there is no need to consider the positive and negative errors caused by noise.Therefore, we divide it into two sub-cases for consideration.(1) When the current pilot pulse angle is less than 0, and the subsequent pilot pulse angle is greater than 0, we need to reverse it after the rotation (multiply the rotation result by −1).
(2) In other cases, direct rotation is sufficient.
The experimental results of the above method are shown in figure 14.As can be seen in (a), before phase compensation, the values and trends of the received data and the modulated Gaussian data are completely inconsistent, indicating that the phase deviation is large; in (b), after phase compensation, the recovered data and the original Gaussian data overlap in value, indicating the effectiveness of phase compensation.

D.5. Frame synchronization
After the phase compensation, the correspondence of sending and receiving data needs to be established.Realizing data alignment at different distances is very important in CV-QKD.The whole frame structure is shown in figure 15, and the Zadoff-Chu (ZC) sequence is used as the frame header of the Gaussian modulation data.The ZC sequence can be expressed as [63] z r [n] = exp where N ZC is the length of the ZC sequence.r is the root index of the ZC sequence and is set as 1.In order to ensure the success of synchronization, we set the length of the ZC sequence to 1000. Figure 16(a) shows the real part of the frame synchronization sequence, which has good zero autocorrelation and low cross-correlation [64].In figure 16(b), we analyze the requirements for the SNR of the frame synchronization sequence under different transmission distances, where 1 represents synchronization success, and 0 represents synchronization failure.It can be seen that as the transmit power of the ZC frame increases, its synchronization will turn from failure to success.Moreover, the longer the transmission distance, the higher the power requirement for the frame synchronization sequence.Therefore, we need to set the transmit power of the frame synchronization sequence according to the transmission distances to ensure accurate synchronization, and it can be controlled directly by adjusting the intensity of the electronic modulation signal.

D.6. Post-processing
In post-processing, we will perform the parameter estimation, then execute the reconciliation according to the estimated parameters, and distill the secure key through privacy amplification.The most critical part is  the decoding part, directly related to reconciliation efficiency β and FER.In the actual implementation of CV-QKD, the received signal-to-noise ratio (SNR) is generally below 0 dB, so achieving decoding under low SNR is particularly important.In addition, it is also very challenging to guarantee β and β simultaneously at different distances.Here, the rate-adaptive LDPC using puncturing and shortening with multidimensional reconciliation [65,66] is adopted to achieve low-SNR decoding at different distances to meet the decoding requirements of CV-QKD within the 100 km transmission distance.Four codes with different code rates (R = 0.5, 0.1, 0.02, 0.01) are combined to realize rate-adaptive LDPC, guaranteeing high-performance decoding under different transmission distances.
In the decoding process, the frame error rate (FER) is a statistical parameter in CV-QKD, which requires many frame samples to obtain an accurate value of FER.In figure 17, we simulate analyzing the FER under different β and SNR with rate-adaptive LDPC algorithm.It can be seen that when the reconciliation efficiency reaches 95%, the FER when the SNR changes.When the reconciliation efficiency is β < 94%, the FER fluctuates less and can be controlled mostly Below 10%.The reason is that.However, since the reconciliation efficiency β directly determines the information that Alice and Bob can finally extract from the mutual information, it cannot be lower than the amount Eve can acquire.For this, we have selected β = 95% to guarantee the secure key rate beyond 0 and got the corresponding FER through 1000 times Monte-Carlo simulation.The SNR of the signal is according to the experiment (see table 2 in the main text).The obtained FER is 15.30%@0.52dB, 18.60%@−5.85dB, 19.10%@−8.13dB and 17.40%@−15.38dB.These decoders finally guarantee the secret key distillation at different distances.

Figure 1 .
Figure 1.The schematic diagram of CV-QKD with Gaussian modulation.CS source: coherent state source; Mod.: modulation; Att.: attenuation.Shot noise is the uncertainty of the quadrature component of a coherent state, and VA is the variance of data modulated on the quadrature component.
are the modulated signal and bias voltage for I path and Q path respectively.The

Figure 2 .
Figure 2. (a) Experimental setup of the GMCS CV-QKD scheme using high-speed mature telecom components; (b) quantum signal modulation module; (c) quantum signal integrated coherent receiver module; (d) quantum random number generator module.The blue and red lines denote the single-mode and polarization-maintaining fibers, respectively.Iso: isolator; BS: beam splitter; VOA: variable optical attenuator; IQM: IQ modulator; PC: polarization controller; MA: microwave amplifier; DC: DC stabilized power supply; ICR: integrated coherent receiver; BHD: balanced homodyne detector.

Figure 3 .
Figure 3. Afterglow of the collected signal.The yellow afterglow represents the X component electronic signal, and the red afterglow represents the P component electronic signal.The higher pulse represents the pilot signal, and the lower pulse between the two pilot signals represents the encoded signal.Afterglow is tested under clock synchronization, and the white parts indicate the sampling point.

