Appl. Phys. Rev. 6, 041303 (2019); https://doi.org/10.1063/1.5115814 6, 041303

© 2019 Author(s).

Photonic quantum information processing: Aconcise review Cite as: Appl. Phys. Rev. 6, 041303 (2019); https://doi.org/10.1063/1.5115814Submitted: 20 June 2019 . Accepted: 16 September 2019 . Published Online: 14 October 2019

Sergei Slussarenko , and Geoff J. Pryde

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Note: This paper is part of the Special Topic on Quantum Computing.

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Photonic quantum information processing: Aconcise review

Cite as: Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814Submitted: 20 June 2019 . Accepted: 16 September 2019 .Published Online: 14 October 2019

Sergei Slussarenkoa) and Geoff J. Prydeb)

AFFILIATIONS

Centre for Quantum Dynamics and Centre for Quantum Computation and Communication Technology, Griffith University,Brisbane, Queensland 4111, Australia

Note: This paper is part of the Special Topic on Quantum Computing.a)Electronic mail: s.slussarenko@griffith.edu.aub)Electronic mail: g.pryde@griffith.edu.au

ABSTRACT

Photons have been a flagship system for studying quantum mechanics, advancing quantum information science, and developing quantumtechnologies. Quantum entanglement, teleportation, quantum key distribution, and early quantum computing demonstrations werepioneered in this technology because photons represent a naturally mobile and low-noise system with quantum-limited detection readilyavailable. The quantum states of individual photons can be manipulated with very high precision using interferometry, an experimental sta-ple that has been under continuous development since the 19th century. The complexity of photonic quantum computing devices and proto-col realizations has raced ahead as both underlying technologies and theoretical schemes have continued to develop. Today, photonicquantum computing represents an exciting path to medium- and large-scale processing. It promises to put aside its reputation for requiringexcessive resource overheads due to inefficient two-qubit gates. Instead, the ability to generate large numbers of photons—and the develop-ment of integrated platforms, improved sources and detectors, novel noise-tolerant theoretical approaches, and more—have solidified it as aleading contender for both quantum information processing and quantum networking. Our concise review provides a flyover of some keyaspects of the field, with a focus on experiment. Apart from being a short and accessible introduction, its many references to in-depth articlesand longer specialist reviews serve as a launching point for deeper study of the field.

VC 2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5115814

TABLE OF CONTENTSI. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

A. Optical quantum computing. . . . . . . . . . . . . . . . . . . 1B. Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

II. PHOTON TECHNOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . 2A. Detecting a photon . . . . . . . . . . . . . . . . . . . . . . . . . . . 2B. Generating a photon. . . . . . . . . . . . . . . . . . . . . . . . . . 3C. Generating a photon deterministically . . . . . . . . . . 4D. Manipulating a photon . . . . . . . . . . . . . . . . . . . . . . . 5E. Integrated quantum photonics . . . . . . . . . . . . . . . . . 6

III. QUANTUM COMPUTING. . . . . . . . . . . . . . . . . . . . . . . . 7A. Intermediate quantum computing . . . . . . . . . . . . . . 7B. Cluster-state based computing . . . . . . . . . . . . . . . . . 9

IV. NETWORKING QUANTUM PROCESSORS. . . . . . . . . 9V. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

I. INTRODUCTIONA. Optical quantum computing

With the invention of the quantum computing (QC) concept, thedevelopment of suitable optical quantum technology became both aninteresting approach to the problem and a necessity. On the one hand,the advantages of using photons as information carriers seem to beobvious: photons are clean and decoherence-free quantum systems forwhich single-qubit operations can be easily performed with incrediblyhigh fidelity.1 On the other hand, quantum information handling withphotons as “flying qubits” is required for communication-based quan-tum information science tasks, such as networking quantum com-puters and enabling distributed processing.

In terms of the traditional DiVincenzo criteria of a quantumcomputer,2 five out of seven are essentially satisfied by choosing

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photons. The remaining criteria are harder to satisfy because photonsdo not easily interact, making deterministic two-qubit gates a chal-lenge. Among the additional technical considerations is photon loss,which arises from currently imperfect detection and photon genera-tion techniques, and from scattering and absorption in optical compo-nents comprising the computation circuits. Although photons arealways flying, computing and networking tasks may need them to bedelayed or stored, so an extra device—an optical quantum memory—may sometimes be needed. Addressing each of these considerationsrequires additional resources, creating a notionally large optical QCoverhead that has sometimes led to negative perceptions of the pho-tonic approach.

Of course, there is intense research under way in the develop-ment of deterministic optical (but matter-mediated) quantum gates,3–5

which could take photonic quantum computing (PQC) in a new direc-tion. Meanwhile, the idea of linear optical quantum computing(LOQC) that relies on simple, but probabilistic, quantum operationshas increasing promise as it has continued development over the last20 years. The earlier history of the field is covered in previousreviews6–9 that have appeared regularly in the literature. Here, we donot provide a typical review—that is, we do not present a comprehen-sive encapsulation of all the achievements of the field during the pastdecade. Instead, we concentrate on the few technological, experimen-tal, and theoretical advances that we think play key roles on the pathtoward a universal quantum computer operating with individual pho-tons and linear operations. On the technology side, we look at photondetection and generation tools, and integrated waveguide technol-ogy—and some new intermediate quantum computing demonstra-tions that these enable. On the conceptual side, we discuss a fewpromising ways toward a realistic universal linear optical quantumcomputer. We will concentrate on photonic10 quantum computing(PQC) that relies on qubits encoded in discrete variables, noting, how-ever, that quantum computing with continuous variables has nowbecome an important part of LOQC.11–14 But before that, we startwith a brief refresher on the basic conceptual elements and history ofPQC.

B. Basics

A qubit can be encoded as probability amplitudes correspondingto the photon occupation of two modes of some degree of freedom ofthe optical field. This method is known as dual-rail encoding. Themost commonly used mode pairs are orthogonal polarizations or non-overlapping propagation paths, but recently, other degrees of freedomsuch as transverse spatial,15–17 frequency mode,18–21 temporal bin-,22–25 and temporal mode-23,26–29 encoding are attracting attention.One-qubit operations—i.e., the shifting of single-photon populationbetween the two modes that comprise the dual-rail qubit, and applica-tions of phase shifts between them—are easily and reliably imple-mented using interferometry in the degree of freedom of choice. Agreat advantage of optical quantum computing is that it does not haveto be confined to qubits: Many of the degrees of freedom listed providea natural way to encode multilevel qudits. Moreover, several degrees offreedom of the same photon can be used simultaneously.30–34 (As wewill discuss later, these tools provide a natural advantage for optics,allowing for simpler logical circuits even when working with qubits asthe basic logical elements.) A way to realize an arbitrary n-dimensionalunitary transformation on the mode space, with linear optics, has been

outlined by Reck et al.35 quite some time ago, with recent improve-ments36 and expansions.37 In principle, Reck-type schemes could per-form universal processing with a single photon in many modes usedto represent multiple qubits. Unfortunately, that encoding leads toexponential scaling in the number of optical components, and thuscannot be used to build a scalable quantum computer. Thus, the use ofmultiple single photons is required for circuits with two-qubit gatesand beyond.

It is natural, then, to implement one qubit per photon, with adual-rail encoding. Two-qubit operations require the ability to apply ap phase shift rotation on one of the qubits depending on the state ofthe remaining qubit.38 These are trickier to implement than singlequbit operations, since this is a nonlinear optical interaction, and suchoptical nonlinearities, at the single photon level, are extremely weak.An alternative is to mimic nonlinear operations with linear optics andmeasurement, resulting in a probabilistic gate that provides the correctoperation after an appropriate postselection, or with an additional her-alding signal.

Historically, a variety of approaches to efficient optical quantumcomputing were discussed and investigated, for example Ref. 39 andreferences therein. However, the field of LOQC took off with the pro-posal of Knill, Laflamme, and Milburn (KLM),40 who invented a scal-able photonic scheme that required linear optics components, singlephoton detection, and classical feed-forward only (the reader mayenjoy reading Ref. 41 for comprehensive lecture notes and Ref. 7 for ahistorical overview of KLM). The KLM scheme essentially works byusing nonclassical interference to generate a phase shift that is nonlin-ear with respect to the photon number, conditioned on photonsappearing at certain heralding modes. These operations are then builtinto nondeterministic logical gates. The gates are used in a repeat-until-success mode, and the operation of a successful gate is teleportedonto the logical qubits. The use of a large number of concatenatedsteps, and lots of ancilla photons, leads to essentially deterministicgates. The KLM scheme theoretically allowed for a resource-efficientimplementation of two- and multiqubit gates—unlike encoding a sin-gle photon across many modes, the resource scaling was not exponen-tial in the number of qubits, but rather linear. Thus the KLM schemeprovides a pathway to build a universal quantum computer, albeitwith a large overhead of ancilla qubits (and their associated circuitry)to deal with the use of nondeterministic two-qubit gates. With theadvent of a viable theoretical approach, photonic quantum computingbecame the subject of extensive theoretical and experimental develop-ment. As well finding approaches that reduce the overhead due tonondeterminism, making this scheme practical also requires high-quality technological components to make, manipulate, and measure42

the photon qubits. We first turn our attention to these technologyconsiderations.

II. PHOTON TECHNOLOGYA. Detecting a photon

A photon’s life in a quantum experiment starts with its genera-tion and concludes with its detection. Both processes need to be effi-cient, and their performance and properties play essential roles inPQC. In this section, we start from the end—with a look at single pho-ton detection43 technology.

An ideal photon detector (PD) clicks every time a photon hits itand immediately restarts its operation. It does not produce false

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positive signals when no real photons were detected (so-called “darkcounts”) and it also tells exactly how many photons were detected inthe same spatiotemporal mode. Such ideal photon detectors do notexist yet. Existing PDs are correspondingly characterized by detectionefficiency gd, reset time sR (that sets the maximum detection rate),detection time jitter sj, dark count rate Cd, and photon-number-resolving (PNR) capabilities. While a “perfect” PD is not actuallyrequired for PQC,44 improving the PD performance to very high levelsis important for a realistic and scalable platform.

Setting aside historical and exotic approaches, the PD of choicefor optical quantum information science experiments has been the Siavalanche photodiode (APD) operating in Geiger mode. These are rel-atively fast (sR � 100 ns), low-noise (typical Cd � 100 counts per sec-ond) detectors. Unfortunately, their limited quantum efficiency,typically up to gd � 65%, sets a practical limit on the number of pho-tons that can be used simultaneously in an experiment. A probabilityof detecting, say, ten photons with ten detectors is already less than2%, and things get exponentially worse with increasing photon num-ber. Si APDs do not possess PNR capabilities45 and their maximumefficiency wavelength range is quite limited. In particular, it does notcover the telecommunications bands around 1310 and 1550 nm. Theequivalent detector for 1550 nm, the InGaAs APD, suffers from lowerquantum efficiency and higher dark counts.

Inefficient detection was a significant limiting factor for PQC forquite some time. Things started to turn for the better with the adventof superconducting nanowire single-photon detectors46,47 (SNSPDs).These provided something close to a direct substitute for the usualAPDs: They have comparable (sR � 40 ns) reset times, yet can achievedetection efficiencies of up to gd � 0:93 (Ref. 48) [and recently evengd � 0:95 (Ref. 49)] in the telecom wavelength range. SNSPDs workby passing a current though a superconducting nanowire close to thecritical current—then, the energy absorbed from even a single photoncan transition the device to normal resistivity. The subsequent voltagespike is filtered and amplified, and registered as a detection. SNSPDsare a bit more complicated to operate than APDs, as they require cryo-genic temperatures of 0.8K–3K (depending on the superconductingmaterial), but the massive enhancement in detection efficiency justifiesthe inconvenience. SNSPD performance can also be optimized to anywavelength by selecting the appropriate material and designing a suit-able optical cavity that envelops the nanowire. They can also bedesigned to efficiently interface with fiber-optic inputs. In short,besides providing an enormous increase in detection efficiency,SNSPDs have enabled operation at telecom wavelength that benefitsfrom previous development of optical materials and efficient photonictools. This detector performance is also beneficial for quantum com-munication and other low-loss applications, e.g., Refs. 50–55.

