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A quantum network provides an infrastructure connecting quantum devices with revolutionary computing, sensing, and communication capabilities. As the best-known application of a quantum network, quantum key distribution (QKD) shares secure keys guaranteed by the laws of quantum mechanics. A quantum satellite constellation offers a solution to facilitate the quantum network on a global scale. The Micius satellite has verified the feasibility of satellite quantum communications, however, scaling up quantum satellite constellations is challenging, requiring small lightweight satellites, portable ground stations and real-time secure key exchange. Here we tackle these challenges and report the development of a quantum microsatellite capable of performing space-to-ground QKD using portable ground stations. The quantum microsatellite features a payload weighing approximately 23 kg, while the portable ground station weighs about 100 kg. These weights represent reductions by more than an order and two orders of magnitude, respectively, compared to the Micius satellite. Additionally, we multiplex bidirectional satellite-ground optical communication with quantum communication, enabling key distillation and secure communication in real-time. Using the microsatellite and the portable ground stations, we demonstrate satellite-based QKD with multiple ground stations and achieve the sharing of up to 0.59 million bits of secure keys during a single satellite pass. The compact quantum payload can be readily assembled on existing space stations or small satellites, paving the way for a satellite-constellation-based quantum and classical network for widespread real-life applications., Comment: 40 pages, 8 figures

We introduce JuliVQC: a light-weight, yet extremely efficient variational quantum circuit simulator. JuliVQC is part of an effort for classical simulation of the \textit{Zuchongzhi} quantum processors, where it is extensively used to characterize the circuit noises, as a building block in the Schr$\ddot{\text{o}}$dinger-Feynman algorithm for classical verification and performance benchmarking, and for variational optimization of the Fsim gate parameters. The design principle of JuliVQC is three-fold: (1) Transparent implementation of its core algorithms, realized by using the high-performance script language Julia; (2) Efficiency is the focus, with a cache-friendly implementation of each elementary operations and support for shared-memory parallelization; (3) Native support of automatic differentiation for both the noiseless and noisy quantum circuits. We perform extensive numerical experiments on JuliVQC in different application scenarios, including quantum circuits, variational quantum circuits and their noisy counterparts, which show that its performance is among the top of the popular alternatives., Comment: 12 pages, 8 figures

We propose the piecemeal quantum telescope through bit-by-bit iteration using different baselines. It improves precision exponentially with number of baselines, and it works robustly under large observation errors such as statistical errors, channel noise, operational errors and so on. For example, under the noisy channel of random phase drifts, our method can provide an exponential precision provided that the detected data is not entirely noisy. Being fault tolerant to statistical error, it requests only a small number of incident single-photons in detecting the star angle with exponential precision. As a result, it requests to detect only a few hundreds of photons from the target star for a precision breaking classical limit by 4 to 5 magnitude orders. This demonstrates a super efficiency., Comment: 6 figures

Quantum digital signatures (QDSs) can provide information-theoretic security of messages against forgery and repudiation. Compared with previous QDS protocols that focus on signing one-bit messages, hash function-based QDS protocols can save quantum resources and are able to sign messages of arbitrary length. Using the idea of likely bit strings, we propose an efficient QDS protocol with hash functions over long distances. Our method of likely bit strings can be applied to any quantum key distribution-based QDS protocol to significantly improve the signature rate and dramatically increase the secure signature distance of QDS protocols. In order to save computing resources, we propose an improved method where Alice participates in the verification process of Bob and Charlie. This eliminates the computational complexity relating to the huge number of all likely strings. We demonstrate the advantages of our method and our improved method with the example of sending-or-not-sending QDS. Under typical parameters, both our method and our improved method can improve the signature rate by more than 100 times and increase the signature distance by about 150 km compared with hash function-based QDS protocols without likely bit strings., Comment: 10 pages, 2 figures

Quantum key distribution (QKD) holds the potential to establish secure keys over long distances. The distance of point-to-point QKD secure key distribution is primarily impeded by the transmission loss inherent to the channel. In the quest to realize a large-scale quantum network, increasing the QKD distance under current technology is of great research interest. Here we adopt the 3-intensity sending-or-not-sending twin-field QKD (TF-QKD) protocol with the actively-odd-parity-pairing method. The experiment demonstrates the feasibility of secure QKD over a 1002 km fibre channel considering the finite size effect. The secure key rate is $3.11\times10^{-12}$ per pulse at this distance. Furthermore, by optimizing parameters for shorter fiber distances, we conducted performance tests on key distribution for fiber lengths ranging from 202 km to 505 km. Notably, the secure key rate for the 202 km, the normal distance between major cities, reached 111.74 kbps., Comment: 18 pages, 3 figures

Gaussian Boson sampling (GBS) plays a crucially important role in demonstrating quantum advantage. As a major imperfection, the limited connectivity of the linear optical network weakens the quantum advantage result in recent experiments. Here we present a faster classical algorithm to simulate the GBS process with limited connectivity. In this work, we introduce an enhanced classical algorithm for simulating GBS processes with limited connectivity. It computes the loop Hafnian of an $n \times n$ symmetric matrix with bandwidth $w$ in $O(nw2^w)$ time which is better than the previous fastest algorithm which runs in $O(nw^2 2^w)$ time. This classical algorithm is helpful on clarifying how limited connectivity affects the computational complexity of GBS and tightening the boundary of quantum advantage in the GBS problem., Comment: 2 figures

