Applications of single photons to quantum communication and computing
单光子在量子通信和计算中的应用

The interest in quantum information emerged in the 1990s and 2000s leading to what was originally called quantum information processing and from the quantum technologies of late 2010. During the evolution of the field, single photons have become necessary building blocks for different types of quantum hardware. In this Review, we first discuss the definition and characterization of single photons and then continue with their generation and detection. Next, we overview the application of single photons in quantum communications and quantum computing. Further applications to quantum metrology, biology and experiments probing the foundations of quantum physics are discussed in an accompanying review in ref. ¹. To keep the main text reasonably short, we make use of the Supplementary Information and will guide the reader towards more specific or technical points when required or provide references for deeper insights. We also direct the readers to other existing reviews^2,3,4,5,6. This Review refers to, but does not discuss, entanglement in detail, but the reader can consult the Supplementary Information and review articles such as refs. ^7,8.
量子信息的兴趣出现在 1990 年代和 2000 年代，最初被称为量子信息处理，并源于 2010 年代末的量子技术。在该领域的发展过程中，单光子已成为不同类型量子硬件的必要构建块。在本综述中，我们首先讨论单光子的定义和特征，然后继续讨论其生成和检测。接下来，我们概述单光子在量子通信和量子计算中的应用。关于量子计量学、生物学以及探讨量子物理基础的实验的进一步应用将在参考文献 1 的附带综述中讨论。为了保持正文的合理简短，我们利用了补充信息，并在需要时引导读者关注更具体或技术性的要点，或提供更深入见解的参考文献。我们还引导读者查看其他现有的综述 2,3,4,5,6。本综述提及但未详细讨论纠缠，读者可以查阅补充信息和如参考文献 7,8 的综述文章。

What is a single photon and how do we characterize it?

Defining and characterizing a single photon is not an easy task and despite decades of research on the subject, the concept of a photon can still be puzzling, which sometimes makes its use in applications quite tricky.

The simplest definition is that a photon is the smallest, discrete amount of electromagnetic radiation or, as introduced by Albert Einstein in 1905 (ref. ⁹), photons are quanta of light. Photons are massless bosons and, as with any quantum object, they have both particle and wave character. Depending on the experimental conditions, one can either probe the particle or the wave behaviour of photons, keeping in mind that both are always present. For the corpuscular nature, one can use the so-called Fock state formalism in which the occupation of a mode (a solution of Maxwell’s equations) is quantized in eigenvalues of the photon number operator $\hat{N}$. A single photon is then the state of the mode with the eigenvalue N = 1 (see Supplementary Information).

For the wave nature, the spatiotemporal photon wavefunction formalism is used. In the 1960s, Glauber¹⁰ unified what is now called quantum optics by laying out the grounds for explaining the photon statistics of a laser, of the black-body radiation and of a single-photon source (SPS). Glauber’s formalism describes the wave behaviour of a photon by the first-order correlation function ${\psi }^E(x,t)$. In this formalism, one defines the wavefunction of a single photon, which again obeys Maxwell equations (see Supplementary Information).

An SPS can be characterized and identified by experimental means. Figure 1 depicts how two characteristic properties of an SPS are measured. Interestingly, these two experiments do test the corpuscular and wave properties of the light emitted by the source, respectively.

a, Measurement of g⁽²⁾/Hanbury Brown and Twiss experiment: the g⁽²⁾ function (that is, the intensity autocorrelation function) parameter is proportional to the number of simultaneous clicks in two detectors at the output of a beamsplitter (R, reflected; T, transmitted); when only one photon is present at a time t, the number of coincident clicks goes to zero when the time delay τ at the outputs is zero (see Supplementary Information); I is the classical light intensity and n is the quantum equivalent light intensity in terms of photon number. b, The Hong–Ou–Mandel effect can be used to characterize the distinguishability of photons¹⁴². It measures the coincident clicks at the output of a beamsplitter when one photon enters through each of the input port. When the photons share the same state (mode) and arrive at the same time at the beamsplitter, they are indistinguishable and the number of coincident clicks goes to zero (see Supplementary Information).

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The Hanbury Brown and Twiss experiment (Fig. 1a), in which a 50/50 beamsplitter is used, measures the so-called g⁽²⁾ function, the intensity autocorrelation function of the source, which is given by the expression:

Here I is the light intensity and τ is the time difference between the photocurrent of the two output ports of the beamsplitter. The two detectors will trigger or not depending on the arrival of a photon. For a single photon (N = 1 Fock state), no coincident (simultaneous) detections between the two output channels should occur at τ = 0 and, therefore, $g^{(2)}(0)=0$. The standard criterion for confirming an SPS is $g^{(2)}(0)<\frac{1}{2}$ (see Supplementary Information). This criterion is the one chosen in this Review to define an SPS.

The second measurement uses the so-called Hong–Ou–Mandel (HOM) interference effect to establish the degree of indistinguishability between two photons (Fig. 1b). In this measurement, one photon is sent into each of the two input ports of a beamsplitter; for perfect indistinguishability, one expects zero coincident detections between the two output ports of the beamsplitter, that is, the probability to measure one photon at each of the output ports is zero. When there is partial distinguishability in any degree of freedom (spatial mode, momentum, energy, time or polarization), one will get partial suppression of the coincidences. This is then a measure of the degree of mutual coherence (indistinguishability) of the two photons, which can originate either from the same or from different sources (see Supplementary Information).

In general, single-photon emission from an SPS is defined by four parameters: its purity (whether there is at most one photon at a time determined by the Hanbury Brown and Twiss experiment), its fidelity (how different are two photons from each other, measured by the HOM experiment), its generation rate (how many photons can the SPS provide per second) and the efficiency of the source (how many photons do I actually get at the end of my experimental set-up)¹¹.

As described in the subsequent sections, different applications will have different requirements for the properties of the used single photons. We should stress that for some cases, one can use weak coherent states as an approximation to a single photon. Weak coherent states are strongly attenuated laser beams, which, however, retain their statistical properties, in particular, g⁽²⁾(τ) = 1 and thus do not fall into the SPS category as such (see Supplementary Information). What is a constant though, for all the applications, is the fact that a source of single photons can provide zero, one or more than one photon. The more-than-one-photon case is the detrimental one for all the applications described in this Review and in ref. ¹.

How to produce single photons

There are at least three ways of producing single photons. The first one consists of using an emitter in which one can identify and isolate two energy levels that will deliver single photons on spontaneous emission (Fig. 2a). The second one is to use a so-called heralded SPS^12,13,14,15, as outlined in Fig. 2b. Here, via the mechanism of spontaneous parametric down-conversion (SPDC), one creates a pair of photons simultaneously in two spatially separate modes¹⁶. The detection of one of the photons in one place (trigger photons in Fig. 2b) signals that the other one will be there in the other path (heralded photons in Fig. 2b) within a very short time window. The role of the fast switch is to let pass (or not) the signal on the heralded arm when the trigger arm has registered photon on its side; this ensures that there is a high probability of having a single photon on the heralded arm (Fig. 2b). The third one uses an extremely nonlinear filter, often called the ‘photon blockade’¹⁷. Here one sends a weak laser beam into a medium that will only transmit single photons, but not pairs or higher-order states. The medium could be a cavity with an embedded quantum emitter. If the two-photon state of the cavity is out of resonance with the incoming field, only the single-photon part of the incoming photon state is transmitted.

a, Two-level system: a possible way to produce a single photon consists of exciting a two-level system and wait until it decays by emitting exactly one photon (see Supplementary Information). b, Principle of a heralded single-photon source^12,13. A nonlinear crystal source of ‘red’ photon pairs is pumped by a UV pulsed laser. At low pump power, the crystal emits mostly empty pair pulses with a small fraction (µ) of single pairs and an even smaller fraction (µ²) of double pairs. A detection by the heralding detector is used to gate single-photon pulses using a fast switch. This random train of pulses will be slightly degraded by false triggers arising from dark counts and by higher-order terms, where the latter can be reduced by lowering µ.

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For each of these ways of producing single photons, there are several technologies, materials and techniques to implement them, of which some are more interesting for some applications than for others, as we see in the next sections. One can classify two main categories for SPSs: the ‘natural’ category and the ‘engineered’ category. The natural category makes use of single atoms^18,19 or single molecules^20,21 given to us by nature, and the ‘engineered’ category involves solid-state sources of light from semiconductor, high-bandgap materials and 0D, 1D to 2D materials such as semiconductor quantum dots (QDs) (grown by physical methods) and colloid nanocrystals (grown by chemical methods), but also colour centres in a diamond matrix and defects in 2D materials. For example, an SPS that seems to be ticking all the boxes, using a semiconductor InGaAs quantum dot, achieved an efficiency of up to 57% at the output of the final optical fibre (where ‘useful’ photons are) with a fidelity of 97.5% for the average two-photon interference visibility (HOM) and with all the properties maintained up to a clock rate (under pulsed excitation and thus pulsed emission) of up to 1 GHz (ref. ²²).