Research on superconducting detectors is still ongoing, aimed atunderstanding detection mechanisms in different types of nanowirematerials,56–60 improving its performance in terms of reset times61 andtime jitter,62,63 and developing new methods of accurate detection effi-ciency measurements.64 Although intrinsic dark counts are low,SNSPDs are susceptible to picking up background thermal radiationfrom the input fiber’s room-temperature environment—this can beovercome by spectral filtering.

The key remaining limitation of this technology is the lack ofPNR capability. While schemes that turn SNSPDs into PNR detectorsare being investigated,65 a different type of detector, based on

transition-edge sensors (TESs),66 can be also employed in experimentswhere photon number counting is essential. TES detectors work asbolometers with single-photon-level resolution: Absorption near thesuperconducting transition changes the resistance of the device mono-tonically with the photon number, which can be read out through anintegrating circuit. TESs have excellent PNR skills:67 In recent experi-ments, they were able to efficiently discriminate up to �20 photons inthe same spatiotemporal mode.68 At the same time, they have shownto be able to reach gd � 0:95 in the telecom wavelength range,69 withfurther developments leading to even higher gd � 0:98,70,71 closelyapproaching the ideal gd ¼ 1. TESs can also be optimized to anywavelength in the visible and IR ranges. A critical drawback of a TESdetector is its slow operation, with �microsecond reset times and�50 ns time jitter. Efforts in improving TES time performance areongoing, with reset times as fast as sR � 460 ns72 and time jitters ofdown to sj ¼ 2:3 ns73 (for 775 nm photons) having been demon-strated. Still, these numbers are at least two orders of magnitude higherthan what might be considered practical for PQC, where clock cycles of�10 ns are likely required for the practical switching of flying photons.

B. Generating a photon

Having exceptional detectors is not much use if one cannot effi-ciently make high-quality photons on which to encode qubits.

Computing tasks in the near and long term requires the capabil-ity of simultaneously generating a large74 (N � 10� 1011) number ofsingle photon states. The obvious way to achieve this is to have a large(N � 10� 1011) number of deterministic sources that can simulta-neously produce one and only one photon each at the push of a button(i.e., on a trigger event). Moreover, these photons must necessarily be:(a) efficiently collected so to be sent into the PQC processor and notlost (e.g., by absorption, scattering, diffraction, or mode mismatch dur-ing the generation and fiber in-coupling process); (b) in a pure quan-tum state and indistinguishable from one another; and (c) compatiblewith the low-loss material and high-efficiency detection technologyfrom above. At present, sources that properly satisfy this list do notexist. However, truly deterministic, high-quality photon sources likethis are being developed using diverse physical systems,45 such astrapped ions and atoms, color centers in diamonds, semiconductors,quantum dots,75 and other, more exotic, methods (e.g., Refs. 76–78).Some of these rely on the use of a single emitter that, in principle, nat-urally provides on-demand single-photon emission, while others—such as atomic ensemble79 and parametric nonlinear processes80—require heralding signals and switching to make them so. (Therequirements for achieving deterministic operation in practice will beconsidered Subsection IIC.)

In the meantime, the key enabling technology for experimentalquantum optics, spontaneous parametric downconverson81–83

(SPDC), remains a practical way to generate high-quality single pho-tons nondeterminstically. Developments in this technology have effec-tively addressed the feature list (a)–(c) above. In this three-wavemixing nonlinear vð2Þ process, a pump photon from a laser has a smallprobability to be converted into a pair of “daughter” (signal and idler)photons. The process must obey the momentum (~kp ¼~ks þ~ki, phasematching) and energy (xp ¼ xs þ xi) conservation laws, with~kn andxn, n ¼ p; s; i, being the wavevectors and angular frequencies ofpump, signal and idler photons, respectively. SPDC is probabilistic,but it can be used to produce “heralded” single photon (and more

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complex multiphoton84–87) states, where the presence of a photon isheralded by the detection of its twin. Alternatively, SPDC can producephoton pairs that are naturally entangled in polarization,88 transversespatial modes,15,89 or frequency.90,91 With modest effort, it is possibleto produce photon pairs with entanglement in a time-bin encoding,22

or even in multiple degrees of freedom simultaneously.92

SPDC can be a simple and cost-effective way to get single pho-tons and (entangled) photon pairs but, in its original and simplestform, it is far from an ideal photon source for PQC. Ongoing techno-logical development is changing that. Among the immense variety ofSPDC-based sources that have been developed and reviewed over pastyears,45,93–95 we concentrate here on some advances that directly serverealistic PQC.

A typical SPDC output from a simple, critically phase-matched,bulk-crystal source88 is not compatible with efficient coupling intosingle-mode optical fiber, because its transverse spatial profile is farfrom a Gaussian mode, resulting in coupling loss. Also, the twin pho-tons are intrinsically entangled in frequency. This means that detectionof one photon—to herald the presence of another—without resolvingits wavelength degrades the purity of the heralded photon.96 The spec-tral filtering necessary to remove this entanglement adds even moreloss to the source. Moreover, traditional SPDC photon wavelengths sitaround 800 nm, due to the standard use of Si APDs and compatiblewith readily available pump laser wavelengths. At these wavelengths,the material loss (e.g., in fibers) is significant, and detection efficiencyis limited. A typical experiment involving more than one photon pairwould have heralding efficiency (probability of a heralded photon tosuccessfully travel from a source to a detector and produce a click97) of�10� 15%, although some experiments report �30% (Ref. 98).Under these conditions, setting up several photon-pair sources allowedthe creation of complex photonic states of up to ten photons,99 but thelow collective detection rates, and achievable state quality, limited thelong-term prospects of these sources.

A significant step forward was the application of quasiphasematching100 (QPM), via periodically poled nonlinear crystals. Thisexpanded the range of possible phasematching wavelengths and emis-sion geometries101–104 and enabled collinear, beam-like downconver-sion in the telecom wavelength range. With both photons emitted intoan almost-single, almost-identical, almost-Gaussian spatial mode, themode-matching loss and fiber propagation loss could be kept very low,leading to high heralding efficiencies. Using type-II phase matchingmeant that degenerate photons could be deterministically separatedwith polarization optics. With the addition of interferometric schemesto generate polarization entanglement,105,106 QPM SPDC sourcescould deliver entangled photon pairs with either continuouswave107,108 or pulsed laser pumps.109,110

There remained the need to remove the residual spectral entan-glement in downconverted photons. This was recently solved byapplying the concept of group velocity matching (GVM).96,111,112 Bycarefully engineering the relative group velocities of the pump, signal,and idler photons, and adjusting the pump laser bandwidth and SPDCcrystal length, the joint spectrum of the daughter photons can be con-trolled. It can be arranged that the signal photon is in a single spectralmode, and the idler photon is in a single spectral mode, to high fidelity.(Note that the photons do not need to be in the same spectral mode asone another.) This technique provides photon pairs that are inherentlyuncorrelated in their spectrum,113,114 and reduces or removes the need

for spectral filtering. GVM at specific donwconversion wavelength setsis attained by selecting an appropriate nonlinear material—KTP(potassium titanyl phosphate) proved to be suitable for degeneratedownconversion in the telecom region. Using GVM, a number of fre-quency uncorrelated,115,116 nondegenerate117 and degenerate indistin-guishable106,118–120 pure photon-pair sources at telecom wavelengthhave been demonstrated. Combined with optimized mode matchingwith the optical fiber121 and high efficiency detection technology inthe telelcom wavelength range, GVM allowed realization of pulsed tel-ecom photon-pair sources that are simultaneously pure, highly effi-cient and (if desired) entangled in a chosen degree offreedom.52,53,55,122 Further tailoring of the crystal’s nonlinearity pro-file123–125 provides photons that are fully uncorrelated in their spec-trum,126–128 completely removing the need for lossy spectral filtering.Investigation of the performance and limitations of periodically poledSPDC sources continues129–132 and even tools for complete SPDCoptimization are now available.133

These developments have provided an enormous leap forwardfor SPDC technology, helping it to get close to satisfying many ofthe criteria (a)–(c) for ideal photon generation. Heralding efficien-cies jumped55,134 to above 0.8. The entangled state quality is harderto survey, because of the variety of figures of merit that are used.Focussing on a couple of standard ones, quantum state puritiesover � 0:997 (Ref. 135) have been observed, and entangled statequalities—equivalent to the fidelity136 with a maximally entangledstate—above 0.99 have been achieved in the lab.52,135,137–139 Thesehigh-performance sources have also allowed realization of impor-tant experiments in entanglement verification52,140 and quantummetrology.55

However, these advances relate to what happens when a photonpair is generated—the pair generation process itself is probabilistic. InSec. II C, we consider how SPDC or other technologies may be used toprovide deterministic single photon generation.

C. Generating a photon deterministically

Photon-pair sources from SPDC and related processes—likespontaneous four-wave mixing (SFWM)141–144—are not only nonde-terministic but generally operate at low generation probabilities. Inorder to keep the single photon state quality high, pump powers haveto be kept low; otherwise, multiple photon pairs will be generated atthe same time.145 This limits practical photon-pair generation proba-bility n, for SPDC and similar processes, to n � 1%. Directly combin-ing an array of n such sources (that will together produce nsimultaneous pairs with probability nn) to generate a larger quantumstate is essentially not a viable option for a scalable photonic quantumcomputer.

A more feasible alternative is to employ a deterministic photonsource. In recent years, photon-on-demand sources based on quantumdots,146 both free-space75,147,148 and integrated149–151 into opticalwaveguides, have demonstrated a significant increase in brightness,enabling new quantum computation experimental demonstrations.152

(It is worth noting that although quantum dots are usually assumed toprovide single photons on demand, quantum dots can also generateentangled photon pairs24,153–157 and superpositions of photon numberstates.158) Although quantum dots159 can couple to optical cavitieswith very high efficiency,148,160,161 a currently outstanding problem iscoupling light efficiently into single mode optical fibers, with present

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coupling efficiencies162 �33% (Ref. 163). Moreover, each quantumdot is usually spectrally different from others due to structural andenvironmental inhomogeneities, so the photons emitted by two dotsare distinguishable from each other. PQC relies on nonclassical inter-ference, and the lack of indistinguishability makes it complicated toincrease the number of photons used simultaneously in an experiment.One way to fix this is to tune the emission spectrum of different quan-tum dots to make indistinguishable.164,165 Alternatively, a single quan-tum dot can be used to generate all the required photons. For this, apulsed output stream of photons from the dot is demultiplexed intodifferent spatial channels via a free-space166,167 or integrated168 activeoptical network. The multiplexed photons are then each delayed byappropriate amounts, so as to be output simultaneously from thesource setup.