Twin-field quantum key distribution (TF-QKD) overcomes the linear rate-loss limit, which promises a boost of secure key rate over long distance. However, the complexity of eliminating the frequency differences between the independent laser sources hinders its practical application. Here, taking the saturated absorption spectroscopy of acetylene as an absolute reference, we propose and demonstrate a simple and practical approach to realize TF-QKD without requiring relative frequency control of the independent laser sources. Adopting the 4-intensity sending-or-not-sending TF-QKD protocol, we experimentally demonstrate the TF-QKD over 502 km, 301 km and 201 km ultra-low loss optical fiber respectively. We expect this high-performance scheme will find widespread usage in future intercity and free-space quantum communication networks., Comment: 13 pages, 5 figures, 7 tables

Measurement-device-independent quantum key distribution (MDI-QKD) can resist all attacks on the detection devices, but there are still some security issues related to the source side. One possible solution is to use the passive protocol to eliminate the side channels introduced by active modulators at the source. Recently, a fully passive QKD protocol has been proposed that can simultaneously achieve passive encoding and passive decoy-state modulation using linear optics. In this work, we propose a fully passive MDI-QKD scheme that can protect the system from both side channels of source modulators and attacks on the measurement devices, which can significantly improve the implementation security of the QKD systems. We provide a specific passive encoding strategy and a method for decoy-state analysis, followed by simulation results for the secure key rate in the asymptotic scenario. Our work offers a feasible way to improve the implementation security of QKD systems, and serves as a reference for achieving passive QKD schemes using realistic devices.

Quantum advantage is the core of quantum computing. Grover's search algorithm is the only quantum algorithm with proven advantage to any possible classical search algorithm. However, realizing this quantum advantage in practice is quite challenging since Grover's algorithm is very sensitive to noise. Here we present a noise-tolerant method that exponentially improves the noise threshold of Grover's algorithm. We present a lower bound for average fidelity of any quantum circuit with O(log D log D) cost under time-independent noise, where D is the dimension of Hilbert space. According to this bound value, we determine the number of iterates which will be applied in Grover's algorithm. Numerical simulation shows that the noise threshold of quantum advantage of Grover's algorithm by our noise-tolerant method is improved by an exponential factor with qubit amount rise., Comment: 6 figures

The decomposition for controlled-$ZX$ gate in [Phys. Rev. A, 87, 062318 (2013)] has a shallow circuit depth $8n-20$ with no ancilla. Here we modify this decomposition to decompose $n$-qubit Toffoli gate with only $2n-3$ additional single-qubit gates. The circuit depth is unchanged and no ancilla is needed. We explicitly show that the circuit after decomposition can be easily constructed in present physical systems., Comment: 6 pages, 9 figures

The original Grover's algorithm has a success probability to output a correct solution, while deterministic Grover's algorithms improve the success probability to 100%. However, the success probability of deterministic Grover's algorithm decreases in noisy environment. Here we improve the deterministic two-parameter (D2p) Grover's algorithm to reach the upper bound for success probability under phase noise. We prove that it is not possible to design any deterministic Grover's algorithm whose success probability is higher than our improved D2p protocol's under phase noise., Comment: 7 pages, 8 figures

We present a result of side-channel-secure (SCS) quantum key distribution (QKD) under fully realistic conditions. Our result is not only measurement-device independent but also effective with imperfect (and unstable) source devices including imperfect vacuum and imperfect coherent-state source. Applying the virtual mapping idea, we present a general security proof under whatever out-side-lab attack, including whatever side-channel coherent attack. As a byproduct, we also present an improved method for SCS protocols which can raise the key rate by 1-2 orders of magnitude. Using these results, we obtain a non-asymptotic key rate which is instantly useful with full realistic conditions., Comment: Typos corrected

Quantum key distribution (QKD) aims to generate secure private keys shared by two remote parties. With its security being protected by principles of quantum mechanics, some technology challenges remain towards practical application of QKD. The major one is the distance limit, which is caused by the fact that a quantum signal cannot be amplified while the channel loss is exponential with the distance for photon transmission in optical fiber. Here using the 3-intensity sending-or-not-sending protocol with the actively-odd-parity-pairing method, we demonstrate a fiber-based twin-field QKD over 1002 km. In our experiment, we developed a dual-band phase estimation and ultra-low noise superconducting nanowire single-photon detectors to suppress the system noise to around 0.02 Hz. The secure key rate is $9.53\times10^{-12}$ per pulse through 1002 km fiber in the asymptotic regime, and $8.75\times10^{-12}$ per pulse at 952 km considering the finite size effect. Our work constitutes a critical step towards the future large-scale quantum network., Comment: 47 pages, 17 figures