There is a major difference between heralded and non-heralded SPSs. Heralded SPSs are usually obtained from SPDC (Fig. 2a), which follows a thermal statistics emission, meaning that for a given probability of obtaining a single pair of photons (and thus a single heralded photon in one arm), there is always a non-zero probability of obtaining two pairs of photons (or more). Thus, the generation rate  of such SPSs is always limited as one needs to work in low excitation levels. Multiplexing can circumvent this issue at the cost of a more complex system (we refer to ref. ²³ for more information).

More information about the various practical ways of producing single photons can be found in Supplementary Information, but we also refer the reader to previous reviews^2,3,5.

How to detect single photons

This Review cannot cover every aspect, so single-photon detectors are not discussed in depth. However, it is important to keep in mind that detecting single photons for applications is as important as being able to produce or to manipulate them efficiently. Depending on the applications, one must aim for different considerations as follows:

• detect all the incoming photons

• be able to detect any photon energy

• be able to resolve the number of incoming photons

• be able to detect them without background noise

The quest for the ‘perfect’ detector is still on, and many aspects have to be taken into account such as the required wavelength range, the speed, a wide dynamic range and so on. To fulfil one’s needs, one will have to choose different materials, electronics, optical interfaces with cryogenic temperatures, and so forth.

At present, most commercial detectors are solid-state detectors made of semiconductor or superconductor materials. There are currently four families of such single-photon detectors, which are photomultiplier tubes, single-photon avalanche photodiodes (made of Si for the visible range and InGaAs for the near-infrared (IR) range), superconducting nanowire single-photon detectors²⁴ and transition-edge sensors (TES; usually made of tungsten). Photomultiplier tubes are based on the photoelectric effect and used mostly in the UV to visible range. Single-photon avalanche photodiodes are semiconductor photodiodes working in the so-called Geiger mode, whereas superconducting nanowire single-photon detectors (near-IR range) and TES (mid-IR range) are superconducting materials working at low temperature close to the critical phase transition with the specificity for TES to be bolometers and to be able to resolve the number of incoming photons (up to 2 at the telecom wavelength of 1,550 nm so far²⁵).
目前，大多数商业探测器是由半导体或超导材料制成的固态探测器。目前有四类单光子探测器，包括光电倍增管、单光子雪崩光电二极管（可见光范围使用硅，近红外（IR）范围使用 InGaAs）、超导纳米线单光子探测器和过渡边缘传感器（TES；通常由钨制成）。光电倍增管基于光电效应，主要用于紫外到可见光范围。单光子雪崩光电二极管是以所谓的盖革模式工作的半导体光电二极管，而超导纳米线单光子探测器（近红外范围）和 TES（中红外范围）是工作在接近临界相变的低温下的超导材料，TES 的特性是作为辐射计并能够分辨入射光子的数量（截至目前在电信波长 1,550 nm 下可分辨最多 2 个光子）。

Introduction to quantum communication
量子通信导论

The field of quantum communication generally deals with the transmission of quantum information between different points in space. Photons are already the carriers of ‘classical’ information via optical fibres. They are also suitable to encode quantum bits (or qubits, see Supplementary Information)²⁶ enabling the transmission of quantum information over long distances owing to their favourable properties in terms of speed, coherence and low propagation loss.
量子通信领域通常涉及在空间不同点之间传输量子信息。光子已经是通过光纤传递“经典”信息的载体。它们也适合编码量子比特（或量子位，见补充信息）26，能够由于其在速度、相干性和低传播损耗方面的有利特性，实现长距离的量子信息传输。

Quantum cryptography²⁷, a method to secure classical data communication, which is to date the most advanced subfield in quantum communications, is based on a fundamental quantum phenomenon: the collapse of the wavefunction induced by a measurement of an observable (in time, energy or polarization). This results in the so-called no-cloning theorem and enables reliable detection of any eavesdropping attempt by a third party on transmitted streams of qubits — something that is impossible to do when transmitting classical bits encoded in laser pulses that contain large numbers of photons each (see Supplementary Information). Quantum key distribution (QKD) allows for the secure generation of keys between two distant points in a network, which are then used for encryption of classical data, being transmitted over a public channel. Many different schemes have been developed in this field with the first and most popular one being the BB84 protocol²⁸, in which qubits are generated and encoded in a random sequence of states in two different bases on one end (often called Alice) and detected on the other (often called Bob) with again a random choice between the two bases as used by Alice. Alice and Bob then publicly share their choice of basis for each transmitted qubit and only keep the ones where these were identical, allowing them to generate a secret key^29,30.
量子密码学是一种保护经典数据通信的方法，至今是量子通信中最先进的子领域，基于一个基本的量子现象：由对可观测量（在时间、能量或极化方面）的测量引起的波函数坍缩。这导致了所谓的不可克隆定理，并使得可靠检测第三方对传输的量子比特流的任何窃听尝试成为可能——这是在传输包含大量光子的激光脉冲编码的经典比特时无法做到的（见补充信息）。量子密钥分发（QKD）允许在网络中两个远程点之间安全生成密钥，这些密钥随后用于对通过公共通道传输的经典数据进行加密。在这一领域已经开发了许多不同的方案，其中第一个也是最流行的是 BB84 协议，在该协议中，量子比特在一端（通常称为爱丽丝）以随机状态序列在两个不同基底中生成和编码，并在另一端（通常称为鲍勃）被检测，鲍勃同样随机选择爱丽丝使用的两个基底中的一个。 爱丽丝和鲍勃随后公开分享他们对每个传输量子比特的基选择，并仅保留那些相同的选择，从而使他们能够生成一个秘密密钥 29,30。

More general-purpose quantum communication networks often referred to as ‘the quantum internet’^31,32 are required for interlinking distant quantum-computational devices such as processors and simulators (see the next sections) and the scalable transmission of qubits on a global scale. Although, using an attenuated laser, QKD protocols can be conveniently implemented using weak coherent states (WCSs)³³ (see Supplementary Information) and operation with true SPS would yield a boost in efficiency at most, implementations of high-level quantum network infrastructures based on quantum teleportation³⁴ such as quantum relays³⁵ and quantum repeaters³⁶ heavily rely on the availability of deterministic photon sources. The generation, distribution and storage of entanglement (see Supplementary Information) lies at the heart of these systems, with suitable single or entangled photon-pair sources being one of the key enabling technologies.
更通用的量子通信网络通常被称为“量子互联网”31,32，旨在连接远程量子计算设备，如处理器和模拟器（见下节），并在全球范围内可扩展地传输量子比特。尽管使用衰减激光，量子密钥分发（QKD）协议可以方便地利用弱相干态（WCSs）33（见补充信息）来实现，而使用真正的单光子源（SPS）则在效率上会有所提升，但基于量子隐形传态 34 的高水平量子网络基础设施的实现，如量子中继 35 和量子中继器 36，严重依赖于确定性光子源的可用性。纠缠的生成、分配和存储（见补充信息）是这些系统的核心，合适的单光子或纠缠光子对源是关键的使能技术之一。

Needs and requirements 需求与要求

As quantum communication protocols are usually linked to single-photon transmission (the quantum channel) at some point, photon loss is the limiting factor in implementations, giving rise to an increase in error rates and drop in efficiencies. Therefore, keeping the losses low by operating in the optimum wavelength bands is of utmost importance. For transmission over existing single-mode fibre networks, lowest loss is achieved in the so-called telecom S-band, C-band and L-band between 1,460 nm and 1,625 nm (ref. ³⁷), with the C-band centred around 1,550 nm being the most widely implemented choice. Operation in the telecom O-band (1,260–1,360 nm) has only slightly higher losses, but has the benefits of zero dispersion and freeing longer wavelengths for data traffic.
量子通信协议通常与某个时刻的单光子传输（量子通道）相关，因此光子损失是实现中的限制因素，导致错误率增加和效率下降。因此，通过在最佳波长范围内操作来保持损失低至关重要。在现有单模光纤网络中，最低损失发生在所谓的电信 S 波段、C 波段和 L 波段，波长范围在 1,460 nm 到 1,625 nm 之间（参考文献 37），其中以中心波长约为 1,550 nm 的 C 波段是最广泛实施的选择。电信 O 波段（1,260–1,360 nm）的损失仅略高，但具有零色散的优点，并为数据流量释放了更长的波长。

The transmission of single photons over free space is relevant for quantum communication links to satellites in low earth orbit. Here, operation at the long-wavelength end of the visible spectrum around 800 nm seems most favourable for lowest loss³⁸; however, operation in the telecom C-band has the benefits of less noise during daylight operation³⁹.
单光子在自由空间中的传输与低地球轨道卫星的量子通信链路相关。在这里，800 纳米附近的可见光谱长波长端的操作似乎最有利于最低损耗；然而，在通信 C 波段的操作在白天运行时具有较少噪声的优势。