Similar active optical circuits can also, in principle, turn probabil-istic sources such as SPDC into deterministic ones. To realize this, anarray of sources is used—see Fig. 1(a). Detecting the heralding signalsfrom such an array will label which source has successfully generated aphoton pair. Then, the corresponding heralded photon can be activelyrerouted through an optical network toward the output, while otherphotons, if generated, would be discarded by the same network. Usingn sources, this way theoretically boosts the generation efficiency tonmulti ¼ 1� ð1� nÞn, ideally, without increasing the pump power

that impinges on a single nonlinear crystal and thus without increasingthe amount of high-photon-number noise from multiple-pair genera-tion events. (In principle, the network can also filter out multiple-pairgeneration events if photon-number resolving detectors are used.)This concept,169,170 experimentally demonstrated in 2011 (Ref. 171),has moved significantly toward practicality since then,172–174 in partbecause of the use of fiber- and waveguide-based integrated platformsto help scaling.

Another method, that does not require multiple separate sources,is to use time169,175,176 (or frequency177,178) multiplexing of a singlesource.169,175,176 In the time multiplexing approach, shown in Fig.1(b), a heralded photon pair is generated in a random time bin, butthe timing is recorded through detection of the heralding signal. Theheralded photons are sent into an active temporal delay network andswitched so as to exit the network at a fixed, although lower, repetitionrate. The number n of time bins that is used to output one single pho-ton plays the role of n sources in a spatial multiplexing scheme. Thus,the improvement in generation probability scales with the size of thedelay network, but is affected negatively by the loss in optical compo-nents in it. This multiplexing idea has been recently implemented in anumber of experiments, demonstrating multiplexing with large-scale,179 or large-scale and low-loss180 networks, or with devices thatproduce indistinguishable output photons.181 The experimental dem-onstration that includes all of these features182 produced single pho-tons in the output fiber with a probability of �0:6, and these photonsdisplayed a nonclassical interference visibility �0:9. A more in-depthlook at near-deterministic sources can be found in Ref. 93.

Interesting preliminary work has also been done toward combin-ing these kinds of techniques to simultaneously generate more thanone single photon at a time. The multiplexing approach can be appliedto more than one probabilistic source to generate states with one pho-ton in each of N> 1 modes.183,184 An alternative method is to use anoptical quantum memory to synchronize several probabilistic sour-ces.185 Although quantum memory might be as simple as a switchableoptical delay (in a free-space, fiber, or waveguide loop, for example),there is also extensive theoretical and experimental development ofmemories based on matter systems,186 with recent achievementsincluding but not limited to broadband,187,188 high-speed,189 multi-mode,79,190 telecom-compatible,51,191 or configurable192 memories,capable of storing vector-,193 vortex-,194 or entangled-51,190 qubits, andstorage with long coherence times195 and high storage efficiency196–198

and fidelity.199,200

Over the span of slightly more than a decade, photon detectionand the probabilistic generation of high-quality photons have under-gone transformational advances, and the development of deterministicsources is well under way, with no in-principle barriers to their realiza-tion. (There are also other interesting advances, such as spectrally nar-rowband sources201 for metrology and fundamental physicsapplications202 that we do not cover here.)

D. Manipulating a photon

Thus, before proceeding to Sec. II E, we briefly turn our attentionto technologies for manipulating photons for PQC. Precise and accu-rate control of photon’s polarization, path, or time-bin stat has alwaysbeen the strength of PQC.203 Recently, this has been extended to per-forming reconfigurable mode transformations in integrated quantumoptics.204 Modern electro-optic elements, such as Pockels cells or

FIG. 1. Schematic representation of two types of triggered photon sources basedon SPDC. (These concepts can be adapted to nondeterministic sources based onother technologies.) (a) Multiplexing scheme that combines multiple (here, four)probabilistic photon sources to provided an increased brightness. Detectors (upperarms) are used to herald the production of a photon by one of the sources, which isthen switched into the output mode by some logic (e.g., a field-programmable gatearray) and switch array. Since there are multiple sources in parallel, this schemeincreases such as probability of having a photon in an appropriate time bin, withoutincreasing the probability of having more that one photon in that bin. With enoughsources in parallel and with low loss, the arrangement can approach an ideal, deter-ministic single photon source. (b) Triggered source that uses only one probabilisticsource and an active delay network. The network rearranges the generated pho-tons in time, so that they are output at a stable, although lower, repetition rate. Thisscheme also provides a deterministic source, in principle.

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integrated electro-optic modulators, allow fast polarization switchingsufficient perform rigorous Bell tests with locality and freedom ofchoice loopholes closed52,140 or spatial mode switching for source mul-tiplexing purposes.166–168 Efficient tools for manipulation of moreexotic degrees of freedom, such as frequency-time205 or transverse spa-tial modes,206 are also being developed, including the techniques thattransfer information from one degree of freedom to another, such aspolarization to spatial transverse mode,16 discrete variable to continu-ous variable,207 frequency conversion,208 and so on.

E. Integrated quantum photonics

While introducing the relevant advances in photon detection andgeneration technology, we mostly limited ourselves to the “bulk”optics environment, with separate optical components sitting on atabletop. As the scale of PQC demonstrations grows to larger numbersof photons and gates, the importance of technological scalability andminiaturization becomes increasingly apparent. Integrated waveguidesand optical chips offer an obvious path to implementing circuits atscale, i.e., with huge numbers of components packaged compactly.Thus, these technologies are now playing a significant role in thefield.209 Several characteristics are important for a waveguide platform:The achievable density of optical components; low propagation andcoupling losses; and the ability and speed of active reconfiguration, forexample. It is also desirable to integrate sources and detectors onto theoptical chip.

Different materials offer their own strengths and advantagesfor realizing a practical integrated quantum photonics platform.Femtosecond-laser-written waveguides (typically in a glass) sup-port polarization qubits210 and are not restricted to a 2D geometry,allowing realization of complex couplings in 3D interferometricnetworks.211 Lithium niobate, a material that is already well estab-lished in classical integrated photonics, is an efficient and flexibleplatform for photon sources and fast switchable electro-opticalcomponents operating at the gigahertz rates. Both ion-indiffusedand high-index-contrast etched waveguides are being developedand employed.150,168,212–215 Silicon-based optical chips offer highcomponent density, low loss, the ability or potential to integrateevery necessary component, and compatibility with existingfoundry processes.216 An enormous range of other material plat-forms are also under consideration.

On the integrated detection front, a lot of work has been done217

in embedding SNSPDs into optical chips since the first demonstrationin 2011 (Ref. 218). This ongoing effort has already provided fast andefficient,219 low-noise,220 fast and low-noise,221 or low-noise, efficientand fast222 (and even faster223) detection at telecom wavelength.Significant effort is being put into turning waveguided SNSPDs intowaveguided PNR detectors, see for example Ref. 213 and referencestherein, and Ref. 65. Similar developments are happening on the TESintegration side.224,225

The situation is even more vivid regarding integrated photonsources. QPM-based downconversion, which now plays the key role inheralded photon and photon-pair generation, was in fact first demon-strated in fiber101 and integrated waveguides.102–104 An importantadvantage here is the transverse spatial confinement of the three(pump, signal, and idler) propagating optical modes along the entirelength of the nonlinear material. This confinement allows constructionof a photon-pair source with both high brightness (absolute generation

rate calculated in pairs per second per mode per unit of pump power)and high heralding efficiency. This is advantageous compared to bulkSPDC, where the spatial mode configuration for high brightness is dif-ferent from the one that provides high heralding efficiency.121 Usingintegrated technology, efficient sources in the telecom wavelengthrange,226 including ones with GVM,227,228 have also been realized,leading to the development of a fully packaged, banana-sized,229 andhighly efficient photon-pair source and similar sources in a variety ofmaterial platforms.230,231 Techniques have been demonstrated fordirect and practical characterization of nonlinear operations (likeSPDC) in integrated quantum photonics.232 Integrated optics has alsoshown the capability of using more than one degree of freedom of aphoton.230

A number of materials for integrated optical components haveno vð2Þ nonlinearity, making them unsuitable for SPDC-based pho-ton-pair sources. In this case, a practical alternative is SFWM. It is avð3Þ nonlinear parametric process where two pump photons (degener-ate or otherwise) are converted into two daughter photons (degenerateor otherwise), conserving energy and momentum. Historically investi-gated in optical fibers141–144 due to the isotropic nature of amorphoussilica, this method is now commonly adopted in those integrated plat-forms where vð3Þ nonlinearities dominate.233,234 A GVM-like approachfor controlling the joint spectrum of daughter photons was also subse-quently generalized to SFWM235 and implemented experimentally infiber236 and on a chip.237 The scalability of the integrated opticsapproach allows one to fabricate arrays of nearly identical photonsources238,239 that are now actively used in PQC experiments in sili-con.204 On the more technical side, a number of SFWM obstacles,including the challenge of strongly filtering out the strong pump fieldfrom the generated photon field, have been overcome in recentyears.240,241 The interested reader can find more information on inte-grated probabilistic sources in Ref. 93 and on recent advances in GVMbulk and waveguided sources in Ref. 94.

The rapid development of quantum integrated photonics is per-haps most obvious in the growth in the scale, complexity and perfor-mance of optical circuits for one- and multiqubit operations. The firstoptical chips with path-242 and polarization-210,243 qubit encoding didnot immedaitely surpass the performance previously achieved withbulk optics244,245 (in repeating the factoring of 15 by a compiled Shor’salgorithm, for example Ref. 246) but emphatically demonstrated thepromise of the integrated approach. Subsequent devices, and the appli-cations they implemented, started to increase in complexity reallyquickly.247 This included increasing the number of interferometers ona chip (Fig. 2) and adding slow or fast active phase212,248–250 and spec-tral215,251 controls in various waveguide platforms. These capabilitieshave led to a realization of fully reconfigurable optical processors foran increasing number of optical modes.252 It has been observed that,for the moment at least, the number of components on integratedquantum photonics chips is undergoing a Moore’s-law-like exponen-tial growth with time.253

A challenge of integrated platforms is optical loss caused bymaterial absorption, waveguide roughness, and coupling onto and offchip. These are actively investigated by a variety of techniques includ-ing: Improved materials (e.g., higher purity); moving to high-index-contrast platforms where devices can be smaller (e.g., Ref. 214), byintegrating sources and detectors directly on chip. Modular architec-tures are also being investigated.254

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III. QUANTUM COMPUTING

The advent of the KLM scheme40 in 2001, with its proof of thescalability of optical processing, inspired a worldwide push toward auniversal quantum computer with photons. Of course, a full-scaleerror-corrected version could not be built at that time, and indeed auniversal quantum computer remains a challenging quest today in anyquantum system. The KLM scheme led to the development andimprovement of a variety of photonic encodings, schemes for quan-tum gates, and protocol and algorithm demonstrations.6–9 Circuit-based approaches, having evolved from KLM, continue to be an activearea of theoretical and experimental research as a path towardintermediate-scale and universal quantum processors.

A significant development for PQC was the realization that thecluster-state model of quantum computing (also known as one-wayquantum computing)255 was well-suited to photon qubits.256 This isprimarily because large cluster states257 can, in principle, be built effi-ciently using entangled photon sources and teleportation gates of thekind used in the KLM scheme. It is also important that photon mea-surements are easy to perform reliably, and because cluster stateschemes can be made tolerant to photon loss, the primary source ofnoise in an optical environment. For these reasons, cluster stateschemes are widely viewed as offering a realistic path to scalable PQC.

As the development of universal PQC has continued, a numberof intermediate goals have emerged, providing short- to medium-termtargets and a path toward full-scale devices. These include: The

development of individual quantum gates of increasing complexity inthe circuit model; the implementation small-scale quantum algorithmsand nonuniversal circuits or clusters for them; the development ofsimplifying and supporting techniques within the circuit and cluster-state models; and the advent of algorithms for sampling problemsbased on the fundamental properties of bosons. Intermediate quantumcomputing research is helpful for optimization of the general schemesfor PQC, and for developing and testing of individual components of afuture quantum computer.