Twin-field quantum key distribution (TF-QKD) has emerged as a promising solution for practical quantum communication over long-haul fiber. However, previous demonstrations on TF-QKD require the phase locking technique to coherently control the twin light fields, inevitably complicating the system with extra fiber channels and peripheral hardware. Here we propose and demonstrate an approach to recover the single-photon interference pattern and realize TF-QKD \emph{without} phase locking. Our approach separates the communication time into reference frames and quantum frames, where the reference frames serve as a flexible scheme for establishing the global phase reference. To do so, we develop a tailored algorithm based on fast Fourier transform to efficiently reconcile the phase reference via data post-processing. We demonstrate no-phase-locking TF-QKD from short to long distances over standard optical fibers. At 50-km standard fiber, we produce a high secret key rate (SKR) of 1.27 Mbit/s, while at 504-km standard fiber, we obtain the repeater-like key rate scaling with a SKR of 34 times higher than the repeaterless secret key capacity. Our work provides a scalable and practical solution to TF-QKD, thus representing an important step towards its wide applications., Comment: Published version

The quantum approximate optimization algorithm (QAOA) is a promising method for solving certain classical combinatorial optimization problems on near-term quantum devices. When employing the QAOA to 3-SAT and Max-3-SAT problems, the quantum cost exhibits an easy-hard-easy or easy-hard pattern respectively as the clause density is changed. The quantum resources needed in the hard-region problems are out of reach for current NISQ devices. We show by numerical simulations with up to 14 variables and analytical arguments that the adaptive-bias QAOA (ab-QAOA) greatly improves performance in the hard region of the 3-SAT problems and hard region of the Max-3-SAT problems. For similar accuracy, on average, ab-QAOA needs 3 levels for 10-variable 3-SAT problems as compared to 22 for QAOA. For 10-variable Max-3-SAT problems, the numbers are 7 levels and 62 levels. The improvement comes from a more targeted and more limited generation of entanglement during the evolution. We demonstrate that classical optimization is not strictly necessary in the ab-QAOA since local fields are used to guide the evolution. This leads us to propose an optimization-free ab-QAOA that can solve the hard-region 3-SAT and Max-3-SAT problems effectively with significantly fewer quantum gates as compared to the original ab-QAOA. Our work paves the way for realizing quantum advantages for optimization problems on NISQ devices., Comment: 18 pages, 15 figures

Based on the idea that there is no side channel in the vacuum state, the side-channel-free quantum key distribution (SCFQKD) protocol was proposed, which is immune to all attacks in the source side-channel space and all attacks in the detectors. In the original SCFQKD protocol, an important assumption is that Alice and Bob can produce the perfect vacuum pulses. But due to the finite extinction ratio of the intensity modulators, the perfect vacuum pulse is impossible in practice. In this paper, we solve this problem and make the quantum key distribution side-channel secure with real source device which does not emit perfect vacuum pulses. Our conclusion only depends on the upper bounds of the intensities of the sources. No other assumptions such as stable sources and stable side channels are needed. The numerical results show that, comparing with the results of SCFQKD protocol with perfect vacuum sources, the key rates and secure distance are only slightly decreased if the upper bound of the intensity of the imperfect vacuum source is less than $10^{-8}$ which can be achieved in experiment by two-stage intensity modulator. We also show that the two-way classical communication can be used to the data post-processing of SCFQKD protocol to improve the key rate. Specially, the active odd-parity pairing method can improve the key rates in all distances by about two times and the secure distance by about 40 km. Give that the side channel security based on imperfect vacuum, this work makes it possible to realize side channel secure QKD with real devices.

We propose to adopt redundant space such as polarization mode in the sending-or-not-sending Twin-Field quantum key distribution (TF-QKD) in the Fock space. With the help of redundant space such as photon polarization, we can post-select events according to the outcome of the observation to the additional quantity. This compresses the bit-flip error rate in the post-selected events of the SNS protocol. The calculation shows that the method using redundant space can greatly improve the performance in practical TF-QKD, especially when the total number of pulses is small., Comment: This manuscript has been accepted for publication as a Regular Article in Physical Review Research

Gaussian boson sampling is originally proposed to show quantum advantage with quantum linear optical elements. Recently, several experimental breakthroughs based on Gaussian boson sampling pointing to quantum computing supremacy have been presented. However, due to technical limitations, the outcomes of Gaussian boson sampling devices are influenced severely by photon loss. Here, we present an efficient and practical method to reduce the negative effect caused by photon loss. With no hardware modifications, our method takes the data post-selection process that discards low-quality data according to our criterion to improve the performance of the final computational results, say part is better than whole. As an example, we show that the post-selection method can turn a GBS experiment that would otherwise fail in a ``non-classical test" into one that can pass that test. Besides improving the robustness of computation results of current GBS devices, this photon loss mitigation method may also benefit the further development of GBS-based quantum algorithms.