For high data capacity of quantum channels and compatibility with state-of-the-art QKD technology, operation at gigahertz clock rates^40,41,42 of photon sources of <200 ps for efficient use of detector gate widths is a further important criterion. With future deployment of sources in compact end-user hardware and quantum network infrastructure consisting of a widely spread grid of (most likely inaccessible) nodes in mind, focus should be on developing robust device platforms guaranteeing simple and reliable long-term operation.
为了实现量子通道的高数据容量以及与最先进的量子密钥分发（QKD）技术的兼容性，光子源在低于 200 皮秒的高达千兆赫的时钟频率下运行，以有效利用探测器的门宽，是一个重要的标准。考虑到未来在紧凑的终端用户硬件和由广泛分布的（很可能无法接触的）节点组成的量子网络基础设施中部署源，重点应放在开发能够保证简单可靠的长期运行的稳健设备平台上。

Weak coherent state sources and photon-pair sources based on spontaneous processes such as SPDC (see Supplementary Information) have to run at very low generation rate for operation in the single-photon regime, typically limiting their efficiencies to well below 50%. This results in low bit rates especially when being used in schemes that rely on two-photon interference and coincidence detection such as projective Bell state measurements (see Supplementary Information and refs. ^34,35,36) and measurement device-independent QKD⁴³. Scalable quantum communication applications such as quantum repeaters (see Supplementary Information) ideally require deterministic operation with the generation of single or entangled photon pairs on demand. These advanced schemes, furthermore, rely on high visibility two-photon interference, making photon indistinguishability a crucial parameter for scalable quantum communication applications as well (see Supplementary Information). A summary of the most established physical systems for single-photon generation and their experimentally confirmed performance regarding the parameters as just discussed is shown in Table 1.
弱相干态源和基于自发过程（如自发参量下转换，见补充信息）的光子对源在单光子区域的操作中必须以非常低的生成速率运行，通常将其效率限制在 50%以下。这导致在依赖于双光子干涉和重合检测的方案中，尤其是投影贝尔态测量（见补充信息及参考文献 34,35,36）和测量设备无关的量子密钥分发（QKD）中，位速率较低。可扩展的量子通信应用，如量子中继（见补充信息），理想情况下需要在需求时生成单个或纠缠光子对的确定性操作。此外，这些先进方案依赖于高可见度的双光子干涉，使得光子不可区分性成为可扩展量子通信应用中的一个关键参数（见补充信息）。表 1 总结了最成熟的单光子生成物理系统及其在刚讨论的参数方面的实验确认性能。

Achieved milestones 达成的里程碑

In the following, we mainly focus on three well-established classes of single/entangled photon sources that have great potential for certain aspects of quantum communication applications and discuss conceptually different schemes using these sources, a small selection of which is shown in Fig. 3. The most widely studied sources have been based on SPDC⁴⁴ or four wave mixing⁴⁵ (see Supplementary Information). Although these sources are nowadays rather simple to operate, achieve high entanglement fidelities and can emit at any of the discussed communication windows, they are not sub-Poissonian (the statistics has smaller fluctuations than Poissonian statistics, see Supplementary Information) and require engineering of the spectral properties when it comes to photon indistinguishability^46,47. Maximum efficiencies are limited to 25% owing to the laws of thermal statistics, with progress in overcoming this limitation with a multiplexing approach for heralded SPSs⁴⁸.
在以下内容中，我们主要关注三类成熟的单光子/纠缠光子源，这些光子源在某些量子通信应用方面具有巨大潜力，并讨论使用这些光子源的概念上不同的方案，其中一小部分在图 3 中展示。最广泛研究的光子源基于自发参量下转换（SPDC）或四波混频（见补充信息）。尽管这些光子源如今操作相对简单，能够实现高纠缠保真度，并且可以在讨论的任何通信窗口发射，但它们并不是亚泊松的（其统计波动小于泊松统计，见补充信息），并且在光子不可区分性方面需要对光谱特性进行工程设计。由于热统计定律，最大效率限制在 25%，但通过多路复用方法克服这一限制的进展正在进行中，适用于预示单光子源（SPS）。

a, Quantum key distribution with time-bin quantum bits (qubits) over 35-km fibre using telecom quantum dot (QD) single-photon source (SPS)⁷³. b, Quantum key distribution using polarization-entangled photon pairs produced by spontaneous parametric down-conversion⁷⁴. c, Quantum relay using electrically driven QD-entangled photon-pair source (E-LED) with sender, Bell state analyser (BSM) and receiver separated by 350 m of optical fibre each. d, Photonic entanglement swapping with two entangled photon pairs subsequently generated from a QD⁸². e, Heralded entanglement of two distant atom SPSs (rubidium-87) via projective measurement⁹³. B, biexciton photon; BS, non-polarizing beamsplitter; D1–D4, detectors; FS, fibre stretcher; M, mirror; P, polarizer; PBS, polarizing beamsplitter; PC, polarization controller; SF, spectral filter; VBG, volume Bragg grating (spectral filter); WDM, wavelength division multiplexer; WP, wave plate; X, exciton photon; XX, biexciton photon. Panel a reprinted with permission from ref. ⁷³, AIP; panel b reprinted with permission from ref. ⁷⁴, APS; panel c reprinted from ref. ⁸¹, Springer Nature Limited; panel d reprinted from ref. ⁸² under a Creative Commons licence CC BY 4.0; panel e reprinted with permission from ref. ⁹³, AAAS.
量子密钥分发，使用电信量子点（QD）单光子源（SPS）在 35 公里光纤上进行时间-bin 量子比特（qubit）分发。使用由自发参数下转换产生的偏振纠缠光子对进行量子密钥分发。量子中继，使用电驱动的 QD 纠缠光子对源（E-LED），发送方、贝尔态分析仪（BSM）和接收方之间各隔 350 米光纤。光子纠缠交换，两个纠缠光子对随后由一个 QD 生成。通过投影测量实现两个远程原子 SPS（铷-87）的预示纠缠。B，双激子光子；BS，非偏振分束器；D1–D4，探测器；FS，光纤拉伸器；M，镜子；P，偏振器；PBS，偏振分束器；PC，偏振控制器；SF，光谱滤波器；VBG，体布拉格光栅（光谱滤波器）；WDM，波长分复用器；WP，波片；X，激子光子；XX，双激子光子。面板 a 经授权转载自参考文献 73，AIP；面板 b 经授权转载自参考文献 74，APS；面板 c 转载自参考文献 81，Springer Nature Limited；面板 d 转载自参考文献。 在创意共享许可证 CC BY 4.0 下使用；面板 e 经 93 号参考文献 AAAS 许可转载。

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The most promising deterministic counterparts are sources on the basis of single semiconductor QDs (see Supplementary Information), which can be operated for emission of single or entangled photon pairs as well^49,50,51. Emission in the most important communication windows has been demonstrated^52,53,54,55, as well as operation at clock rates up to 3 GHz (refs. ^56,57). Optically resonantly excited QD SPSs have set a new standard in photon indistinguishability^58,59 and efficiency^3,22,60. The so-called Ekert protocol⁶¹ for QKD using entangled photons was also realized using QDs with the biexciton/exciton cascade for the creation of the entangled pairs^62,63 (see Supplementary Information). A major advantage of QD-based sources is their potential for simple electrical operation without the need of lasers. Even though this generally comes at the cost of reduced photon coherence, critical for scalable quantum communication schemes, the impact on timescales involved in high-speed pulsed operation is likely to be less significant, making these prime candidates for robust and safely deployable devices in quantum communication applications.
最有前景的确定性对应物是基于单个半导体量子点（见补充信息）的源，这些源可以用于发射单个或纠缠光子对。已证明在最重要的通信窗口中发射光子，并且在高达 3 GHz 的时钟频率下运行。光学共振激发的量子点单光子源在光子不可区分性和效率方面设定了新的标准。使用纠缠光子的量子密钥分发（QKD）所称的 Ekert 协议也通过量子点的双激子/激子级联实现了纠缠对的创建（见补充信息）。基于量子点的源的一个主要优势是其无需激光即可简单电气操作的潜力。尽管这通常以降低光子相干性为代价，而光子相干性对于可扩展的量子通信方案至关重要，但在高速脉冲操作中涉及的时间尺度上的影响可能不那么显著，使这些源成为量子通信应用中稳健且安全可部署设备的主要候选者。

Another class of photon source, especially important for quantum communication, is based on single laser-cooled atoms or defect centres in diamond, which have generally in common that a static spin qubit is coupled to the quantum state of an emitted single photon^64,65,66 (see Supplementary Information). Despite the lack of compatibility with fibre wavelength telecommunication bands and only low operation speeds owing to the used long-lived excited states, these sources have a great potential for quantum repeater applications as they are in fact static quantum memories often with the option for deterministic high-fidelity state manipulation and readout, a key task for implementation of quantum repeater chains with entanglement swapping over large distances (see  Supplementary Information).
另一类光子源，特别是对量子通信至关重要的，基于单个激光冷却原子或钻石中的缺陷中心，这些源通常具有共同点，即静态自旋量子比特与发射的单光子的量子态耦合（见补充信息）。尽管与光纤波长电信频段不兼容，并且由于使用了长寿命激发态而仅具有低操作速度，这些源在量子中继应用中具有巨大潜力，因为它们实际上是静态量子存储器，通常具有确定性高保真状态操控和读出的选项，这是实现大距离纠缠交换的量子中继链的关键任务（见补充信息）。