A. Intermediate quantum computing

Photons can be readily and accurately manipulated at the single-qubit level—very high fidelity one-qubit gates can be constructed1

because of excellent optical mode control.252 Initially, particular attentionfell on the controlled-NOT (CNOT) gate to complete a universal gate setin the quantum circuit model. Theoretical proposals for nondeterminis-tic CNOT gates,259,260 demonstrating the basic measurement-inducednonlinearity concept of KLM, were quickly followed by experimentalCNOT demonstrations261,262 and characterizations.263,264 These wereexpanded to include heralded KLM-style265 and teleportation266-based39

schemes.267 A number of proof of principle algorithms followed the earlydemonstrations of photonic gates.6 While using CNOTs to build arbi-trary unitary circuits is, of course, a working theoretical method, it is farfrom optimal. This is because, for example, the decomposition of a

FIG. 2. A circuit diagram of the multidimensional silicon quantum photonic circuit. Reprinted with permission from Wang et al., Science 360, 285–291 (2018). Copyright 2018AAAS. The device monolithically integrates 16 photon-pair sources, 93 thermo-optical phase shifters, 122 multimode interferometer beamsplitters, 256 waveguide crossers,and 64 optical grating couplers. A photon pair is generated by SFWM in superposition across 16 optical modes, producing a tunable multidimensional bipartite entangled state.The two photons, signal and idler, are separated by an array of asymmetric Mach–Zehnder-Interferometer (MZI) filters and routed by a network of crossers, allowing the localmanipulation of the state by linear optical circuits. Triangular networks of MZIs perform arbitrary local projective measurements. The photons are coupled off the chip into fibersby means of grating couplers, and are detected by two SNSPDs. See Ref. 204 for details.

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three-qubit gate, such as the Toffoli gate, into one- and two-qubit opera-tions may require a large number of such gates.268 An alternative wouldbe to look for ways of implementing gates that can operate on a largernumber of qubits directly.

An interesting and important class of arbitrary-scale quantumlogic is the family of controlled-Unitary (CU) gates. In these, a (possi-bly multiqubit) unitary operation acts or not—depending on the stateof a control qubit—on the target qubits. CU gates are important invarious computational tasks, for example the phase estimation algo-rithm that underlies Shor’s algorithm244,245 and in quantum chemis-try.269,270 A key realization is that implementing the unitary operationU alone may be possible or even easy, but adding the control opera-tion—i.e., conditional action—is difficult.

A general scheme for adding a control operation to an arbitraryunitary transformation was proposed in 2009 (Ref. 271). In thismethod, given the unitary to be controlled, the Hilbert space dimen-sionality of the incoming target qubits is first doubled by using someauxiliary degree of freedom of the corresponding photons. Half of themodes of each target qubit pass through the unitary, while the remain-ing half bypass it. Then, the control qubit state is used to route the tar-get qubits to either pass the unitary or bypass it, via the correspondingmodes. After that, the modes are recombined, so the Hilbert space isshrunk to its original dimensionality. This effectively creates a CU gate.(The scheme can be simplified even further, by substituting Hilbert-space-expanding gates with photon sources that generate entanglementin the auxiliary degree of freedom. The term “entanglement-based” isusually used in the literature to describe these types of gates, which arenot completely general due to the need to generate the initial entangle-ment, but can be useful at circuit inputs.) This overall method is partic-ularly suitable for optical quantum computing, because highdimensional systems, multiple degrees of freedom, and means of trans-fering information between them are readily available. Moreover, theo-retical studies also highlighted that adding control to arbitrary unitarygates is generally impossible for matter-based qubits,272,273 so themethod demonstrates a benefit of using fields to quantum compute.

This general approach was used to experimentally realize arbi-trary controlled-single-qubit unitaries, a CNOT gate,274 and three-qubit gates—namely the Toffoli271 and Fredkin (controlled-SWAP,see Fig. 3)258 gates. It was also employed in experimentally implement-ing a number of quantum computing tasks, such as solving systems oftwo linear equations275 (this was also done without entanglement-based gates276) factoring 21 by a version of Shor’s algorithm (Ref.277), measuring state overlaps and state purity,258 and eigenstate wit-nessing for simple quantum algorithms.270 Entanglement-based gatesare now also used in larger quantum circuits, including the ones real-ized in an integrated platform.270,278

The use of various photonic quantum gate architectures hasallowed realization of a variety of intermediate scale simulations,implemented in bulk and integrated optics platforms. Among these279

are spin chain simulation,280 calculating molecular ground-state ener-gies,281 Hamiltonian learning282 and eigenstates witnessing,270 andcomplex state transformations such as Fourier211,283 or Kravchuk284

transforms.A highly topical intermediate photonic quantum computing task

is that of BosonSampling,285–292 which is an example of sampling-typecomputational problems more generally.293 BosonSampling is a non-universal protocol for which there is strong theoretical evidence that aquantum advantage can be observed. Consider n single photons inputinto m� n optical modes, which are subjected to a random unitaryoperation on the mode space. It is classically computationally hard toobtain samples from the probability distribution representing wherethe photons appear at the output. By contrast, photons (and otherbosons) traversing a unitary on the mode space perform this calcula-tion naturally. Interestingly, the same quantum-classical performancedivide exists even if the photons are allowed to arrive at random inputsof the circuit.294 It is thought that better-than-classical BosonSamplingperformance may be achieved with 50–100 photons, promoting theidea that this system could well provide the first rigorous experimentaldemonstration of a quantum computational advantage. Nevertheless,challenging constraints on photon loss and other noise still need to be

FIG. 3. An optical quantum Fredkin. Reprinted with permission from Patel et al., Sci. Adv. 2, e1501531 (2016)]. Copyright 2016 Author(s), licensed under a Creative CommonsAttribution 4.0 NonCommercial License. The Fredkin (or controlled-SWAP) gate uses the method of adding control to an arbitrary untiary operation. Entangled photons are pro-duced in BBO (beta-Barium borate) crystals via SPDC. The control qubit is encoded into modes 1B and 1R, target 1 is encoded on modes 2R and 2B, and target 2 is encodedon modes 1G and 1Y. The control circuit consists of a polarization beam displacer interferometer. The path-entangled state, required for the Fredkin operation, is produced aftereach target photon enters a displaced Sagnac interferometer and the which-path information is erased on a nonpolarizing beam splitter. Quarter-wave plates and half-wave platesencode the target qubits’ input state. Successful operation is heralded by fourfold coincidence events between the control, target, and trigger detectors. See Ref. 258 for details.

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met to achieve this goal.134,295,296 Recent reviews95,293 cover conceptualand experimental aspects of the topic in more detail.

Intermediate quantum computing is likely to lack fully fledgederror correction. Thus, photon loss and noise in PQC will need to becontrolled by other methods. One prominent approach being investi-gated for NISQ297 (noisy intermediate-scale quantum devices) ismachine learning (ML). ML provides a method to work with quantumprotocols operating in an environment of unknown or uncharacter-ized noise, or where the full ab initio modeling of the protocol isintractable,298 and can be applied to PQC and other systems.299 Theflip side to ML helping quantum computation by controlling noise isthe hope that quantum computers can enhance ML for other applica-tions, possibly even in the NISQ regime.297 Other relevant proposalsfor ML quantum applications include long-distance quantum commu-nication300 or metrology.301,302 Experimental demonstrations of MLapplication to quantum information science have recently started toappear, too.282,303–305

B. Cluster-state based computing

In conventional PQC, uncorrelated input qubits are processed bya complicated quantum circuit of one-, two-, three-, and many-qubitgates (which in turn can be decomposed to one- and two-qubit gates).Here, generating many uncorrelated photonic qubits is considered the“easy” part of the problem, and the logical circuit does the “hard”306

task of performing the computation. An alternative approach is one-way (or cluster-state) quantum computing.255,256,307 In one-way com-puting, a hard-to-make, highly entangled multiphoton state is sentinto an easy-to-implement processing circuit that consists only ofsingle-qubit operations, measurements, and classical feed-forward.6,308

The key idea is that, in the absence of deterministic two-photon opera-tions, the cluster state can be built up offline using nondeterministicinteractions, and then the computation progresses via those determin-istic single-qubit operations for which optics is especially suited.

Follow-up development showed how to create cluster states moreefficiently,309 leading to significantly reduced resource requirements(characterized as Bell-pairs per effective two-qubit gate, a metric ofPQC overhead; smaller $ better) compared to many other opticalschemes. The one-way computing approach is also more tolerant tolosses, compared to KLM.310 Since the first experimental demonstra-tion of the essentials of one-way quantum computing,311 considerablesteps have been made toward making larger cluster states,291,312 dem-onstrating larger computing networks,313 and improving the feed-forward performance.314 A type of one-way-based computing, wherethe computer cannot determine the input data and performs the com-putation blindly but correctly, has also been demonstrated.315

Developments in the theory of optical one-way computing have drivenincreasingly realistic schemes for large-scale photonic quantumcomputing.

Indeed, recent theoretical developments suggest that cluster-based quantum computation may be a more realistic approach towardthe future photonic quantum computer than gate-based models.There are a number of key advantages to a cluster-state approach. Oneconcerns the way that clusters are built, through progressive nondeter-ministic fusion operations309,316 that seek to merge two smallerentangled states into a larger one. The key point is that the failure ofthe nondeterministic operation slightly reduces the sizes of the initialentangled states, but does not destroy them.309 In fact, it has been

shown theoretically that missing links and nodes (e.g., due to fusionfailures or optical loss) in the constructed cluster state need not beproblematic. As long as their prevalence is below a certain threshold,percolation theory can be used to reshape the entangled state and per-form universal computation.317,318 The percolation operation corre-sponds, roughly, to a classically efficient relabeling of the cluster.Furthermore, error correction for fault-tolerant quantum operationsseems achievable, especially given modest loss thresholds.310,319–321

In principle, cluster states can be generated and processed (viaadaptive measurement) on the fly, without the need to store photonsin an optical quantum memory. This is known as ballistic cluster statecomputing.322 In this scheme, an array of sources and simple circuitsproduce entangled photons at each time step—these photons areentangled together to produce a 3D cluster where each layer representsa generation step in time. It has been shown that the depth of the clus-ter that needs to exist at any time is only of the order of a few tens ofphotons.323 In this case, given a suitably small number of faults in thecluster, the computation can proceed indefinitely in principle, with thesource array continuing to make new cluster layers at each time stepand detectors measuring a layer at each step.

Ongoing theoretical and experimental research on photonic clus-ters and ballistic schemes is also addressing many technical details(e.g., optimal cluster geometry, error correction schemes, and sourcesdesigned for cluster generation). However, it is emerging that photoniccluster schemes, and closely related ideas, are extremely plausibleapproaches for realizing universal quantum computers.320

In summary, in the span of less than two decades, photonicquantum information science has matured immensely. New photongeneration and detection technologies have enormously improved theefficiency and quality of photonic quantum states. Integrated circuitsgrew from a simple demonstration of a beam splitter to massively mul-timode reconfigurable circuits. The number of photons simultaneouslyused in experiments has grown, from 2 to 4 up to 12 (Ref. 291).Overall, experimental PQC is steadily moving toward the major goalof universal quantum computing and theoretical PQC is steadily pro-gressing toward more resource-efficient and noise-tolerant schemes.In parallel, nonuniversal quantum computation schemes such asBosonSampling are also rapidly scaling up toward the demonstrationof the true quantum computational advantage over classicalcomputers.