We present the method of decoy-state analysis after bit-flip error correction and using confidential observed numbers. Taking this tool we then construct a universal approach to sending-or-not-sending (SNS) protocol of twin-field quantum key distribution. In this improved protocol, the code bits are not limited to heralded events in time windows participated by pulses of intensity $\mu_z$ and vacuum. All kinds of heralded events can be used for code bits to distill the final keys. The number of intensities (3 or 4) and the kinds of heralded events for code bits are automatically chosen by the key rate optimization itself. Numerical simulation shows that the key rate rises drastically in typical settings, up to 80\% improvement compared with the prior results. Also, larger intensity value can be used for decoy pulses. This makes the protocol more robust in practical experiments.

Digital signatures can guarantee the unforgeability and transferability of the message. Different from classical digital signatures, whose security depends on computational complexity, quantum digital signatures (QDS) can provide information-theoretic security. We propose a general method of random pairing QDS (RP-QDS), which can drastically improve QDS efficiency. In a way, our random pairing method provide a tightened result of security level of QDS. In the method, the parity value of each pair is used for the outcome bit value. We present general formulas for fraction of untagged bits and error rates of the outcome bits. Random pairing can be applied as a fundamental method to improve the QDS efficiency for all existing quantum key distribution (QKD) protocols. We take sending-or-not-sending (SNS) QDS and side-channel-free (SCF) QDS as examples to demonstrate the advantage of random pairing through numerical simulation. Similar advantage with random pairing is also founded with decoy-state MDIQKD and also decoy-state BB84 protocol. We study the RP-SNS-QDS with finite data size through novel optimization. The numerical simulation results show that the signature rate can be increased by more than 100% under noisy channel using our random pairing method.

The sending-or-not-sending (SNS) protocol is one of the most major variants of the twin-field (TF) quantum key distribution (QKD) protocol and has been realized in a 511 km field fiber, the farthest field experiment to date. In practice, however, all decoy-state methods have unavoidable source errors, and the source errors may be non-random, which compromises the security condition of the existing TF-QKD protocols. In this study, we present a general approach for efficiently calculating the SNS protocol's secure key rate with source errors, by establishing the equivalent protocols through virtual attenuation and tagged model. This makes the first result for TF-QKD in practice where source intensity cannot be controlled exactly. Our method can be combined with the two-way classical communication method such as active odd-parity pairing to further improve the key rate. The numerical results show that if the intensity error is within a few percent, the key rate and secure distance only decrease marginally. The key rate of the recent SNS experiment in the 511 km field fiber is still positive using our method presented here, even if there is $\pm 9.5\%$ intensity fluctuation. This shows that the SNS protocol is robust against source errors.

Nuclear-spin qubits have long coherence time and are desirably applied into quantum information processing. However, the existing methods either fail to address single nucleus (such as nuclear magnetic resonance), or severely affect nuclear coherence time (such as electrical nuclear manipulation based on hyperfine stark effect, ENMHSE). Here we propose an electrical nuclear manipulation called nuclear electric resonance which can on the one hand address the single nuclear qubit, and on the other keep the long coherence time. Applying this, we construct universal quantum gates with external electric field. These universal gates are practicable for arbitrary $S\ge 1$ spin nuclei. Given the much longer coherence time of nuclear electric resonance, we improve the number of single-qubit operations by three orders of magnitude compared with that of ENMHSE., Comment: 6 pages

Twin-field quantum key distribution (TF-QKD) promises ultra-long secure key distribution which surpasses the rate distance limit and can reduce the number of the trusted nodes in long-haul quantum network. Tremendous efforts have been made towards implementation of TF-QKD, among which, the secure key with finite size analysis can distribute more than 500 km in the lab and in the field. Here, we demonstrate the sending-or-not-sending TF-QKD experimentally, achieving a secure key distribution with finite size analysis over 658 km ultra-low-loss optical fiber, improve the secure distance record by around 100 km. Meanwhile, in a TF-QKD system, any phase fluctuation due to temperature variation and ambient variation during the channel must be recorded and compensated, and all these phase information can then be utilized to sense the channel vibration perturbations. With our QKD system, we recovered the external vibrational perturbations on the fiber generated by an artificial vibroseis and successfully located the perturbation position with a resolution better than 1 km. Our results not only set a new distance record of QKD, but also demonstrate that the redundant information of TF-QKD can be used for remote sensing of the channel vibration, which can find applications in earthquake detection and landslide monitoring besides secure communication., Comment: 20 pages, 4 figures and 1 table

The quantum approximate optimization algorithm (QAOA) transforms a simple many-qubit wavefunction into one which encodes a solution to a difficult classical optimization problem. It does this by optimizing the schedule according to which two unitary operators are alternately applied to the qubits. In this paper, the QAOA is modified by updating the operators themselves to include local fields, using information from the measured wavefunction at the end of one iteration step to improve the operators at later steps. It is shown by numerical simulation on MaxCut problems that, for a fixed accuracy, this procedure decreases the runtime of QAOA very substantially. This improvement appears to increase with the problem size. Our method requires essentially the same number of quantum gates per optimization step as the standard QAOA, and no additional measurements. This modified algorithm enhances the prospects for quantum advantage for certain optimization problems., Comment: 16 pages, 13 figures