Quantum cryptography with single-photon sources
单光子源的量子密码学

Owing to the lack of portable SPSs for many years, the field of quantum cryptography has adapted to the use of WCS laser sources, which are nowadays the standard workhorse in almost every QKD system. The implementation of the decoy protocol³³ allows for the detection of an eavesdropping attempt, even though weak coherent laser pulses in the single-photon regime are used for transmission of qubits (see Supplementary Information). Current state of the art with this technology are systems operating at gigahertz clock rates with secure (quantum) bit rates above 10 Mb s^–1 (ref. ⁶⁷) and maximum transmission distances of more than 240 km in fibre⁶⁸ and 1,200 km in free space⁶⁹.
由于多年来缺乏便携式单光子源，量子密码学领域已适应使用宽谱激光源，这些激光源如今几乎成为每个量子密钥分发系统的标准工作马。诱饵协议的实施允许检测窃听尝试，即使在单光子范围内使用弱相干激光脉冲进行量子比特的传输（见补充信息）。目前该技术的最新进展是以千兆赫时钟频率运行的系统，安全（量子）比特率超过 10 Mb s–1（参考文献 67），在光纤中最大传输距离超过 240 公里，在自由空间中达到 1200 公里。

An important point for the integration of quantum communication hardware in existing communication networks is the capability of multiplexing with classical data traffic^70,71,72. This can be reliably achieved by sticking to wavelength division multiplexing methods as they are being used in classical communication, requiring future SPSs to comply with the same classical wavelength standards.
在现有通信网络中集成量子通信硬件的一个重要点是能够与经典数据流量进行复用 70,71,72。这可以通过坚持使用经典通信中采用的波分复用方法来可靠地实现，这要求未来的单光子源（SPS）遵循相同的经典波长标准。

Even though the security of current QKD technology is not compromised using WCS sources, their operation is intrinsically probabilistic, resulting in limited efficiencies around 30–40% for typically used mean photon numbers of 0.4–0.5 for the signal state. As soon as deterministic SPSs are surpassing this boundary, there is a clear benefit for their use in quantum cryptography, resulting in higher secure bit rates especially over longer transmission distances⁷³. Figure 3a shows an example for the implementation of point-to-point QKD, where qubits are encoded in the differential phase of time-bin qubits that are generated with a QD SPS on one end (Alice) and detection on the other end (Bob), much the same way as it is being done using WCS sources. Note that the use of nonlinear entangled photon pair sources such as SPDC has also proven to be suitable for QKD, protecting from the so-called photon-number splitting attacks by the inherent randomness of the quantum mechanical measurement process^74,75. Figure 3b shows an example of entanglement-based QKD using an SPDC source that symmetrically distributes entangled quantum bits between two points (Alice and Bob). Here, measurements in random bases at both ends of the link are being performed to generate encryption keys. Even though the fundamentally limited efficiency for entangled pair production in these can be overcome by multiplexing schemes²³, an obvious advantage is expected from deterministically operated sub-Poissonian entangled photon-pair sources such as QDs.
尽管当前量子密钥分发（QKD）技术在使用弱相干光源（WCS）时的安全性并未受到影响，但其操作本质上是概率性的，导致在通常使用的信号态平均光子数为 0.4-0.5 时，效率仅限于 30-40%。一旦确定性单光子源（SPS）超越这一界限，其在量子密码学中的应用将显现出明显的优势，尤其是在较长的传输距离上，能够实现更高的安全比特率。图 3a 展示了点对点 QKD 的实现示例，其中量子比特编码在由量子点单光子源（QD SPS）在一端（爱丽丝）生成的时间-bin 量子比特的差分相位中，另一端（鲍勃）进行检测，方式与使用 WCS 光源时相似。需要注意的是，使用非线性纠缠光子对源（如自发参量下转换，SPDC）也被证明适合用于 QKD，通过量子力学测量过程的固有随机性来防护所谓的光子数分裂攻击。图 3b 展示了使用 SPDC 源的基于纠缠的 QKD 示例，该源在两个点（爱丽丝和鲍勃）之间对纠缠量子比特进行对称分配。 在这里，链接两端的随机基测量正在进行，以生成加密密钥。尽管在这些情况下，纠缠对的产生效率的根本限制可以通过多路复用方案克服，但显然可以从确定性操作的亚泊松纠缠光子对源（如量子点）中获得明显的优势。

Entanglement-based quantum communication
基于纠缠的量子通信

Quantum communication applications going beyond simple point-to-point encryption mostly require the distribution of entanglement in some form (see Supplementary Information). In a chain of quantum relays³⁵, qubits are consecutively teleported from relay station to relay station with the projective Bell state measurement (see Supplementary Information) in one node acting as heralding event for the arrival of a single qubit in the following node. This effectively reduces some of the noise mainly induced by detector dark counts, resulting in longer overall transmission distances.
量子通信应用超越简单的点对点加密，通常需要以某种形式分发纠缠（见补充信息）。在一系列量子中继中，量子比特通过投影贝尔态测量从一个中继站连续传送到下一个中继站，其中一个节点的测量结果作为下一个节点中单个量子比特到达的信号。这有效地减少了主要由探测器暗计数引起的一些噪声，从而实现更长的整体传输距离。

First demonstrations of this scheme were done using SPDC sources^34,76,77 with the drawback that suppression of multipair emissions requires operation at very low generation rate, resulting in a low efficiency. Proof-of-principle experiments have also been done using QD-entangled photon-pair sources^{78,79,80,81,82,83}, which are lacking this fundamental imperfection and have the potential for fully deterministic operation in the near future⁸⁴. An example of experimental implementation of a quantum relay using an entangled light emitting diode as sub-Poissonian photon-pair source is shown in Fig. 3c, in which WCS quantum bits generated at the sender on the left are teleported to the receiver on the right.
该方案的首次演示是使用自发参量下转换（SPDC）源进行的，缺点是抑制多对发射需要在非常低的生成率下操作，导致效率低下。原理验证实验也使用了量子点纠缠光子对源进行，这些源缺乏这种基本缺陷，并且在不久的将来有潜力实现完全确定性的操作。图 3c 展示了使用纠缠发光二极管作为亚泊松光子对源的量子中继的实验实现示例，其中在左侧发送器生成的 WCS 量子比特被传送到右侧接收器。

Purely photonic quantum relays do not solve the scalability problem owing to photon loss as single qubits still have to physically propagate through optical fibres. By contrast, quantum memory-assisted repeaters are expected to solve this problem via the teleportation of quantum bits over large distances after previous distribution of entanglement between the end points of a link. Long-distance entanglement is generated via entanglement swapping between intermediate quantum repeater nodes (see Supplementary Information). An experimental scheme for demonstrating photonic entanglement swapping using two entangled photon pairs generated from a QD-entangled photon-pair source is shown in Fig. 3d. Note that if quantum repeaters do help to ensure secure communication over greater distances, they do not help in increasing the data rate as this is still provided by the original source of photonic qubits.
纯光子量子中继并未解决可扩展性问题，因为光子损失使得单量子比特仍需通过光纤进行物理传播。相比之下，量子存储辅助的中继预计将通过在链接端点之间先前分发的纠缠后，进行量子比特的远距离传送来解决此问题。远距离纠缠是通过中间量子中继节点之间的纠缠交换生成的（见补充信息）。图 3d 展示了一种使用来自量子点纠缠光子对源生成的两个纠缠光子对进行光子纠缠交换的实验方案。需要注意的是，如果量子中继确实有助于确保更大距离的安全通信，它们并未帮助提高数据传输速率，因为这仍由原始光子量子比特源提供。

One can distinguish between three different approaches for the implementation of quantum repeaters. The first approach keeps photon sources and memories for repeaters completely separate, making it essential to achieve a good interface between both technologies for efficient entanglement transfer from photons to static qubits. Preliminary experiments for controlled absorption of single photons by quantum memory systems have been carried out using a heralded SPS (see Supplementary Information) and a single atom⁸⁵ and rare-earth doped crystal quantum memories^86,87, an atom-based SPS and quantum memory⁸⁸ and a QD-based SPS and quantum memory⁸⁹. Clearly, high efficiency and therefore deterministic operation of both photon sources and quantum memories are as important as good bandwidth matching for scalable implementation.
可以区分三种不同的量子中继器实现方法。第一种方法将光子源和中继器的存储器完全分开，因此必须在这两种技术之间实现良好的接口，以便有效地将纠缠从光子转移到静态量子比特。已经进行了一些初步实验，使用了预示性单光子源（见补充信息）和单原子以及稀土掺杂晶体量子存储器，原子基单光子源和量子存储器，以及量子点基单光子源和量子存储器。显然，高效率，因此光子源和量子存储器的确定性操作与良好的带宽匹配同样重要，以实现可扩展的实施。