IV. NETWORKING QUANTUM PROCESSORS

PQC is strongly interlinked with other optical quantum informa-tion tasks. On the one hand, quantum phase estimation algorithms,used in e.g., Shor’s algorithm and a number of intermediate quantumcomputing schemes (as in Ref. 281), are also useful in quantum-enhanced metrology.324–327 On the other hand, quantum communica-tion is essential for building a distributed quantum processor frominterlinked quantum computers. Flying fast, photons (or other opticalstates) are the obvious way to transmit quantum information. Thus,photonic quantum interconnects can naturally be tasked with interfac-ing remote systems and, perhaps, local processing cores. Optical con-nections make sense regardless of the quantum system chosen forprocessing, but using photonic processing means that the interconver-sion between a stationary and a flying qubit can be skipped. (Indeed,quantum teleportation—an entanglement-based protocol used incommunication—also plays a key role in a number of PQC

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approaches.39,40) Nevertheless, it may be that there is some need toadjust the spectral properties of photons between the communicationand the processor, and ways to do this are being investigated for a vari-ety of different interconversion wavelengths, and photon-carrying andgenerating architectures.208,328–333

Creating verified communication links capable of sharing andtransmitting entanglement is essential for networking quantumcomputers, and also quantum secure communication, smallcommunication-based processing tasks (quantum communicationcomplexity334,335), and quantum networks for distributed metrol-ogy.336 A major step in entanglement verification and distributionwas the experimental implementation of loophole-free Bell tests,executed with photonic52,140 and matter qubits.337 Besides defini-tively showing local realistic explanations of entanglement are notviable, these tests confirmed that entanglement can now be rigor-ously verified in a loophole-free manner, opening the road to theunconditionally secure device-independent protocols (e.g., Ref.338). A remaining challenge is enabling these protocols in thepresence of very high loss in a communication channel used to dis-tribute the entanglement. As in PQC, loss is the predominantsource of added noise that degrades entanglement.

One can neglect the loss by postselecting only on successfuldetection events; however, such experiments do not offer device-independent security or a quantum advantage in metrology.Unfortunately, the no-cloning theorem forbids creation of identi-cal backup copies of unknown quantum states to be used if a pho-ton is lost. A state-independent attempt to amplify a qubit or qudit(i.e., to boost the photon number to its original value) would inevi-tably lead to the degradation of the state purity. Noiseless amplifi-cation can only be performed in probabilistic manner—consistentwith noise reduction being a nonunitary process—and produces awrong output upon failure. Fortunately, heralded amplification(also known as noiseless linear amplification, NLA) is possible: Inthis probabilistic scheme, successful amplification events are her-alded by an independent photon detection signal, allowing them tobe sorted from the failed trials.203 Heralded amplification can beused to distribute entanglement in the presence of loss—even withthe detection loophole closed, in principle. Since the first demon-strations,339–341 NLA has been actively researched in both thediscrete-variable (photon) and continuous-variables communities.It has shown the ability to amplify polarization,342 path,343 andtime-bin344 qubits and has been used to restore mode entangle-ment that was degraded due to loss339,343,345 and versions of thescheme have been applied to quantum communication346 andcloning.347 There have been proposals and experiments related toimplementing NLAs with quantum logic gates.348,349

Many other communication protocols for sharing high-qualityentanglement in lossy environments (i.e., for realizing quantumrepeaters) are based on entanglement swapping.350 Recent advanceshave used entanglement swapping for sharing entanglement with thedetection loophole closed, even over high-loss channels.337,351 Otherpotential tools include quantum nondemoliton measurements of thephoton number352,353 and, of course, a variety of error-correction-code protocols (e.g., Ref. 354). Ultimately, entanglement-based net-works will likely also require local processing (i.e., small quantumcomputers) for distilling entanglement and quantum memory for syn-chronizing operations.

V. CONCLUSION

Our short review has only touched briefly on other PQC ele-ments including error correction in photonic schemes,355 opticalquantum memories, and algorithms and protocols. There is also abroad range of related research that is beyond our immediate scope,including other qubit or qudit encodings—such as single-rail,6,356 par-ity state,6,357 continuous-variable,13,358–360 and hybrid207,361—as wellas other source and detector technologies. Some of these techniquesare also promising in terms of resource use and scalability. Instead, wehave covered technologies and methods that are the main focus of theexperimental development of photonic (Fock-state) quantum infor-mation processing in the medium term, and provided a firm founda-tion for the development of large-scale devices.

There is significant promise for the long term. Improvements incluster-state schemes designed specifically for photonics are providinga reduction in the overhead (from nondeterminism) and in errorthresholds—especially for loss. In conjunction, exceptional qualitysources, detectors, and gates—and large-scale integrated platforms—are providing the hardware advances required to build processorscomprising very many elements. Intermediate tasks likeBosonSampling provide a path to demonstrating a true quantum com-puting advantage sooner rather than later. Photonics continues to bethe dominant platform for connecting processors separated by dis-tance and for remote entanglement sharing, in general.

There remain other potentially transformational technologies forphotonic processing. We have only touched briefly on nonlinear inter-actions at the single-photon level—mediated by atoms, for example.Such schemes, applied at scale, could massively reduce the overhead of“linear plus measurement” approaches. However, there remain signifi-cant research and development required to capitalize on their promise.In the meantime, or perhaps in their stead, the convergence of techno-logical performance and theoretical requirements in photonic linearoptics is pointing to a bright future for photon processing.

ACKNOWLEDGMENTS

This work was conducted by the ARC Centre of Excellence forQuantum Computation and Communication Technology underGrant No. CE170100012.

REFERENCES1N. Peters, J. Altepeter, E. Jeffrey, D. Branning, and P. Kwiat, “Precise creation,characterization, and manipulation of single optical qubits,” Quantum Inf.Comput. 3, 503 (2003).

2D. P. DiVincenzo, “The physical implementation of quantum computation,”Fortschr. Phys. 48, 771–783 (2000).

3T. G. Tiecke, J. D. Thompson, N. P. de Leon, L. R. Liu, V. Vuletic, and M. D.Lukin, “Nanophotonic quantum phase switch with a single atom,” Nature508, 241 (2014).

4D. Tiarks, S. Schmidt, G. Rempe, and S. D€urr, “Optical p phase shift createdwith a single-photon pulse,” Sci. Adv. 2, e1600036 (2016).

5N. K. Langford, S. Ramelow, R. Prevedel, W. J. Munro, G. J. Milburn, and A.Zeilinger, “Efficient quantum computing using coherent photon conversion,”Nature 478, 360 (2011).

6T. Ralph and G. Pryde, “Optical quantum computation,” Prog. Opt. 54,209–269 (2010).

7P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J.Milburn, “Linear optical quantum computing with photonic qubits,” Rev.Mod. Phys. 79, 135–174 (2007).

Applied Physics Reviews REVIEW scitation.org/journal/are

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-10

VC Author(s) 2019

8J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum tech-nologies,” Nat. Photonics 3, 687–695 (2009).

9T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L.O’Brien, “Quantum computers,” Nature 464, 45 (2010).

10In this article, we use the term “photonic” to refer to schemes with countedphotons, i.e. to discrete-variable schemes such as those with qubit � photonor qudit � photon.

11S. Lloyd and S. L. Braunstein, “Quantum computation over continuous varia-bles,” Phys. Rev. Lett. 82, 1784–1787 (1999).

12S. L. Braunstein and P. van Loock, “Quantum information with continuousvariables,” Rev. Mod. Phys. 77, 513–577 (2005).

13N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A.Nielsen, “Universal quantum computation with continuous-variable clusterstates,” Phys. Rev. Lett. 97, 110501 (2006).

14F. Lenzini, J. Janousek, O. Thearle, M. Villa, B. Haylock, S. Kasture, L. Cui, H.-P. Phan, D. V. Dao, H. Yonezawa, P. K. Lam, E. H. Huntington, and M.Lobino, “Integrated photonic platform for quantum information with contin-uous variables,” Sci. Adv. 4, eaat9331 (2018).

15A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson,“Experimental high-dimensional two-photon entanglement and violations ofgeneralized bell inequalities,” Nat. Phys. 7, 677 (2011).

16V. D’Ambrosio, E. Nagali, C. H. Monken, S. Slussarenko, L. Marrucci, and F.Sciarrino, “Deterministic qubit transfer between orbital and spin angularmomentum of single photons,” Opt. Lett. 37, 172–174 (2012).

17M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: Newquantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).

18J. Roslund, R. M. de Ara�ujo, S. Jiang, C. Fabre, and N. Treps, “Wavelength-multiplexed quantum networks with ultrafast frequency combs,” Nat.Photonics 8, 109 (2013).

19Y. Cai, J. Roslund, G. Ferrini, F. Arzani, X. Xu, C. Fabre, and N. Treps,“Multimode entanglement in reconfigurable graph states using optical fre-quency combs,” Nat. Commun. 8, 15645 (2017).

20M. Kues, C. Reimer, P. Roztocki, L. R. Cort�es, S. Sciara, B. Wetzel, Y. Zhang,A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Aza~na, and R.Morandotti, “On-chip generation of high-dimensional entangled quantumstates and their coherent control,” Nature 546, 622 (2017).

21J. M. Lukens and P. Lougovski, “Frequency-encoded photonic qubits for scal-able quantum information processing,” Optica 4, 8–16 (2017).

22J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-timeentangled twin-photon source for quantum communication,” Phys. Rev. Lett.82, 2594–2597 (1999).

23P. C. Humphreys, B. J. Metcalf, J. B. Spring, M. Moore, X.-M. Jin, M. Barbieri,W. S. Kolthammer, and I. A. Walmsley, “Linear optical quantum computingin a single spatial mode,” Phys. Rev. Lett. 111, 150501 (2013).

24H. Jayakumar, A. Predojevic, T. Kauten, T. Huber, G. S. Solomon, and G.Weihs, “Time-bin entangled photons from a quantum dot,” Nat. Commun. 5,4251 (2014).

25F. Samara, A. Martin, C. Autebert, M. Karpov, T. J. Kippenberg, H. Zbinden,and R. Thew, “High-rate photon pairs and sequential time-bin entanglementwith si3n4 microring resonators,” Opt. Express 27, 19309–19318 (2019).

26D. Kielpinski, J. F. Corney, and H. M. Wiseman, “Quantum optical waveformconversion,” Phys. Rev. Lett. 106, 130501 (2011).

27B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporalmodes: A complete framework for quantum information science,” Phys.Rev. X 5, 041017 (2015).

28V. Averchenko, D. Sych, G. Schunk, U. Vogl, C. Marquardt, and G. Leuchs,“Temporal shaping of single photons enabled by entanglement,” Phys. Rev. A96, 043822 (2017).

29V. Ansari, E. Roccia, M. Santandrea, M. Doostdar, C. Eigner, L. Padberg, I.Gianani, M. Sbroscia, J. M. Donohue, L. Mancino, M. Barbieri, and C.Silberhorn, “Heralded generation of high-purity ultrashort single photons inprogrammable temporal shapes,” Opt. Express 26, 2764–2774 (2018).

30W.-B. Gao, C.-Y. Lu, X.-C. Yao, P. Xu, O. G€uhne, A. Goebel, Y.-A. Chen,C.-Z. Peng, Z.-B. Chen, and J.-W. Pan, “Experimental demonstration of ahyper-entangled ten-qubit Schr€odinger cat state,” Nat. Phys. 6, 331–335(2010).