Quantum key distribution can provide unconditionally secure key exchange for remote users in theory. In practice, however, in most quantum key distribution systems, quantum hackers might steal the secure keys by listening to the side channels in the source, such as the photon frequency spectrum, emission time, propagation direction, spatial angular momentum, and so on. It is hard to prevent such kinds of attacks because side channels may exist in any of the encoding space whether the designers take care of or not. Here we report an experimental realization of a side-channel-free quantum key distribution protocol which is not only measurement-device-independent, but also immune to all side-channel attacks in the source. We achieve a secure key rate of 4.80e-7 per pulse through 50 km fiber spools., Comment: 23 pages, 5 figures

We present methods to strictly calculate the finite-key effects in quantum key distribution (QKD) with error rejection through two-way classical communication (TWCC) for the sending-or-not-sending twin-field protocol. Unlike the normal QKD without TWCC, here the probability of tagging or untagging for each two-bit random group is not independent. We rigorously solve this problem by imagining a virtual set of bits where every bit is independent and identical. We show the relationship between the outcome starting from this imagined set containing independent and identical bits and the outcome starting with the real set of non-independent bits. With explicit formulas, we show that simply applying Chernoff bound in the calculation gives correct key rate, but the failure probability changes a little bit.

The basic principle of quantum mechanics guarantee the unconditional security of quantum key distribution (QKD) at the cost of inability of amplification of quantum state. As a result, despite remarkable progress in worldwide metropolitan QKD networks over the past decades, long haul fiber QKD network without trustful relay has not been achieved yet. Here, through sending-or-not-sending (SNS) protocol, we complete a twin field QKD (TF-QKD) and distribute secure keys without any trusted repeater over a 511 km long haul fiber trunk linking two distant metropolitans. Our secure key rate is around 3 orders of magnitudes greater than what is expected if the previous QKD field test system over the same length were applied. The efficient quantum-state transmission and stable single-photon interference over such a long distance deployed fiber paves the way to large-scale fiber quantum networks., Comment: 32 pages, 8 figures

Quantum key distribution endows people with information-theoretical security in communications. Twin-field quantum key distribution (TF-QKD) has attracted considerable attention because of its outstanding key rates over long distances. Recently, several demonstrations of TF-QKD have been realized. Nevertheless, those experiments are implemented in the laboratory, remaining a critical question about whether the TF-QKD is feasible in real-world circumstances. Here, by adopting the sending-or-not-sending twin-field QKD (SNS-TF-QKD) with the method of actively odd parity pairing (AOPP), we demonstrate a field-test QKD over 428~km deployed commercial fiber and two users are physically separated by about 300~km in a straight line. To this end, we explicitly measure the relevant properties of the deployed fiber and develop a carefully designed system with high stability. The secure key rate we achieved breaks the absolute key rate limit of repeater-less QKD. The result provides a new distance record for the field test of both TF-QKD and all types of fiber-based QKD systems. Our work bridges the gap of QKD between laboratory demonstrations and practical applications, and paves the way for intercity QKD network with high-speed and measurement-device-independent security.

The purpose of fingerprinting is to compare long messages with low communication complexity. Compared with its classical version, the quantum fingerprinting can realize exponential reduction in communication complexity. Recently, the multi-party quantum fingerprinting is studied on whether the messages from many parties are the same. However, sometimes it is not enough just to know whether these messages are the same, we usually need to know the relationship among them. We provide a general model of quantum fingerprinting network, defining the relationship function $f^R$ and giving the corresponding decision rules. In this work, we take the four-party quantum fingerprinting protocol as an example for detailed analysis. We also choose the optimal parameters to minimize communication complexity in the case of asymmetric channels. Furthermore, we compare the multi-party quantum fingerprinting with the protocol based on the two-party quantum fingerprinting and find that the multi-party protocol has obvious advantages, especially in terms of communication time. Finally, the method of encoding more than one bit on each coherent state is used to further improve the performance of the protocol., Comment: 12 pages, 4 figures

We study the sending-or-not-sending (SNS) protocol with discrete phase modulation of coherent states. We first make the security of the SNS protocol with discrete phase modulation. We then present analytic formulas for key rate calculation. We take numerical simulations for the key rate through discrete phase modulation of both the original SNS protocol and the SNS protocol with two way classical communications of active-odd-parity pairing (AOPP). Our numerical simulation results show that only with $6$ phase values, the key rates of the SNS protocol can exceed the linear bound, and with $12$ phase values, the key rates are very close to the results of the SNS protocol with continuously modulated phase-randomization.

We propose a method named as double-scanning method, to improve the key rate of measurement-device-independent quantum key distribution (MDI-QKD) drastically. In the method, two parameters are scanned simultaneously to tightly estimate the counts of single-photon pairs and the phase-flip error rate jointly. Numerical results show that the method in this work can improve the key rate by $35\%-280\%$ in a typical experimental set-up. Besides, we study the optimization of MDI-QKD protocol with all parameters including the source parameters and failure probability parameters, over symmetric channel or asymmetric channel. Compared with the optimized results with only the source parameters, the all-parameter-optimization method could improve the key rate by about $10\%$.