The second approach makes use of SPSs that can act as a quantum memory at the same time, emitting single photons that are entangled with a single or collective spin, for example. Distant entanglement between two nodes is then generated via a projective measurement of the photons^90,91, generating a heralding event based on coincidence detection in this purely probabilistic approach, effectively suppressing noise. This makes it feasible even with non-deterministic low-efficiency sources, which is reflected in the large number of successful implementations of this scheme using various types of SPSs^{64,92,93,94,95}, with the most prominent recent demonstrations being loop-hole free violation of the Bell inequality^96,97. Figure 3e illustrates an example for an experiment, generating distant entanglement between two single-atom SPSs (in this case rubidium atoms) via a projective measurement of emitted photons. Sources based on nitrogen-vacancy (NV) centres and single atoms have the advantage of long spin coherence times with deterministic manipulation and readout, but are not suitable for gigahertz operation speeds such as QDs, owing to typically long excited state lifetimes. Atoms and QDs deliver best photon indistinguishabilities required for high success probabilities, whereas solid-state based sources as QDs and NV centres are best to integrate. For more on this topic, we refer to a review on using QDs for quantum cryptography⁹⁸.
第二种方法利用可以同时作为量子存储器的单光子源（SPS），例如，发射与单个或集体自旋纠缠的单光子。然后，通过对光子的投影测量生成两个节点之间的远程纠缠，通过这种纯粹的概率方法基于重合检测生成预示事件，有效抑制噪声。这使得即使在非确定性低效率源的情况下也可行，这在使用各种类型的单光子源实施该方案的大量成功案例中得到了体现，最近最显著的演示是无漏洞违反贝尔不等式。图 3e 展示了一个实验示例，通过对发射光子的投影测量，在两个单原子单光子源（在此情况下为铷原子）之间生成远程纠缠。基于氮空位（NV）中心和单原子的源具有长自旋相干时间、确定性操控和读出等优点，但由于通常较长的激发态寿命，不适合千兆赫兹的操作速度，如量子点（QDs）。 原子和量子点提供了高成功概率所需的最佳光子不可区分性，而基于固态的源，如量子点和氮空位中心，则更适合集成。有关此主题的更多信息，请参阅关于使用量子点进行量子密码学的综述。

The third, more recent, approach for a quantum repeater is purely photonic and requires no quantum memories⁹⁹, at least for all intermediate nodes in a long-distance link when establishing a universal quantum communication channel that goes beyond QKD. The striking difference between previously discussed approaches and this scheme, which requires the distribution of large-scale photonic cluster states similar to the ones used in all-optical quantum computation²⁶, is that no classical communication between repeater nodes is required; this is one of the main reasons for the need of long qubit storage times in the other repeater schemes. To date, this repeater scheme is the only one that has been successfully implemented in an experiment¹⁰⁰, using six SPDC photon-pair sources. Regarding the generation of cluster states using deterministic SPSs, there has been promising progress using QD SPSs as well^101,102,103. Note that there are promising intermediate solutions to the scalability problem of QKD to long distances via trusted nodes on satellites¹⁰⁴ or new QKD schemes based on untrusted nodes such as measurement device independent-QKD⁴³ or twin-field QKD¹⁰⁵.
第三种较新的量子中继器方法是纯光子型的，不需要量子存储器 99，至少在建立超越量子密钥分发（QKD）的通用量子通信通道时，对于长距离链路中的所有中间节点都是如此。之前讨论的方法与该方案之间显著的区别在于，该方案需要分发大规模的光子簇态，类似于在全光量子计算中使用的状态 26，而不需要中继节点之间的经典通信；这是其他中继方案中需要长量子比特存储时间的主要原因之一。迄今为止，这种中继方案是唯一在实验中成功实现的方案 100，使用了六个自发参量下转换（SPDC）光子对源。关于使用确定性单光子源（SPS）生成簇态，使用量子点（QD）单光子源的研究也取得了有希望的进展 101,102,103。请注意，针对量子密钥分发（QKD）在长距离上的可扩展性问题，存在通过卫星上的可信节点 104 或基于不可信节点的新 QKD 方案（如测量设备独立 QKD43 或双场 QKD105）的一些有希望的中间解决方案。

Field demonstrations of quantum communication
量子通信的现场演示

There have been many demonstrations of QKD on existing optical fibre networks. As truly portable and efficient deterministic SPSs are still not widely available, the vast majority of these has been done with either WCS sources or probabilistic heralded sources. Regarding the most basic application — QKD with WCS — there have been multiple demonstrations ranging from fibre-based point-to-point links^106,107,108 to multi (trusted)-node networks^{109,110,111,112,113,114} and free-space links^115,116,117, including demonstrations of satellite-based QKD^118,119. Despite the higher complexity of SPDC sources, there have been several field trials of entanglement-based QKD^{74,120,121,122}. These have been successfully implemented using probabilistic sources over free space on earth^123,124, from a satellite¹²⁵ and over installed fibres^126,127,128. Photonic quantum teleportation over macroscopic distances has also been demonstrated in several experiments^{129,130,131,132}.
在现有光纤网络上已经进行了许多量子密钥分发（QKD）的演示。由于真正便携且高效的确定性单光子源（SPS）仍未广泛可用，因此绝大多数演示都是使用光子数态（WCS）源或概率性预示源进行的。关于最基本的应用——使用 WCS 的 QKD——已经进行了多次演示，涵盖了从基于光纤的点对点链接到多（可信）节点网络和自由空间链接的多个示例，包括基于卫星的 QKD 的演示。尽管自发参量下转换（SPDC）源的复杂性更高，但也进行了几次基于纠缠的 QKD 的现场试验。这些试验成功地在地球的自由空间、从卫星以及通过已安装的光纤上使用概率性源实施。光子量子传送在宏观距离上的实验也已在多个实验中得到验证。

Owing to the currently much higher complexity of sources generating true single photons in the right wavelength bands, there are very few field demonstrations so far. Although not using a standard telecom network, macroscopic entanglement of distant¹³³ matter qubits via two-photon interference has been demonstrated over distances of few hundreds of metres^96,97. The incompatibility with the standard telecommunication bands is currently the main limitation for extension of these schemes to larger distances. In this regard, quantum frequency conversion is a possible solution^134,135,136, albeit at the cost of increased complexity. Another promising, and potentially simpler, approach uses QD single-photon and entangled pair sources emitting directly at telecom wavelengths, which have been demonstrated in standard telecom networks on a metropolitan scale^137,138. Table 1 provides the current state of the art of various SPSs in terms of wavelength emission, clock rate, photon purity, transmission distance (distance between the source and the detector) as well as the type of excitation and the operation mode whether it is deterministic (det.) or probabilistic (prob.).
由于目前在正确波长范围内生成真正单光子的源的复杂性大大增加，迄今为止现场演示非常少。尽管没有使用标准电信网络，但通过双光子干涉在几百米的距离上已经演示了远程物质量子比特的宏观纠缠。与标准电信波段的不兼容性目前是将这些方案扩展到更大距离的主要限制。在这方面，量子频率转换是一种可能的解决方案，尽管这会增加复杂性。另一种有前景且可能更简单的方法是使用量子点单光子和纠缠对源，直接在电信波长下发射，这已在城市规模的标准电信网络中得到验证。表 1 提供了各种单光子源在波长发射、时钟频率、光子纯度、传输距离（源与探测器之间的距离）、激发类型以及操作模式（确定性或概率性）方面的最新技术状态。

Future improvements 未来的改进

Although all building blocks required for quantum communication have been demonstrated using different kinds of SPSs, real-world applications with these sources are still lacking far behind technology on the basis of WCS or nonlinear photon sources. This situation currently prevents the implementation of scalable quantum communication links, requiring deterministic sub-Poissonian SPS or entangled photon-pair sources, which would finally culminate in the establishment of a global-scale quantum network¹³⁹.
尽管已经使用不同类型的单光子源（SPS）展示了量子通信所需的所有构建块，但基于波长压缩（WCS）或非线性光子源的技术仍然远远领先于这些源的实际应用。这种情况目前阻碍了可扩展量子通信链路的实现，要求确定性亚泊松单光子源或纠缠光子对源，最终将促成全球规模量子网络的建立。

Research still needs to be done on the physical systems to get a deterministic SPS with, at the same time, high efficiency (>50%), high photon indistinguishability, gigahertz repetition rate and emission at telecom wavelength in a laboratory. Apart from that, the main challenge for application in communication networks will be the deployability of the surrounding (currently often laboratory filling) technology required to operate the emitting element in an SPS. This mainly requires further miniaturization and integration of stable laser systems as they are being required for resonantly optically excited or laser-cooled emitters (QDs, NV centres and atoms), miniaturization of frequency converters that are required for SPS emitting in the visible range (NV centres and atoms) and development of compact cryocoolers as they are required for most solid-state-based sources (QDs and NV centres). Although these are only technical challenges, currently emerging technologies, such as, for example, carbon nanotubes¹⁴⁰ or 2D materials¹⁴¹ (Table 1), might not require any of these surrounding technologies and could therefore provide a significant boost for quantum communications. However, current research on these systems is still at a too-early stage to make any predictions.
研究仍需在物理系统上进行，以获得具有确定性光子源（SPS），同时具备高效率（>50%）、高光子不可区分性、千兆赫兹重复率和在实验室中以通信波长发射的能力。除此之外，应用于通信网络的主要挑战将是部署所需的周边技术（目前通常是实验室设备），以操作 SPS 中的发射元件。这主要需要进一步的小型化和稳定激光系统的集成，因为这些系统是共振光学激发或激光冷却发射器（量子点、氮缺陷中心和原子）所需的，还需要小型化在可见光范围内发射 SPS 所需的频率转换器（氮缺陷中心和原子），以及开发紧凑型制冷机，因为大多数基于固态的光源（量子点和氮缺陷中心）都需要这些设备。尽管这些仅是技术挑战，但目前新兴的技术，例如碳纳米管或二维材料，可能不需要任何这些周边技术，因此可能为量子通信提供显著的推动。 然而，目前对这些系统的研究仍处于过早的阶段，无法做出任何预测。