31T. M. Graham, H. J. Bernstein, T.-C. Wei, M. Junge, and P. G. Kwiat,“Superdense teleportation using hyperentangled photons,” Nat. Commun. 6,7185 (2015).

32X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y.Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom ofa single photon,” Nature 518, 516 (2015).

33M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger,“Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248(2016).

34X.-L. Wang, Y.-H. Luo, H.-L. Huang, M.-C. Chen, Z.-E. Su, C. Liu, C. Chen,W. Li, Y.-Q. Fang, X. Jiang, J. Zhang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan,“18-qubit entanglement with six photons’ three degrees of freedom,” Phys.Rev. Lett. 120, 260502 (2018).

35M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realiza-tion of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).

36W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A.Walmsley, “Optimal design for universal multiport interferometers,” Optica3, 1460–1465 (2016).

37N. Tischler, C. Rockstuhl, and K. Słowik, “Quantum optical realization ofarbitrary linear transformations allowing for loss and gain,” Phys. Rev. X 8,021017 (2018).

38G. J. Milburn, “Quantum optical Fredkin gate,” Phys. Rev. Lett. 62,2124–2127 (1989).

39D. Gottesman and I. L. Chuang, “Demonstrating the viability of universalquantum computation using teleportation and single-qubit operations,”Nature 402, 390 (1999).

40E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantumcomputation with linear optics,” Nature 409, 46 (2001).

41C. R. Myers and R. Laflamme, “Linear optics quantum computation: An over-view,” preprint arXiv:quant-ph/0512104 (2004).

42H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control(Cambridge University Press, 2009).

43Single-Photon Generation and Detection: Physics and Applications,Experimental Methods in the Physical Sciences Vol. 45, edited by A. Migdall,S. V. Polyakov, J. Fan, and J. C. Bienfang (Academic Press, 2013).

44M. Silva, M. R€otteler, and C. Zalka, “Thresholds for linear optics quantumcomputing with photon loss at the detectors,” Phys. Rev. A 72, 032307(2005).

45M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article:Single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).

46G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K.Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski,“Picosecond superconducting single-photon optical detector,” Appl. Phys.Lett. 79, 705–707 (2001).

47K. M. Rosfjord, J. K. W. Yang, E. A. Dauler, A. J. Kerman, V. Anant, B. M.Voronov, G. N. Gol’tsman, and K. K. Berggren, “Nanowire single-photondetector with an integrated optical cavity and anti-reflection coating,” Opt.Express 14, 527–534 (2006).

48F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I.Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detectingsingle infrared photons with 93% system efficiency,” Nat. Photonics 7,210–214 (2013).

49D. V. Reddy, R. R. Nerem, A. E. Lita, S. W. Nam, R. P. Mirin, and V. B.Verma, “Exceeding 95% system efficiency within the telecom c-band in super-conducting nanowire single photon detectors,” in CLEO (OSA, 2019), p.FF1A.3.

50F. Bussieres, C. Clausen, A. Tiranov, B. Korzh, V. B. Verma, S. W. Nam, F.Marsili, A. Ferrier, P. Goldner, H. Herrmann, C. Silberhorn, W. Sohler, M.Afzelius, and N. Gisin, “Quantum teleportation from a telecom-wavelengthphoton to a solid-state quantum memory,” Nat. Photonics 8, 775 (2014).

51E. Saglamyurek, J. Jin, V. B. Verma, M. D. Shaw, F. Marsili, S. W. Nam, D.Oblak, and W. Tittel, “Quantum storage of entangled telecom-wavelengthphotons in an erbium-doped optical fibre,” Nat. Photonics 9, 83 (2015).

52L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M.J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S.D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L.Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A.

Applied Physics Reviews REVIEW scitation.org/journal/are

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-11

VC Author(s) 2019

Stern, C. Abell�an, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G.Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, “Strong loophole-free test of local realism,” Phys. Rev. Lett. 115, 250402 (2015).

53M. M. Weston, H. M. Chrzanowski, S. Wollmann, A. Boston, J. Ho, L. K.Shalm, V. B. Verma, M. S. Allman, S. W. Nam, R. B. Patel, S. Slussarenko,and G. J. Pryde, “Efficient and pure femtosecond-pulse-length source ofpolarization-entangled photons,” Opt. Express 24, 10869–10879 (2016).

54R. Valivarthi, M. G. Puigibert, Q. Zhou, G. H. Aguilar, V. B. Verma, F.Marsili, M. D. Shaw, S. W. Nam, D. Oblak, and W. Tittel, “Quantum telepor-tation across a metropolitan fibre network,” Nat. Photonics 10, 676 (2016).

55S. Slussarenko, M. M. Weston, H. M. Chrzanowski, L. K. Shalm, V. B. Verma,S. W. Nam, and G. J. Pryde, “Unconditional violation of the shot-noise limitin photonic quantum metrology,” Nat. Photonics 11, 700–703 (2017).

56J. J. Renema, R. Gaudio, Q. Wang, Z. Zhou, A. Gaggero, F. Mattioli, R. Leoni,D. Sahin, M. J. A. de Dood, A. Fiore, and M. P. van Exter, “Experimental testof theories of the detection mechanism in a nanowire superconducting singlephoton detector,” Phys. Rev. Lett. 112, 117604 (2014).

57A. Engel, J. J. Renema, K. Il’in, and A. Semenov, “Detection mechanism ofsuperconducting nanowire single-photon detectors,” Supercond. Sci. Technol.28, 114003 (2015).

58R. Gaudio, J. J. Renema, Z. Zhou, V. B. Verma, A. E. Lita, J. Shainline, M. J.Stevens, R. P. Mirin, S. W. Nam, M. P. van Exter, M. J. A. de Dood, and A.Fiore, “Experimental investigation of the detection mechanism in wsi nano-wire superconducting single photon detectors,” Appl. Phys. Lett. 109, 031101(2016).

59F. Marsili, M. J. Stevens, A. Kozorezov, V. B. Verma, C. Lambert, J. A. Stern,R. D. Horansky, S. Dyer, S. Duff, D. P. Pappas, A. E. Lita, M. D. Shaw, R. P.Mirin, and S. W. Nam, “Hotspot relaxation dynamics in a current-carryingsuperconductor,” Phys. Rev. B 93, 094518 (2016).

60J. J. Renema, R. Gaudio, Q. Wang, A. Gaggero, F. Mattioli, R. Leoni, M. P.van Exter, A. Fiore, and M. J. A. de Dood, “Probing the hotspot interactionlength in nbn nanowire superconducting single photon detectors,” Appl.Phys. Lett. 110, 233103 (2017).

61A. J. Kerman, D. Rosenberg, R. J. Molnar, and E. A. Dauler, “Readout ofsuperconducting nanowire single-photon detectors at high count rates,”J. Appl. Phys. 113, 144511 (2013).

62I. Esmaeil Zadeh, J. W. N. Los, R. B. M. Gourgues, V. Steinmetz, G. Bulgarini,S. M. Dobrovolskiy, V. Zwiller, and S. N. Dorenbos, “Single-photon detectorscombining high efficiency, high detection rates, and ultra-high timing reso-lution,” APL Photonics 2, 111301 (2017).

63B. A. Korzh, Q.-Y. Zhao, S. Frasca, J. P. Allmaras, T. M. Autry, E. A. Bersin,M. Colangelo, G. M. Crouch, A. E. Dane, T. Gerrits, F. Marsili, G. Moody, E.Ramirez, J. D. Rezac, M. J. Stevens, E. E. Wollman, D. Zhu, P. D. Hale, K. L.Silverman, R. P. Mirin, S. W. Nam, M. D. Shaw, and K. K. Berggren,“Demonstrating sub-3 ps temporal resolution in a superconducting nanowiresingle-photon detector,” preprint arXiv:1804.06839 (2018).

64J. Tiedau, E. Meyer-Scott, T. Nitsche, S. Barkhofen, T. J. Bartley, and C.Silberhorn, “A high dynamic range optical detector for measuring single pho-tons and bright light,” Opt. Express 27, 1–15 (2019).

65F. Mattioli, Z. Zhou, A. Gaggero, R. Gaudio, S. Jahanmirinejad, D. Sahin, F.Marsili, R. Leoni, and A. Fiore, “Photon-number-resolving superconductingnanowire detectors,” Supercond. Sci. Technol. 28, 104001 (2015).

66B. Cabrera, R. M. Clarke, P. Colling, A. J. Miller, S. Nam, and R. W. Romani,“Detection of single infrared, optical, and ultraviolet photons using supercon-ducting transition edge sensors,” Appl. Phys. Lett. 73, 735–737 (1998).

67I. A. Burenkov, A. K. Sharma, T. Gerrits, G. Harder, T. J. Bartley, C.Silberhorn, E. A. Goldschmidt, and S. V. Polyakov, “Full statistical modereconstruction of a light field via a photon-number-resolved measurement,”Phys. Rev. A 95, 053806 (2017).

68G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn,“Single-mode parametric-down-conversion states with 50 photons as a sourcefor mesoscopic quantum optics,” Phys. Rev. Lett. 116, 143601 (2016).

69A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-pho-tons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008).

70A. E. Lita, B. Calkins, L. A. Pellouchoud, A. J. Miller, and S. W. Nam,“Superconducting transition-edge sensors optimized for high-efficiency pho-ton-number resolving detectors,” Proc. SPIE 7681, 76810D (2010).

71D. Fukuda, G. Fujii, T. Numata, K. Amemiya, A. Yoshizawa, H. Tsuchida, H.Fujino, H. Ishii, T. Itatani, S. Inoue, and T. Zama, “Titanium-based transition-edge photon number resolving detector with 98% detection efficiency withindex-matched small-gap fiber coupling,” Opt. Express 19, 870–875 (2011).

72B. Calkins, A. E. Lita, A. E. Fox, and S. Woo Nam, “Faster recovery time of ahot-electron transition-edge sensor by use of normal metal heat-sinks,” Appl.Phys. Lett. 99, 241114 (2011).

73A. Lamas-Linares, B. Calkins, N. A. Tomlin, T. Gerrits, A. E. Lita, J. Beyer, R.P. Mirin, and S. Woo Nam, “Nanosecond-scale timing jitter for single photondetection in transition edge sensors,” Appl. Phys. Lett. 102, 231117 (2013).

74Y. Li, P. C. Humphreys, G. J. Mendoza, and S. C. Benjamin, “Resource costsfor fault-tolerant linear optical quantum computing,” Phys. Rev. X 5, 041007(2015).

75P. Senellart, G. Solomon, and A. White, “High-performance semiconductorquantum-dot single-photon sources,” Nat. Nanotechnol. 12, 1026 (2017).

76X. He, N. F. Hartmann, X. Ma, Y. Kim, R. Ihly, J. L. Blackburn, W. Gao, J.Kono, Y. Yomogida, A. Hirano, T. Tanaka, H. Kataura, H. Htoon, and S. K.Doorn, “Tunable room-temperature single-photon emission at telecom wave-lengths from sp3 defects in carbon nanotubes,” Nat. Photonics 11, 577(2017).

77T. Vogl, G. Campbell, B. C. Buchler, Y. Lu, and P. K. Lam, “Fabrication anddeterministic transfer of high-quality quantum emitters in hexagonal boronnitride,” ACS Photonics 5, 2305–2312 (2018).

78T. T. Tran, D. Wang, Z.-Q. Xu, A. Yang, M. Toth, T. W. Odom, and I.Aharonovich, “Deterministic coupling of quantum emitters in 2d materials toplasmonic nanocavity arrays,” Nano Lett. 17, 2634–2639 (2017).