Measurement-device-independent quantum key distribution (MDI-QKD), based on two-photon interference, is immune to all attacks against the detection system and allows a QKD network with untrusted relays. Since the MDI-QKD protocol was proposed, fibre-based implementations have been rapidly developed towards longer distance, higher key rates, and network verification. However, owing to the effect of atmospheric turbulence, MDI-QKD over free-space channel remains experimentally challenging. Here, by developing the robust adaptive optics system, high precision time synchronization and frequency locking between independent photon sources located far apart, we realised the first free-space MDI-QKD over a 19.2-km urban atmospheric channel, which well exceeds the effective atmospheric thickness. Our experiment takes the first step towards satellite-based MDI-QKD. Moreover, the technology developed here opens the way to quantum experiments in free space involving long-distance interference of independent single photons., Comment: 14 pages, 3 figures

For the unsorted database quantum search with the unknown fraction $\lambda$ of target items, there are mainly two kinds of methods, i.e., fixed-point or trail-and-error. (i) In terms of the fixed-point method, Yoder et al. [Phys. Rev. Lett. 113, 210501 (2014)] claimed that the quadratic speedup over classical algorithms has been achieved. However, in this paper, we point out that this is not the case, because the query complexity of Yoder's algorithm is actually in $O(1/\sqrt{\lambda_0})$ rather than $O(1/\sqrt{\lambda})$, where $\lambda_0$ is a known lower bound of $\lambda$. (ii) In terms of the trail-and-error method, currently the algorithm without randomness has to take more than 1 times queries or iterations than the algorithm with randomly selected parameters. For the above problems, we provide the first hybrid quantum search algorithm based on the fixed-point and trail-and-error methods, where the matched multiphase Grover operations are trialed multiple times and the number of iterations increases exponentially along with the number of trials. The upper bound of expected queries as well as the optimal parameters are derived. Compared with Yoder's algorithm, the query complexity of our algorithm indeed achieves the optimal scaling in $\lambda$ for quantum search, which reconfirms the practicality of the fixed-point method. In addition, our algorithm also does not contain randomness, and compared with the existing deterministic algorithm, the query complexity can be reduced by about 1/3. Our work provides an new idea for the research on fixed-point and trial-and-error quantum search., Comment: 11 pages, 2 figures, 1 table. Comments are warmly welcome!

We propose a novel entanglement-creation scheme in a multi-atom ensemble, named entanglement amplification, which converts unentangled states into entangled states and amplifies less-entangled ones to maximally-entangled Greenberger-Horne-Zeilinger (GHZ) states. The scheme starts with a multi-atom ensemble initialized in a coherent spin state. By shifting the energy of a particular Dicke state, we break the Hilbert space of the ensemble into two isolated subspaces to tear the coherent spin state into two components so that entanglement is introduced. After that, we utilize the isolated subspaces to further enhance the entanglement by coherently separating the two components. By single-particle Rabi drivings on atoms in a high-finesse optical cavity illuminated by a single-frequency light, 2000-atom GHZ states can be created with a fidelity above 80% in an experimentally achievable system, making resources of ensembles at Heisenberg limit practically available for quantum metrology., Comment: 6 pages and 4 figures

Twin field quantum key distribution promises high key rates at long distance to beat the rate distance limit. Here, applying the sending or not sending TF QKD protocol, we experimentally demonstrate a secure key distribution breaking the absolute key rate limit of repeaterless QKD over 509 km, 408 km ultra-low loss optical fibre and 350 km standard optical fibre. Two independent lasers are used as the source with remote frequency locking technique over 500 km fiber distance; Practical optical fibers are used as the optical path with appropriate noise filtering; And finite key effects are considered in the key rate analysis. The secure key rates obtained at different distances are more than 5 times higher than the conditional limit of repeaterless QKD, a bound value assuming the same detection loss in the comparison. The achieved secure key rate is also higher than that a traditional QKD protocol running with a perfect repeaterless QKD device and even if an infinite number of sent pulses. Our result shows that the protocol and technologies applied in this experiment enable TF QKD to achieve high secure key rate at long distribution distance, and hence practically useful for field implementation of intercity QKD., Comment: 17 pages, 10 figures and 8 tables

The sending-or-not-sending (SNS) protocol of the twin-field quantum key distribution (TFQKD) can tolerant large misalignment error and its key rate can exceed the bound of repeaterless QKD. But the original SNS protocol requires the two users to use the same source parameters. Here we propose a general protocol with asymmetric source parameters and give the security proof of this protocol. Our general protocol has a much better performance than that of the original SNS protocol when the channel of the system is asymmetric., Comment: submitted to PRA on Aug. 12

Odd-parity error rejection (OPER) can drastically improve the asymptotic key rate of sending-or-not-sending twin-field (SNS-TF) quantum key distribution (QKD). However, in practice, the finite key effects have to be considered for security. Here, we propose a zigzag approach to verify the phase-flip error of the survived bits after OPER. Based on this, we can take all the finite key effect efficiently in calculating the non-asymptotic key rate. Numerical simulation shows that our method here produces the highest key rate over all distances among all existing methods, improving the key rate by more than $100\%$ to $3000\%$ in comparison with different prior art methods with typical experimental setting. Also, we show that with the method here, the SNS-TF QKD can by far break the the absolute bound of repeater-less key rate with whatever detection efficiency. We can even reach a non-asymptotic key rate more than $40$ times of the practical bound and $13$ times of the absolute bound with $10^{12}$ pulses. Besides, we apply the McDiarmid inequality to estimate the phase flip error rate, further improving the key rate by more than $20\%$.