Figure 4 .
Figure 4. Gaussian modulation characterization.(a) and (b) show the modulation data and received data, which exhibit Gaussian distribution characteristics.(c) shows the correlation between modulation and received data corresponding to an SNR of 2.58 dB.(d) illustrates both the quantum signal and pilot.The SNR of the pilot is set as 12.53 dB, which is 9.95 dB higher than the quantum signal.

Figure 5 .
Figure 5. Measured phase noise and frequency offset.(a) The residual phase noise under different SNR of the pilot has been tested.The phase noise in each SNR was tested ten times, and both the SNR and the phase noise deviation are displayed.(b) the frequency offset between two lasers has been tested.The FO between the two NKT ultra narrow-linewidth lasers is represented in red.The blue represents the change of the FO of the commercial ITLA lasers for comparison.

Figure 6 .
Figure 6.Measured excess noise.The excess noise under the telecom SMF fiber with the transmission distance of 20 km (3.68 dB attenuation) is tested, with the error bar marked.The lines in the figure show the excess noise threshold of corresponding achievable SKR, and the average value of the excess noise measured can be below the 10 Mbps threshold bound.Relevant parameters are shown in table 1.

Figure 10 .
Figure 10.Optical spectrums of signal laser and LO laser.The red curve represents the LO laser, and the green curve represents the signal laser.In order to show more clearly that the center wavelengths of the two lasers are aligned and are limited by the input optical power, the optical powers of the local oscillator and signal light are both set at [−10, 0] dBm.

Figure 11 .
Figure 11.Power spectrum of the pilot signal.The highest point in the spectrum means that the frequency offset is 10.56 MHz.The SNR of the pilot is 18.65 dB.

Figure 12 .
Figure 12.Phase difference pre-compensation.(a) shows the phase difference after clockwise compensation.(b) shows the phase difference after counterclockwise compensation.It can be seen that the phase angle of the complex amplitude obtained in (a) converges to 0, while the phase angle of the complex amplitude in (b) converges to a non-zero value, so it can be judged that the direction of the FO at this time is clockwise.

Figure 13 .
Figure13.Phase compensation algorithm.(a) shows the execution of the proposed phase compensation scheme in the phase space.The left figure shows that this scheme first rotates 1/2 of the phase angle of the previous pilot clockwise and then rotates 1/2 of the phase angle of the subsequent pilot counterclockwise.The middle figure shows that the phase change from the previous pilot angle to the subsequent pilot angle cannot be determined when the FO direction is unclear.The right figure shows that on the positive x-axis, due to the additive noise, the phase will produce ambiguity, leading to the failure of the original algorithm.(b) describes the flow chart adopted to enhance the algorithm's robustness for exceptional circumstances.The specific process is described in the text.

Figure 14 .
Figure 14.Phase compensation result.(a) describes the comparison between modulated Gaussian data and data before phase compensation.(b) describes the comparison between the modulation data and the data after phase compensation.It can be seen that the phase has been restored.The SNR of the signal here is 7.82 dB.

Figure 15 .
Figure15.The frame structure.A data frame is composed of the frame header, pilot, and raw key data.The length of one frame is 5 × 10 5 Sample, where the frame header occupies 1 × 10 3 sample, data occupies 2.49 × 10 5 sample, and the interleaved pilot is 2.5 × 10 5 sample.

Figure 16 .
Figure 16.Frame syncronization.(a) shows the ZC frame sequence, where the sequence length is 1000.(b) shows the required ZC sequence SNR at different transmission distances, and Monte Carlo simulation is used to determine whether the synchronization is successful. 1 represents synchronization success, and 0 represents synchronization failure.

Figure 17 .
Figure 17.Rate-adaptive LDPC performance.SNR and FER under different beta with the rate-adaptive algorithm.Based on four different code rates R = 0.5, 0.1, 0.02, 0.01, the rate-adaptive LDPC decoding algorithm is formed.Through 1000 Monte-Carlo simulations for each point, the statistical values of FER under different SNR and reconciliation efficiency are obtained.The two points in the figure correspond to the decoding performance of 100 km and 70 km, respectively.

Table 1 .
Practical parameter in CV-QKD experiment.km, 70 km, and 100 km, respectively, and the corresponding attenuation is 3.68 dB, 9.26 dB, 12.94 dB, and 18.96 dB.In order to ensure accurate phase compensation without affecting the quantum signal, the SNR of the pilot is controlled at around 16.00 dB.The modulation variance V A is actively changed slightly with the distance to achieve high SKR performance at different distances.The rate-adaptive LDPC is employed to decode, and FER with β = 95% is obtained through statistics from a large number of samples, which are 15.30%, 18.60%, 19.10%, and 17.40% respectively.Finally, the average SKR in the asymptotic regime achieved in the experiment is 10.37 Mbps, 1.61 Mbps, 337.82 kbps, and 58.06 kbps, respectively.