79K. R. Ferguson, S. E. Beavan, J. J. Longdell, and M. J. Sellars, “Generation oflight with multimode time-delayed entanglement using storage in a solid-state spin-wave quantum memory,” Phys. Rev. Lett. 117, 020501 (2016).

80F. Dell’Anno, S. D. Siena, and F. Illuminati, “Multiphoton quantum opticsand quantum state engineering,” Phys. Rep. 428, 53–168 (2006).

81W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noisein parametric processes. I,” Phys. Rev. 124, 1646–1654 (1961).

82D. N. Klyshko, “Scattering of light in a medium with nonlinear polarizability,”Sov. Phys. JETP 28, 522 (1969).

83D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in paramet-ric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).

84C. Wagenknecht, C.-M. Li, A. Reingruber, X.-H. Bao, A. Goebel, Y.-A. Chen,Q. Zhang, K. Chen, and J.-W. Pan, “Experimental demonstration of a her-alded entanglement source,” Nat. Photonics 4, 549 (2010).

85S. Barz, G. Cronenberg, A. Zeilinger, and P. Walther, “Heralded generation ofentangled photon pairs,” Nat. Photonics 4, 553 (2010).

86D. R. Hamel, L. K. Shalm, H. H€ubel, A. J. Miller, F. Marsili, V. B. Verma, R.P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation ofthree-photon polarization entanglement,” Nat. Photonics 8, 801 (2014).

87S. Krapick, B. Brecht, H. Herrmann, V. Quiring, and C. Silberhorn, “On-chipgeneration of photon-triplet states,” Opt. Express 24, 2836–2849 (2016).

88P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y.Shih, “New high-intensity source of polarization-entangled photon pairs,”Phys. Rev. Lett. 75, 4337–4341 (1995).

89A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, “Entanglement of the angularmomentum states of photons,” Nature 412, 313–315 (2001).

90V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, “Generatingentangled two-photon states with coincident frequencies,” Phys. Rev. Lett. 88,183602 (2002).

91O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. K€artner,“Two-photon coincident-frequency entanglement via extended phasematching,” Phys. Rev. Lett. 94, 083601 (2005).

92J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation ofhyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).

93L. Caspani, C. Xiong, B. J. Eggleton, D. Bajoni, M. Liscidini, M. Galli, R.Morandotti, and D. J. Moss, “Integrated sources of photon quantum statesbased on nonlinear optics,” Light Sci. Appl. 6, e17100 (2017).

94V. Ansari, J. M. Donohue, B. Brecht, and C. Silberhorn, “Tailoring nonlinearprocesses for quantum optics with pulsed temporal-mode encodings,” Optica5, 534–550 (2018).

Applied Physics Reviews REVIEW scitation.org/journal/are

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-12

VC Author(s) 2019

95F. Flamini, N. Spagnolo, and F. Sciarrino, “Photonic quantum informationprocessing: A review,” Rep. Prog. Phys. 82, 016001 (2019).

96W. P. Grice and I. A. Walmsley, “Spectral information and distinguishabilityin type-II down-conversion with a broadband pump,” Phys. Rev. A 56, 1627(1997).

97D. N. Klyshko, “Use of two-photon light for absolute calibration of photoelec-tric detectors,” Sov. J. Quantum Electron. 10, 1112–1116 (1980).

98X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y.Lu, Y.-A. Chen, and J.-W. Pan, “Observation of eight-photon entanglement,”Nat. Photonics 6, 225–228 (2012).

99X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, and J.-W. Pan, “Experimental ten-photon entanglement,”Phys. Rev. Lett. 117, 210502 (2016).

100D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Phys. 8, 180– 198(2007).

101G. Bonfrate, V. Pruneri, P. G. Kazansky, P. Tapster, and J. G. Rarity,“Parametric fluorescence in periodically poled silica fibers,” Appl. Phys. Lett.75, 2356–2358 (1999).

102S. Tanzilli, H. D. Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. D. Micheli,D. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using peri-odically poled lithium niobate waveguide,” Electron. Lett. 37(2), 26–28(2001).

103K. Sanaka, K. Kawahara, and T. Kuga, “New high-efficiency source of photonpairs for engineering quantum entanglement,” Phys. Rev. Lett. 86, 5620–5623(2001).

104K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated pho-tons in controlled spatial modes by downconversion in nonlinear wave-guides,” Opt. Lett. 26, 1367–1369 (2001).

105A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “Awavelength-tunable fiber-coupled source of narrowband entangled photons,”Opt. Express 15, 15377–15386 (2007).

106P. G. Evans, R. S. Bennink, W. P. Grice, T. S. Humble, and J. Schaake, “Brightsource of spectrally uncorrelated polarization-entangled photons with nearlysingle-mode emission,” Phys. Rev. Lett. 105, 253601 (2010).

107T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source ofpolarization-entangled photons using a polarization Sagnac interferometer,”Phys. Rev. A 73, 012316 (2006).

108F. N. C. Wong, J. H. Shapiro, and T. Kim, “Efficient generation ofpolarization-entangled photons in a nonlinear crystal,” Laser Phys. 16,1517–1524 (2006).

109O. Kuzucu and F. N. C. Wong, “Pulsed Sagnac source of narrow-band polari-zation-entangled photons,” Phys. Rev. A 77, 032314 (2008).

110T. Scheidl, F. Tiefenbacher, R. Prevedel, F. Steinlechner, R. Ursin, and A.Zeilinger, “Crossed-crystal scheme for femtosecond-pulsed entangled photongeneration in periodically poled potassium titanyl phosphate,” Phys. Rev. A89, 042324 (2014).

111T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for sponta-neous parametric down-conversion driven by a narrow pump pulse,” Phys.Rev. A 56, 1534 (1997).

112F. K€onig and F. N. C. Wong, “Extended phase matching of second-harmonicgeneration in periodically poled ktiopo4 with zero group-velocity mismatch,”Appl. Phys. Lett. 84, 1644–1646 (2004).

113W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency andspace-time correlations in multiphoton states,” Phys. Rev. A 64, 063815(2001).

114A. U’Ren, C. Silberhorn, K. Banaszek, I. Walmsley, R. Erdmann, W. Grice, andM. Raymer, “Generation of pure-state single-photon wavepackets by condi-tional preparation based on spontaneous parametiric downconversion,” LaserPhys. 15, 146–161 (2005).

115P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C.Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single pho-tons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).

116R.-B. Jin, R. Shimizu, K. Wakui, H. Benichi, and M. Sasaki, “Widely tunablesingle photon source with high purity at telecom wavelength,” Opt. Express21, 10659–10666 (2013).

117F. Kaneda, K. Garay-Palmett, A. B. U’Ren, and P. G. Kwiat, “Heralded single-photon source utilizing highly nondegenerate, spectrally factorable spontane-ous parametric downconversion,” Opt. Express 24, 10733–10747 (2016).

118T. Gerrits, M. J. Stevens, B. Baek, B. Calkins, A. Lita, S. Glancy, E. Knill, S. W.Nam, R. P. Mirin, R. H. Hadfield, R. S. Bennink, W. P. Grice, S. Dorenbos, T.Zijlstra, T. Klapwijk, and V. Zwiller, “Generation of degenerate, factorizable,pulsed squeezed light at telecom wavelengths,” Opt. Express 19, 24434–24447(2011).

119N. Bruno, A. Martin, T. Guerreiro, B. Sanguinetti, and R. T. Thew, “Pulsedsource of spectrally uncorrelated and indistinguishable photons at telecomwavelengths,” Opt. Express 22, 17246–17253 (2014).

120C. Greganti, P. Schiansky, I. A. Calafell, L. M. Procopio, L. A. Rozema, and P.Walther, “Tuning single-photon sources for telecom multi-photonexperiments,” Opt. Express 26, 3286–3302 (2018).

121R. S. Bennink, “Optimal collinear Gaussian beams for spontaneous parametricdown-conversion,” Phys. Rev. A 81, 053805 (2010).

122R.-B. Jin, R. Shimizu, K. Wakui, M. Fujiwara, T. Yamashita, S. Miki, H. Terai,Z. Wang, and M. Sasaki, “Pulsed sagnac polarization-entangled photon sourcewith a ppktp crystal at telecom wavelength,” Opt. Express 22, 11498–11507(2014).

123A. M. Bra�nczyk, A. Fedrizzi, T. M. Stace, T. C. Ralph, and A. G. White,“Engineered optical nonlinearity for quantum light sources,” Opt. Express 19,55–65 (2011).

124B. P. Dixon, J. H. Shapiro, and F. N. C. Wong, “Spectral engineering byGaussian phase-matching for quantum photonics,” Opt. Express 21,5879–5890 (2013).

125F. Graffitti, D. Kundys, D. T. Reid, A. M. Bra�nczyk, and A. Fedrizzi, “Puredown-conversion photons through sub-coherence-length domain engineer-ing,” Quantum Sci. Technol. 2, 035001 (2017).

126C. Chen, C. Bo, M. Y. Niu, F. Xu, Z. Zhang, J. H. Shapiro, and F. N. C. Wong,“Efficient generation and characterization of spectrally factorable biphotons,”Opt. Express 25, 7300–7312 (2017).

127F. Graffitti, P. Barrow, M. Proietti, D. Kundys, and A. Fedrizzi, “Independenthigh-purity photons created in domain-engineered crystals,” Optica 5,514–517 (2018).

128C. Chen, J. E. Heyes, K.-H. Hong, M. Y. Niu, A. E. Lita, T. Gerrits, S. W. Nam,J. H. Shapiro, and F. N. C. Wong, “Indistinguishable single-mode photonsfrom spectrally engineered biphotons,” Opt. Express 27, 11626–11634 (2019).

129E. Meyer-Scott, N. Montaut, J. Tiedau, L. Sansoni, H. Herrmann, T. J. Bartley,and C. Silberhorn, “Limits on the heralding efficiencies and spectral purities ofspectrally filtered single photons from photon-pair sources,” Phys. Rev. A 95,061803 (2017).

130F. Laudenbach, R.-B. Jin, C. Greganti, M. Hentschel, P. Walther, and H.H€ubel, “Numerical investigation of photon-pair generation in periodicallypoled mTiOxo4 (m ¼ K, rb, cs; x ¼ p, as),” Phys. Rev. Appl. 8, 024035 (2017).

131K. Zielnicki, K. Garay-Palmett, D. Cruz-Delgado, H. Cruz-Ramirez, M. F.O’Boyle, B. Fang, V. O. Lorenz, A. B. U’Ren, and P. G. Kwiat, “Joint spectralcharacterization of photon-pair sources,” J. Mod. Opt. 65, 1141–1160 (2018).

132F. Graffitti, J. Kelly-Massicotte, A. Fedrizzi, and A. M. Bra�nczyk, “Design con-siderations for high-purity heralded single-photon sources,” Phys. Rev. A 98,053811 (2018).

133L. K. Shalm, K. Garay, J. Palfree, J. Migdall, A. U’Ren, and S. W. Nam, http://spdcalc.org for “Spontaneous parametric downconversion calculator.”

134J. Renema, V. Shchesnovich, and R. Garcia-Patron, “Classical simulability ofnoisy boson sampling,” preprint arXiv:1809.01953 (2019).

135N. Tischler, F. Ghafari, T. J. Baker, S. Slussarenko, R. B. Patel, M. M. Weston,S. Wollmann, L. K. Shalm, V. B. Verma, S. W. Nam, H. C. Nguyen, H. M.Wiseman, and G. J. Pryde, “Conclusive experimental demonstration of one-way Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 121, 100401 (2018).