Quantum algorithms can enhance machine learning in different aspects. Here, we study quantum-enhanced least-square support vector machine (LS-SVM). Firstly, a novel quantum algorithm that uses continuous variable to assist matrix inversion is introduced to simplify the algorithm for quantum LS-SVM, while retaining exponential speed-up. Secondly, we propose a hybrid quantum-classical version for sparse solutions of LS-SVM. By encoding a large dataset into a quantum state, a much smaller transformed dataset can be extracted using quantum matrix toolbox, which is further processed in classical SVM. We also incorporate kernel methods into the above quantum algorithms, which uses both exponential growth Hilbert space of qubits and infinite dimensionality of continuous variable for quantum feature maps. The quantum LS-SVM exploits quantum properties to explore important themes for SVM such as sparsity and kernel methods, and stresses its quantum advantages ranging from speed-up to the potential capacity to solve classically difficult machine learning tasks., Comment: 9 pages and 0 figures

In [Phys. Rev. Lett. 113, 210501 (2014)], to achieve the optimal fixed-point quantum search in the case of unknown fraction (denoted by $\lambda$) of target items, the analytical multiphase matching (AMPM) condition has been proposed. In this paper, we find out that the AMPM condition can also be used to design the exact quantum search algorithm in the case of known $\lambda$, and the minimum number of iterations reaches the optimal level of existing exact algorithms. Experiments are performed to demonstrate the proposed algorithm on IBM's quantum computer. In addition, we theoretically find two coincidental phases with equal absolute value in our algorithm based on the AMPM condition and that algorithm based on single-phase matching. Our work confirms the practicability of the AMPM condition in the case of known $\lambda$, and is helpful to understand the mechanism of this condition., Comment: 13 pages, 6 figures, 1 table. Comments are warmly welcome!

The decoy-state method has been developed rapidly in quantum key distribution (QKD) since it is immune to photon-number splitting attacks. However, two basis detector efficiency asymmetry, which exists in realistic scenarios, has been ignored in the prior results. By using the recent 4-intensity decoy-state optimization protocol, we report the first implementation of high-rate QKD with asymmetric basis detector efficiency, demonstrating 1.9 to 33.2 times higher key rate than previous protocols in the situation of large basis detector efficiency asymmetry. The results ruled out an implicitly assumption in QKD that the efficiency of Z basis and X basis are restricted to be same. This work paves the way towards a more practical QKD setting.

On the basis of the existing trace distance result, we present a simple and efficient method to tighten the upper bound of the guessing probability. The guessing probability of the final key k can be upper bounded by the guessing probability of another key k', if k' can be mapped from the final key k. Compared with the known methods, our result is more tightened by thousands of orders of magnitude. For example, given a 10^{-9}-secure key from the sifted key, the upper bound of the guessing probability obtained using our method is 2*10^(-3277). This value is smaller than the existing result 10^(-9) by more than 3000 orders of magnitude. Our result shows that from the perspective of guessing probability, the performance of the existing trace distance security is actually much better than what was assumed in the past., Comment: published version (with typing error corrected)

We present improved method of sending-or-not-sending twin-field quantum key distribution (SNS TF-QKD) based on its structure and the application of error rejection. %And we present iteration formula for bit-flip error rate of the survived bits after error rejection. Taking the finite key effect into consideration with only $10^{11}$ total pulses, we show that the method here exceeds the absolute limit of repeater-less key rate with whatever detection efficiency. We also make comparative study of different protocols numerically. It shows that the method here presents advantageous results at long distance regime and large noise regime, asymptotically or non-asymptotically. Applying the de Finetti theorem, we present the iteration formula of phase-flip error rate for survived bits from odd-parity events only. With this, the performance of SNS protocol can be further improved a lot.

The Sending-or-Not-Sending protocol of the twin-field quantum key distribution (TF-QKD) has its advantage of unconditional security proof under any coherent attack and fault tolerance to large misalignment error. So far this is the only coherent-state based TF-QKD protocol that has considered finite-key effect, the statistical fluctuations. Here we consider the complete finite-key effects for the protocol and we show by numerical simulation that the protocol with typical finite number of pulses in practice can produce unconditional secure final key under general attack, including all coherent attacks. It can exceed the secure distance of 500 $km$ in typical finite number of pulses in practice even with a large misalignment error., Comment: Our results with finite number of pulses are secure under general attacks including whatever coherent attack