Introduction to quantum computing
量子计算导论

Another major application of single photons is in quantum computing. Photons can be easily engineered into superposition states of modes (path and polarization), allowing arbitrary states on the so-called Bloch sphere (used to represent any arbitrary qubit, see  Supplementary Information) to be generated. Furthermore, single-qubit and two-qubit gates can be implemented with photons. Once placed into a superposition state, photons are remarkably resilient to decoherence. A photon created in a polarization state in the Crab Nebula will remain polarized for the 8,000 years it takes to get to Earth, for instance. What is challenging, however, is that photons are prone to be lost by absorption or scattering out of the mode.
单光子的另一个主要应用是在量子计算中。光子可以很容易地被工程化为模式（路径和偏振）的叠加态，从而允许在所谓的布洛赫球上生成任意态（用于表示任何任意量子比特，见补充信息）。此外，可以用光子实现单量子比特和双量子比特门。一旦处于叠加态，光子对去相干具有显著的抗性。例如，在蟹状星云中产生的偏振态光子将在到达地球的 8000 年中保持偏振。然而，具有挑战性的是，光子容易因吸收或散射而丢失。

Needs and requirements 需求与要求

Creating a well-defined and (relatively) long-lived qubit is the first step. Next, interactions between qubits to create multiqubit entanglement are needed. But photons only weakly interact with the environment, thus creating photon–photon interactions with just linear elements is almost impossible. However, the realization of high visibility two-photon interference via the HOM effect (see Supplementary Information)¹⁴² has led to the development of two-qubit gates operating on so-called post-selection^143,144. Here, the gate success probability is limited to typically less than 50% (see  Supplementary Information and refs. ^5,145,146). Post-selection in this context means that the gate is successful when a specific measurement pattern is recorded. In principle, high-fidelity gates can be built, and the loss implied by the limited efficiency of the process can then be mitigated by adding additional single-photon resources¹⁴³. However, for quantum computing, there is a requirement for scalability as realistic calculations require many qubits. Although a quantum advantage can be achieved with less than 100 qubits¹⁴⁷, arbitrary and useful algorithms will require millions of qubits. In such theoretical machines based on linear-optical gates^148,149,150, many photons would be used to encode single qubits such that error correction can be used to ‘protect’ qubits against decoherence, low-fidelity gates and loss processes. This scalability then places exceptional requirements on the efficiency of production of single photons, the loss statistics of the optical circuits and the fidelity of the interference operations¹⁵¹. We list a set of requirements in Box 1, noting that these requirements are much more stringent than those for other applications such as communications or metrology¹.
创建一个定义明确且（相对）寿命较长的量子比特是第一步。接下来，需要量子比特之间的相互作用以创建多量子比特纠缠。然而，光子与环境的相互作用非常微弱，因此仅通过线性元件创建光子-光子相互作用几乎是不可能的。然而，通过 HOM 效应实现高可见度的双光子干涉（见补充信息）142，促成了基于所谓后选择的双量子比特门的发展 143,144。在这里，门的成功概率通常限制在 50%以下（见补充信息及参考文献 5,145,146）。在此背景下，后选择意味着当记录到特定的测量模式时，门是成功的。原则上，可以构建高保真度的门，而由过程效率有限所带来的损失可以通过增加额外的单光子资源来减轻 143。然而，对于量子计算，存在可扩展性的要求，因为现实计算需要许多量子比特。尽管可以在少于 100 个量子比特的情况下实现量子优势 147，但任意且有用的算法将需要数百万个量子比特。 在基于线性光学门的理论机器中，许多光子将被用来编码单个量子比特，以便可以使用错误纠正来“保护”量子比特免受退相干、低保真度门和损失过程的影响。这种可扩展性对单光子的生产效率、光学电路的损失统计以及干涉操作的保真度提出了特殊要求。我们在框 1 中列出了一组要求，指出这些要求比其他应用（如通信或计量）的要求要严格得多。

Achieved milestones 达成的里程碑

For single photons to enable quantum computing, one has to make sure that they have the right properties in terms of efficiency, purity and indistinguishability. The most prominent model of quantum computing is gate-based quantum computing (see Supplementary Information), in which quantum algorithms are decomposed into quantum gates acting on single or multiple qubits²⁶. In the context of photonic systems, the challenge, however, is that photons do not interact directly and thus the implementation of photonics-based quantum gates requires some workarounds. In 2001, ref. ¹⁴³ showed that efficient quantum computation with linear optics is possible by using beamsplitters, phase shifters, SPSs and photodetectors.
为了使单光子能够实现量子计算，必须确保它们在效率、纯度和不可区分性方面具有正确的特性。量子计算的最主要模型是基于门的量子计算（见补充信息），在该模型中，量子算法被分解为作用于单个或多个量子比特的量子门。在光子系统的背景下，挑战在于光子不直接相互作用，因此基于光子的量子门的实现需要一些变通方法。2001 年，参考文献 143 表明，通过使用分束器、相位移器、单光子源和光电探测器，线性光学下的高效量子计算是可能的。

The two ways to realize linear-optical quantum gates are via post-selection or by using additional ancilla photons and heralding measurements; the latter approach having the advantage that multiple gates can be applied in a successive manner. Post-selection means that photons enter a linear-optical circuit, pass through it and, at the output, only specific measurement configurations are selected (Fig. 4a). A typical configuration chosen here is that only the events with one photon per spatial output mode are considered. All other events (zero or more than one photon per spatial mode) are discarded. The remaining — post-selected — state has then undergone the targeted transformation. The disadvantage in this approach — besides the scaling — is that no further gates can then be applied to the output (as all the photons must be measured). Here, one possibility is to use additional photons and ancilla modes — ancilla photons — and doing a measurement of some of the output modes only. Upon the detection of a certain measurement pattern of the ancilla, one knows that a certain operation has been implemented — without the need to measure all photons — and thus, another operation can be applied to the remaining photons.
实现线性光学量子门的两种方法是通过后选择或使用额外的辅助光子和预示测量；后者的优点在于可以以连续的方式应用多个门。后选择意味着光子进入线性光学电路，经过电路后，在输出端仅选择特定的测量配置（图 4a）。这里选择的典型配置是仅考虑每个空间输出模式有一个光子的事件。所有其他事件（每个空间模式零个或多个光子）都被丢弃。剩下的——后选择的——状态则经历了目标变换。这种方法的缺点——除了规模问题——是无法对输出应用进一步的门（因为所有光子必须被测量）。在这里，一种可能性是使用额外的光子和辅助模式——辅助光子——并仅对一些输出模式进行测量。 在检测到辅助量子比特的某种测量模式时，可以知道某个操作已经被实施——无需测量所有光子——因此，可以对剩余的光子施加另一个操作。

a, The gate-based model, in which two-qubit gates require use of ancilla photons and/or post-selection. b, The one-way model, which is based on performing single-qubit measurements on a large-scale cluster state. The nodes represent physical qubits, and each row represents a logical qubit. The lines (connections) indicate where CPhase gates have been applied to generate the cluster states. First, the qubits on the very left are measured, the result is then fed-forward into the measurement instruction for the second column, then the third column and so on — until only one column is left. This becomes the output of the quantum circuit. c, Boson sampling is based on letting photons scatter through a linear-optical circuit. Although parts a and b are schemes for universal quantum computing, boson sampling is useful in solving certain types of problems. Figure reprinted with permission from ref. ¹⁴⁶, IOP.
a，基于门的模型，其中两个量子比特门需要使用辅助光子和/或后选择。b，单向模型，基于对大规模簇态进行单量子比特测量。节点代表物理量子比特，每一行代表一个逻辑量子比特。线（连接）指示应用了 C 相位门以生成簇态的位置。首先，最左侧的量子比特被测量，结果随后被反馈到第二列的测量指令中，然后是第三列，依此类推——直到只剩下一列。这成为量子电路的输出。c，玻色取样基于让光子通过线性光学电路散射。尽管部分 a 和 b 是通用量子计算的方案，但玻色取样在解决某些类型的问题中是有用的。图像经 146 号参考文献许可转载，IOP。

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Both CNOT (control-NOT; Supplementary Information) and CPhase (control phase; Supplementary Information) gates have been demonstrated using bulk optical components^144,152,153. As these set-ups require long-term stability and scalability is limited through the number of bulk components needed, a pathway to scaling up linear-optical gate-based approach is integration¹⁵⁴. Here, using circuits of integrated waveguides, optical components can be realized on a small footprint¹⁵⁵ and offer long-term stability that is crucial in this type of experiments. There are various platforms and technologies for realizing integrated optical circuits, for example, femtosecond writing in silica, silica-on-insulator, silicon-based approaches (silicon-on-insulator (Si) and silicon nitride), lithium niobate and gallium arsenide (see ref. ¹⁵⁴ for a concise review of various integrated platforms). Both integrated quantum gates have been demonstrated using different degrees of freedom such as path or polarization¹⁵⁶. Here, CPhase and CNOT gates have been shown as the basic building block of more elaborate quantum circuits or quantum algorithms¹⁵⁷; the required single-qubit gates are implemented using phase shifters in path encoding^{156,158,159,160,161} and wave plates or elements rotating polarization for polarization encoding^{162,163,164,165}. Also, it is possible to convert one encoding into the other such that both approaches can be used flexibly¹⁶⁵.
CNOT（控制非门；补充信息）和 CPhase（控制相位；补充信息）门已通过大宗光学元件得以实现 144,152,153。由于这些设置需要长期稳定性，并且通过所需的大宗元件数量限制了可扩展性，因此基于线性光学门的扩展路径是集成 154。在这里，通过使用集成波导电路，可以在小占地面积上实现光学元件 155，并提供在此类实验中至关重要的长期稳定性。实现集成光学电路有多种平台和技术，例如在二氧化硅中进行飞秒写入、硅基绝缘体、基于硅的方法（硅基绝缘体（Si）和氮化硅）、铌酸锂和砷化镓（参见参考文献 154 以获取各种集成平台的简要综述）。这两种集成量子门已通过不同的自由度（如路径或偏振）得以实现 156。 在这里，C 相位门和 CNOT 门被视为更复杂量子电路或量子算法的基本构建块；所需的单量子比特门通过路径编码中的相位移位器和用于偏振编码的波片或旋转偏振元件来实现。此外，可以将一种编码转换为另一种编码，从而灵活地使用这两种方法。

Linear-optical quantum gates also facilitate the implementation of quantum algorithms using linear-optical systems. Here, proof-of-principle implementation of the well-known quantum algorithms has been performed, for example, Grover’s algorithm and Shor’s algorithm^157,166,167. Recently, implementations of the algorithm for solving systems of linear equations have been demonstrated, and some proof-of-principle implementation of machine-learning tasks has been shown^{168,169,170,171,172}. In the majority of these implementations, photons were used as a testbed to demonstrate the basic functionality of each algorithm using a few qubits.
线性光学量子门还促进了使用线性光学系统实现量子算法。在这里，已经进行了著名量子算法的原理验证实现，例如，Grover 算法和 Shor 算法。最近，已经展示了解决线性方程组的算法实现，并且一些机器学习任务的原理验证实现也已被展示。在这些实现中，大多数使用光子作为测试平台，以展示每个算法的基本功能，使用了少量量子比特。

A further approach to quantum computing are measurement-based models and in particular one-way quantum computing^173,174 (Fig. 4b). Here, first, a highly entangled state is generated, called a cluster state, and then, computation is implemented through single-qubit measurements and feedforward. In the context of photonic systems, entangled states are generated by applying CPhase gates to single photons; when entangled photon sources are used, pre-entangled states can be concatenated to larger entangled states, again by applying entangling gates^153,175. The integrated generation of linear cluster states has also been demonstrated^176,177,178. Another approach to initial state generation is a ‘cluster-state machine gun’, which has been demonstrated with NV centres and quantum dots^101,102.
量子计算的另一种方法是基于测量的模型，特别是一种单向量子计算（图 4b）。在这里，首先生成一个高度纠缠的状态，称为簇态，然后通过单量子比特测量和前馈实现计算。在光子系统的背景下，通过对单光子施加相位门生成纠缠态；当使用纠缠光源时，可以通过施加纠缠门将预先纠缠的状态连接成更大的纠缠态。线性簇态的集成生成也已被证明。另一种初始状态生成的方法是“簇态机枪”，已在氮空位中心和量子点中得到验证。

To perform one-way quantum computing, the photons in a cluster state have to be measured in a particular order and basis. Dependent on the measurement outcome, a feedforward operation is applied, meaning that subsequent measurement angles depend on the previously measured outcomes. Two-qubit gates are realized by using the entanglement of the cluster state — one can show that the ‘vertical’ lines in Fig. 4b correspond to two-qubit gates. As most photonic cluster states have been demonstrated using the polarization degree of freedom, a measurement corresponds to performing a projection onto a certain polarization using wave plates and polarizing beamsplitters¹⁴⁶.
为了进行单向量子计算，簇态中的光子必须按照特定的顺序和基进行测量。根据测量结果，应用前馈操作，这意味着后续的测量角度依赖于先前的测量结果。通过利用簇态的纠缠实现两量子比特门——可以证明图 4b 中的“垂直”线对应于两量子比特门。由于大多数光子簇态是通过极化自由度实现的，因此一次测量对应于使用波片和偏振 beamsplitters 进行对某一极化的投影。

In this one-way model, single-qubit and two-qubit gates as well as various quantum algorithms have been demonstrated^175,179,180. The key is to translate quantum algorithms from the circuit model to a one-way picture and to optimize it according to the available resources.
在这个单向模型中，已经展示了单量子比特和双量子比特门以及各种量子算法。关键在于将量子算法从电路模型转换为单向图像，并根据可用资源进行优化。

The one-way quantum computer is also the basis for blind quantum computing (BQC)¹⁸¹. BQC aims at realizing secure delegated quantum computations in a network¹⁸². Here, a client with no quantum-computational power delegates a computation to a quantum server such that the input, output as well as the computation itself remains secret. The idea behind is to perform one-way quantum computation, but with encoded measurements on an encoded state. A client prepares qubits in a random state only known to the client themselves and encoded each measurement instruction. Both the qubits and the measurement settings are sent to a quantum server. There, an encoded cluster state is generated from the encoded qubits, and encoded measurements on this cluster state perform an encoded quantum computation. The outputs are being sent back to the clients who can decode the results. Photonic systems are well suited to be used for BQC as they allow information processing and transmission of quantum information within one physical system, meaning that photons can be used as carriers not only to send quantum information from a client to a server, but also to perform information processing within a node of a network. The original BQC protocol and variants thereof have been demonstrated with systems consisting of few photons^182,183. Methods to verify quantum computations in this setting have been demonstrated with photonic systems^146,184.
单向量子计算机也是盲量子计算（BQC）的基础。BQC 旨在实现网络中安全的委托量子计算。在这里，一个没有量子计算能力的客户端将计算委托给量子服务器，以确保输入、输出以及计算本身保持秘密。其背后的理念是执行单向量子计算，但在编码状态上进行编码测量。客户端准备处于随机状态的量子比特，这种状态仅客户端自己知道，并对每个测量指令进行编码。量子比特和测量设置都被发送到量子服务器。在那里，从编码的量子比特生成一个编码的簇态，并在该簇态上进行编码测量以执行编码的量子计算。输出结果被发送回客户端，客户端可以解码结果。 光子系统非常适合用于量子计算（BQC），因为它们允许在一个物理系统内处理和传输量子信息，这意味着光子不仅可以作为载体将量子信息从客户端发送到服务器，还可以在网络的一个节点内执行信息处理。原始的 BQC 协议及其变体已在由少量光子组成的系统中得到验证。已经在这种环境下使用光子系统演示了验证量子计算的方法。

Another approach to information processing with photonic systems is boson sampling^185,186. The idea is to take bosons (in our case photons), and let them pass through a passive linear-optical circuit consisting of beamsplitters and phase shifters and then sample from the output distribution of the circuit (Fig. 4c). It has been shown that this task is related to the evaluation of permanents of matrices — a problem that is hard using classical computers. However, photonic systems solve the task naturally by exploiting quantum interference of two bosons at beamsplitters.
另一种利用光子系统进行信息处理的方法是玻色取样 185,186。其思想是将玻色子（在我们的案例中是光子）通过由分束器和相位移器组成的被动线性光学电路，然后从电路的输出分布中进行取样（图 4c）。已经证明，这一任务与矩阵的永久性评估相关——这是一个使用经典计算机难以解决的问题。然而，光子系统通过利用两个玻色子在分束器处的量子干涉，自然地解决了这一任务。