Channel loss seems to be the most severe limitation on the practical application of long distance quantum key distribution. The idea of twin-field quantum key distribution can improve the key rate from the linear scale of channel loss in the traditional decoy-state method to the square root scale of the channel transmittance. However, the technical demanding is rather tough because it requests single photon level interference of two remote independent lasers. Here, we adopt the technology developed in the frequency and time transfer to lock two independent lasers' wavelengths and utilize additional phase reference light to estimate and compensate the fiber fluctuation. Further with a single photon detector with high detection rate, we demonstrate twin field quantum key distribution through the sending-or-not-sending protocol with realistic phase drift over 300 km optical fiber spools. We calculate the secure key rates with finite size effect. The secure key rate at 300 km ($1.96\times10^{-6}$) is higher than that of the repeaterless secret key capacity ($8.64\times10^{-7}$)., Comment: 34 pages, 10 figures and 9 tables

We present a simple protocol where Alice and Bob only needs sending out a coherent state or not-sending out a coherent state to Charlie. There is no bases switching. We show that this protocol is both encoding-state-side-channel free to the source part and measurement-device-independent. We don't have to control exactly the whole space state of the light pulse, which is an impossible task in practice. The protocol is immune to all adverse due to encoding-state imperfections in side-channel space such as the photon frequency spectrum, emission time, propagation direction, spatial angular moment, and so on. Numerical simulation shows that our scheme can reach a side-channel-free result for quantum key distribution over a distance longer than 200 km given the single-photon-interference misalignment error rate of $20\%$., Comment: 12 pages, 2 figures

We present results of practical sending-or-not quantum key distribution. In real-life implementations, we need consider the following three requirements, a few different intensities rather than infinite number of different intensities, a phase slice of appropriate size rather than infinitely small size and the statistical fluctuations. We first show the decoy-state method with only a few different intensities and a phase slice of appropriate size. We then give a statistical fluctuation analysis for the decoy-state method. Numerical simulation shows that, the performance of our method is comparable to the asymptotic case for which the key size is large enough. Our results show that practical implementations of the sending-or-not quantum key distribution can be both secure and efficient.

We study the the optimization of full implementation of the four-intensity decoy-state Measurement-Device-Independent Quantum Key Distribution (MDIQKD) over asymmetric and unstable quantum channel.

Grover's algorithm achieves a quadratic speedup over classical algorithms, but it is considered necessary to know the value of $\lambda$ exactly [Phys. Rev. Lett. 95, 150501 (2005); Phys. Rev. Lett. 113, 210501 (2014)], where $\lambda$ is the fraction of target items in the database. In this paper, we find out that the Grover algorithm can actually apply to the case where one can identify the range that $\lambda$ belongs to from a given series of disjoint ranges. However, Grover's algorithm still cannot maintain high success probability when there exist multiple target items. For this problem, we proposed a complementary-multiphase quantum search algorithm, %with general iterations, in which multiple phases complement each other so that the overall high success probability can be maintained. Compared to the existing algorithms, in the case defined above, for the first time our algorithm achieves the following three goals simultaneously: (1) the success probability can be no less than any given value between 0 and 1, (2) the algorithm is applicable to the entire range of $\lambda$, and (3) the number of iterations is almost the same as that of Grover's algorithm. Especially compared to the optimal fixed-point algorithm [Phys. Rev. Lett. 113, 210501 (2014)], our algorithm uses fewer iterations to achieve success probability greater than 82.71\%, e.g., when the minimum success probability is required to be 99.25\%, the number of iterations can be reduced by 50\%., Comment: Editorially approved for publication in Physical Review A. 12 pages, 5 figures, 2 tables. Comments are warmly welcome!

We show that a side-channel-free (SCF) source does not have to be an ideal source by introducing the idea of mapping from ideal source. We propose a 3-state no-touch protocol for quantum key distribution (QKD) where Alice and Bob does not modulate any light sent out, the only thing the do is to send (or not send, in sending-or-not protocol). The reference light are from independent Lasers. We show that, the protocol is side-channel-free (i.e., both source side channel free and measurement device independent) and there is no modulation to the weak beams for QKD coding, except for sending or not sending. Calculation shows that one can reach a side-channel-free secure distance over 300 km using only coherent-state source. We use worst-case analysis which takes no limitation to the channel or detection loss for security. Our protocol is immune to all adverse due to side channels such as the photon frequency spectrum, emission time, propagation direction, spatial angular moment, and so on. Numerical simulations show that our scheme can reach a side-channel-free result for quantum key distribution over a distance longer than 200 km given the single-photon-interference misalignment error rate of 30%, and a distance longer than 300 km given the single-photon-interference misalignment error rate of 10%. Our no-touch idea can also apply to phase-coding twin-field QKD protocols. The no-touch idea also applies to twin-field QKD with phase coding., Comment: Long distance Side-Channel-Free QKD can be really done without entanglement or memory!

Based on the novel idea of twin-field quantum key distribution, we present a sending-or-not-sending twin-field fault tolerant quantum key distribution protocol. Our protocol can access a secure distance longer than 700 km even though the misalignment error rate is $15\%$. In the case of zero alignment error, our protocol can exceeds a secure distance of 800 km. Thanks to the novel idea of TF-QKD !, Comment: 13 pages, 3 figures