First demonstrations of boson sampling used few photons, and it was also shown in integrated linear-optical circuits and fibre networks^{187,188,189,190}. More recently, larger-scale implementations of boson sampling have been demonstrated using QDs as photon sources^191,192. When using photons generated through SPDC, the emission of photons is non-deterministic, meaning that the preparation of a certain input state would take an exponentially long time and would destroy the computational advantage of boson sampling. However, it has been shown that scattershot boson sampling, where k heralded SPSs (k > n) are used as input into the interferometer, can restore the original advantage^189,193. Other variants are driven boson sampling¹⁹⁴ and Gaussian boson sampling^195,196,197. Gaussian boson sampling uses squeezed states (discussed subsequently) as inputs and has been shown to solve computationally hard problems, such as computing the Hafnian of a matrix, estimating the number of perfect matchings of undirected graphs and the graph isomorphism problem^196,197,198.
首次的玻色取样演示使用了少量光子，并且在集成线性光学电路和光纤网络中也得到了展示。最近，已经展示了使用量子点作为光子源的大规模玻色取样实现。当使用通过自发参量下转换（SPDC）生成的光子时，光子的发射是非确定性的，这意味着准备某个输入状态将需要指数级的时间，并且会破坏玻色取样的计算优势。然而，已经证明，散射玻色取样，其中使用 k 个预示的单光子源（k > n）作为干涉仪的输入，可以恢复原有的优势。其他变体包括驱动玻色取样和 Gaussian 玻色取样。Gaussian 玻色取样使用压缩态作为输入，并已被证明能够解决计算上困难的问题，例如计算矩阵的 Hafnian、估计无向图的完美匹配数量以及图同构问题。

Following this approach, a quantum-computational advantage using photonic systems has been demonstrated¹⁹⁹. In this experiment, coincidences of up to 76 photons have been detected, and a 100-mode interferometer has been realized. Integrated on-chip approaches with squeezed states have also been demonstrated²⁰⁰. A squeezed state of light refers to one of the non-commuting observables to be ‘squeezed’ in measurement uncertainty. A familiar example from quantum mechanics is that when the position of a particle is observed with high confidence, the momentum observation smears out. For photon states, the observables are: the two electric field quadratures (squeezed vacuum), phase and photon number (phase and intensity squeezing).
采用这种方法，已经证明了使用光子系统的量子计算优势。在这个实验中，检测到了多达 76 个光子的重合，并实现了一个 100 模干涉仪。还展示了集成在芯片上的压缩态方法。光的压缩态是指在测量不确定性中被“压缩”的非对易可观测量。量子力学中的一个常见例子是，当以高置信度观察粒子的位置时，动量的观察会变得模糊。对于光子态，可观测量包括：两个电场象限（压缩真空）、相位和光子数（相位和强度压缩）。

Boson sampling has been shown to be useful in quantum simulation in the way that using squeezed states of light, molecular vibronic spectra can be generated²⁰¹. Figure 4 gives a schematic overview of the various photonic quantum computing configurations. The feasibility of hybrid approaches, such as the variational quantum eigensolver method, was first demonstrated with an example from quantum chemistry, in which the ground-state molecular energy for He–H+ was calculated using photons²⁰².
波色采样已被证明在量子模拟中是有用的，通过使用压缩光态，可以生成分子振动光谱。图 4 给出了各种光子量子计算配置的示意图。混合方法的可行性，例如变分量子特征求解器方法，首次通过量子化学中的一个例子得到了验证，其中使用光子计算了氦-氢离子的基态分子能量。

Future improvements 未来的改进

Scaling up photonic quantum technologies to larger systems requires integration of sources and elements for information processing. An extensive review on integrated photonic quantum technologies is given in refs. ^154,203. Here, we focus on a few specific points that are particularly relevant in the context of quantum computing.
将光子量子技术扩展到更大系统需要集成信息处理的源和元件。参考文献 154 和 203 中对集成光子量子技术进行了广泛的综述。在这里，我们关注几个在量子计算背景下特别相关的具体点。

In the context of photon sources, one important development has been photon sources on the basis of QDs with high generation rates and near-unity indistinguishability^58,204,205. This brings these sources close to satisfying the requirements for linear-optical quantum computing applications with proof-of-principle experiments showing boson sampling with 4–5 single photons^206,207,208 with the latest results claiming 20 photons in 60 modes²⁰⁹. So far, these experiments are performed using single QD sources routing sequential single photons into different optical paths in mainly bulk optical experiments. The development of deterministically placed solid-state sources with well-defined spectra that can be easily overlapped to demonstrate high visibility interference between separate dots is essential for scalability.
在光子源的背景下，一个重要的发展是基于量子点（QDs）的光子源，其具有高生成率和接近完美的不可区分性。这使得这些光子源接近满足线性光学量子计算应用的要求，原则性实验显示出 4-5 个单光子的玻色取样，最新的结果声称在 60 个模式中实现了 20 个光子。目前，这些实验主要使用单个量子点源，将顺序单光子引导到不同的光学路径中，主要是在体材料光学实验中。开发具有良好定义光谱的确定性放置的固态光源，并能够轻松重叠以展示不同点之间的高可见度干涉，对于可扩展性至关重要。

Another approach to integrated photon sources is using silicon photonics by using the intrinsic nonlinearity of silicon. Dispersion-engineered silicon waveguides^210,211,212 are now producing high spectro-temporal purity heralded single photons²¹³ suitable for high-fidelity linear quantum gates built on chip. The primary fidelity limitation is now coming from higher-order number terms in the spontaneous pair photon emission process. Many sources producing occasional heralded single photons can be actively multiplexing to obtain high-efficiency heralded single photons with very low higher order multiphoton terms²¹⁴. The challenge here is to build low loss time delays and fast-switching on-chip to gate out the successfully heralded photons.
另一种集成光子源的方法是利用硅光子学，通过利用硅的内在非线性。经过色散工程设计的硅波导 210,211,212 目前正在产生高光谱时间纯度的预示单光子 213，适用于在芯片上构建高保真线性量子门。主要的保真度限制现在来自自发对光子发射过程中的高阶数项。许多偶尔产生预示单光子的源可以通过主动多路复用来获得高效率的预示单光子，同时具有非常低的高阶多光子项 214。这里的挑战是构建低损耗的时间延迟和快速切换的芯片，以成功筛选出预示的光子。

Once an ideal high-efficiency single-photons generation on-chip is available, the biggest problem facing the development of quantum computing with photons will be the non-deterministic nature of the entangling gates, which essentially brings another efficiency factor reducing the throughput. Theoretical scalable quantum computing proposals envision the notion of flexible cluster state computation modes^148,149,150, in which heralded single photons are assembled into three-photon Greenberger–Horne–Zeilinger states and then merged into complex 3D cluster states developing fusion gate success rates above the percolation threshold. In general, the scalability is achieved by adding extra resources and we envisage millions of identical SPSs emitting into circuits containing millions of elements. To date, quantum photonic circuits containing up to 550 elements have been reported²¹⁵. This approach is also followed by some start-up companies that aim at developing million-element circuits.
一旦理想的高效单光子片上生成技术可用，量子计算中面临的最大问题将是纠缠门的非确定性特性，这本质上带来了另一个效率因素，降低了吞吐量。理论上可扩展的量子计算提案设想了灵活的簇态计算模式，在这些模式中，预示的单光子被组装成三光子格林伯格-霍恩-齐林格态，然后合并成复杂的三维簇态，发展出超过渗流阈值的融合门成功率。一般来说，扩展性是通过增加额外资源来实现的，我们设想数百万个相同的单光子源(SPS)发射到包含数百万个元件的电路中。迄今为止，已报告的量子光子电路包含多达 550 个元件。这种方法也得到了某些初创公司的追随，这些公司旨在开发百万元件电路。

This scaling overhead can be reduced when gate and entanglement generation become deterministic. This can be achieved in SPSs in which the emitted (or scattered) single photon is inextricably correlated with the emitter ground state spin (see Supplementary Information). Such schemes begin to be realized in the laboratory with trapped ions and NV centres in diamond. Local clusters of up to 10-spin qubits (reviewed elsewhere²¹⁶) could be used as processing nodes linked by single-photon channels to make a distributed computing architecture. However, this requires further development of high efficiency SPSs, in which the emitted photons are entangled with a local qubit (or cluster of qubits) in the source allowing distributed quantum computers to be developed.
当门和纠缠生成变得确定性时，这种缩放开销可以减少。这可以在发射（或散射）单光子的单光子源（SPSs）中实现，该光子与发射器的基态自旋密不可分地相关（见补充信息）。这样的方案在实验室中开始通过捕获离子和钻石中的氮空位中心得以实现。最多可达 10 个自旋量子比特的局部集群（在其他地方已进行综述 216）可以作为处理节点，通过单光子通道连接，以构建分布式计算架构。然而，这需要进一步开发高效的单光子源，其中发射的光子与源中的局部量子比特（或量子比特集群）纠缠，从而允许开发分布式量子计算机。

In this Review, we have explored the applications of single photons in quantum communication and computing. Research is still needed to obtain single photons that are on-demand, indistinguishable and with the highest emission rate possible. This is reiterated in the accompanying review in ref. ¹, in which we discuss other applications of single photons and provide a wish list to guide future developments.
在本综述中，我们探讨了单光子在量子通信和计算中的应用。仍需进行研究以获得按需、不可区分且具有最高发射率的单光子。参考文献 1 中的附带综述重申了这一点，我们讨论了单光子的其他应用，并提供了一个愿望清单以指导未来的发展。