136We use F ¼ ½Trffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qp

rffiffiffiqpp2 as the definition of the fidelity between the states

q and r.137R. Rangarajan, M. Goggin, and P. Kwiat, “Optimizing type-i polarization-

entangled photons,” Opt. Express 17, 18920–18933 (2009).138P. Bierhorst, E. Knill, S. Glancy, Y. Zhang, A. Mink, S. Jordan, A. Rommal, Y.-

K. Liu, B. Christensen, S. W. Nam, M. J. Stevens, and L. K. Shalm,“Experimentally generated randomness certified by the impossibility of super-luminal signals,” Nature 556, 223–226 (2018).

Applied Physics Reviews REVIEW scitation.org/journal/are

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-13

VC Author(s) 2019

139A. Lohrmann, A. Villar, A. Stolk, and A. Ling, “High fidelity field stop collec-tion for polarization-entangled photon pair sources,” Appl. Phys. Lett. 113,171109 (2018).

140M. Giustina, M. A. M. Versteegh, S. Wengerowsky, J. Handsteiner, A.Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-A. Larsson, C. Abell�an, W.Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K.Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger,“Significant-loophole-free test of bell’s theorem with entangled photons,”Phys. Rev. Lett. 115, 250401 (2015).

141M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pairsource for quantum communications,” IEEE Photonics Technol. Lett. 14,983–985 (2002).

142J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. J. Russell,“Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13,534–544 (2005).

143J. Fulconis, O. Alibart, W. J. Wadsworth, P. S. Russell, and J. G. Rarity, “Highbrightness single mode source of correlated photon pairs using a photoniccrystal fiber,” Opt. Express 13, 7572–7582 (2005).

144J. Fan, A. Migdall, and L. J. Wang, “Efficient generation of correlated photonpairs in a microstructure fiber,” Opt. Lett. 30, 3368–3370 (2005).

145J. Fulconis, O. Alibart, W. J. Wadsworth, and J. G. Rarity, “Quantum interfer-ence with photon pairs using two micro-structured fibres,” New J. Phys. 9, 276(2007).

146O. Gazzano and G. S. Solomon, “Toward optical quantum information proc-essing with quantum dots coupled to microstructures [invited],” J. Opt. Soc.Am. B 33, C160–C175 (2016).

147O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I.Sagnes, L. Lanco, A. Lemaıtre, and P. Senellart, “Bright solid-state sources ofindistinguishable single photons,” Nat. Commun. 4, 1425 (2013).

148N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G.Hornecker, S. L. Portalupi, T. Grange, C. Ant�on, J. Demory, C. G�omez, I.Sagnes, N. D. Lanzillotti-Kimura, A. Lemaıtre, A. Auffeves, A. G. White, L.Lanco, and P. Senellart, “Near-optimal single-photon sources in the solidstate,” Nat. Photonics 10, 340 (2016).

149J.-H. Kim, S. Aghaeimeibodi, C. J. K. Richardson, R. P. Leavitt, D. Englund,and E. Waks, “Hybrid integration of solid-state quantum emitters on a siliconphotonic chip,” Nano Lett. 17, 7394–7400 (2017).

150S. Aghaeimeibodi, B. Desiatov, J.-H. Kim, C.-M. Lee, M. A. Buyukkaya, A.Karasahin, C. J. K. Richardson, R. P. Leavitt, M. Loncar, and E. Waks,“Integration of quantum dots with lithium niobate photonics,” Appl. Phys.Lett. 113, 221102 (2018).

151S. Dutta, T. Cai, M. A. Buyukkaya, S. Barik, S. Aghaeimeibodi, and E. Waks,“Coupling quantum emitters in wse2 monolayers to a metal-insulator-metalwaveguide,” Appl. Phys. Lett. 113, 191105 (2018).

152Y. He, X. Ding, Z.-E. Su, H.-L. Huang, J. Qin, C. Wang, S. Unsleber, C. Chen,H. Wang, Y.-M. He, X.-L. Wang, W.-J. Zhang, S.-J. Chen, C. Schneider, M.Kamp, L.-X. You, Z. Wang, S. H€ofling, C.-Y. Lu, and J.-W. Pan, “Time-bin-encoded boson sampling with a single-photon device,” Phys. Rev. Lett. 118,190501 (2017).

153A. Dousse, J. Suffczy�nski, A. Beveratos, O. Krebs, A. Lemaıtre, I. Sagnes, J.Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photonpairs,” Nature 466, 217 (2010).

154A. J. Bennett, M. A. Pooley, R. M. Stevenson, M. B. Ward, R. B. Patel, A. B. dela Giroday, N. Sk€old, I. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J. Shields,“Electric-field-induced coherent coupling of the exciton states in a singlequantum dot,” Nat. Phys. 6, 947 (2010).

155D. Heinze, A. Zrenner, and S. Schumacher, “Polarization-entangled twin pho-tons from two-photon quantum-dot emission,” Phys. Rev. B 95, 245306(2017).

156M. Prilm€uller, T. Huber, M. M€uller, P. Michler, G. Weihs, and A. Predojevic,“Hyperentanglement of photons emitted by a quantum dot,” Phys. Rev. Lett.121, 110503 (2018).

157D. Huber, M. Reindl, S. F. Covre da Silva, C. Schimpf, J. Mart�ın-S�anchez, H.Huang, G. Piredda, J. Edlinger, A. Rastelli, and R. Trotta, “Strain-tunableGaAs quantum dot: A nearly dephasing-free source of entangled photon pairson demand,” Phys. Rev. Lett. 121, 033902 (2018).

158J. C. Loredo, C. Ant�on, B. Reznychenko, P. Hilaire, A. Harouri, C. Millet, H.Ollivier, N. Somaschi, L. De Santis, A. Lemaıtre, I. Sagnes, L. Lanco, A.Auffeves, O. Krebs, and P. Senellart, “Generation of non-classical light in aphoton-number superposition,” Nat. Photonics (2019).

159We have focused on quantum dots as these are the leading contender toSPDC/SFWM for photon sources in PQC. However, as mentioned earlier, avariety of other single emitters are also suitable in principle.

160D. Englund, A. Faraon, B. Zhang, Y. Yamamoto, and J. Vuckovic, “Generationand transfer of single photons on a photonic crystal chip,” Opt. Express 15,5550–5558 (2007).

161H. Wang, Y.-M. He, T.-H. Chung, H. Hu, Y. Yu, S. Chen, X. Ding, M.-C.Chen, J. Qin, X. Yang, R.-Z. Liu, Z.-C. Duan, J.-P. Li, S. Gerhardt, K. Winkler,J. Jurkat, L.-J. Wang, N. Gregersen, Y.-H. Huo, Q. Dai, S. Yu, S. H€ofling, C.-Y.Lu, and J.-W. Pan, “Towards optimal single-photon sources from polarizedmicrocavities,” Nat. Photonics (2019).

162Here, we mean the ratio of the rate of photons coupled to the rate of triggerpulses.

163H. Wang, Y. He, Y.-H. Li, Z.-E. Su, B. Li, H.-L. Huang, X. Ding, M.-C. Chen,C. Liu, J. Qin, J.-P. Li, Y.-M. He, C. Schneider, M. Kamp, C.-Z. Peng, S.H€ofling, C.-Y. Lu, and J.-W. Pan, “High-efficiency multiphoton bosonsampling,” Nat. Photonics 11, 361 (2017).

164R. B. Patel, A. J. Bennett, I. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J.Shields, “Two-photon interference of the emission from electrically tunableremote quantum dots,” Nat. Photonics 4, 632 (2010).

165D. J. P. Ellis, A. J. Bennett, C. Dangel, J. P. Lee, J. P. Griffiths, T. A. Mitchell,T.-K. Paraiso, P. Spencer, D. A. Ritchie, and A. J. Shields, “Independent indis-tinguishable quantum light sources on a reconfigurable photonic integratedcircuit,” Appl. Phys. Lett. 112, 211104 (2018).

166H. Wang, W. Li, X. Jiang, Y.-M. He, Y.-H. Li, X. Ding, M.-C. Chen, J. Qin, C.-Z. Peng, C. Schneider, M. Kamp, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, J. P.Dowling, S. H€ofling, C.-Y. Lu, and J.-W. Pan, “Toward scalable boson sam-pling with photon loss,” Phys. Rev. Lett. 120, 230502 (2018).

167C. Ant�on, J. C. Loredo, G. Coppola, H. Ollivier, N. Viggianiello, A. Harouri,N. Somaschi, A. Crespi, I. Sagnes, A. Lemaıtre, L. Lanco, R. Osellame, F.Sciarrino, and P. Senellart, “Interfacing scalable photonic platforms: Solid-state based multi-photon interference in a reconfigurable glass chip,” preprintarXiv:1905.00936 (2019).

168F. Lenzini, B. Haylock, J. C. Loredo, R. A. Abr~ahao, N. A. Zakaria, S. Kasture,I. Sagnes, A. Lemaitre, H.-P. Phan, D. V. Dao, P. Senellart, M. P. Almeida, A.G. White, and M. Lobino, “Active demultiplexing of single photons from asolid-state source,” Laser Photonics Rev. 11, 1600297 (2017).

169A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon andmultiphoton probabilities of a single-photon on-demand source,” Phys. Rev.A 66, 053805 (2002).

170J. H. Shapiro and F. N. Wong, “On-demand single-photon generation using amodular array of parametric downconverters with electro-optic polarizationcontrols,” Opt. Lett. 32, 2698–2700 (2007).

171X-s Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental gen-eration of single photons via active multiplexing,” Phys. Rev. A 83, 043814(2011).

172M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss,M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of her-alded single-photon sources,” Nat. Commun. 4, 2582 (2013).

173T. Meany, L. A. Ngah, M. J. Collins, A. S. Clark, R. J. Williams, B. J. Eggleton,M. J. Steel, M. J. Withford, O. Alibart, and S. Tanzilli, “Hybrid photonic circuitfor multiplexed heralded single photons,” Laser Photonics Rev. 8, L42–L46(2014).

174R. J. A. Francis-Jones, R. A. Hoggarth, and P. J. Mosley, “All-fiber multiplexedsource of high-purity single photons,” Optica 3, 1270–1273 (2016).

175T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Single photons on pseudode-mand from stored parametric down-conversion,” Phys. Rev. A 66, 042303(2002).

176E. Jeffrey, N. A. Peters, and P. G. Kwiat, “Towards a periodic deterministicsource of arbitrary single-photon states,” New J. Phys. 6, 100 (2004).

177M. Grimau Puigibert, G. H. Aguilar, Q. Zhou, F. Marsili, M. D. Shaw, V. B.Verma, S. W. Nam, D. Oblak, and W. Tittel, “Heralded single photons based

Applied Physics Reviews REVIEW scitation.org/journal/are

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-14

VC Author(s) 2019

on spectral multiplexing and feed-forward control,” Phys. Rev. Lett. 119,083601 (2017).

178C. Joshi, A. Farsi, S. Clemmen, S. Ramelow, and A. L. Gaeta, “Frequency mul-tiplexing for quasi-deterministic heralded single-photon sources,” Nat.Commun. 9, 847 (2018).

179G. J. Mendoza, R. Santagati, J. Munns, E. Hemsley, M. Piekarek, E. Mart�ın-L�opez, G. D. Marshall, D. Bonneau, M. G. Thompson, and J. L. O’Brien,“Active temporal and spatial multiplexing of photons,” Optica 3, 127–132(2016).

180F. Kaneda, B. G. Christensen, J. J. Wong, H. S. Park, K. T. McCusker, and P.G. Kwiat, “Time-multiplexed heralded single-photon source,” Optica 2,1010–1013 (2015).

181C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel,D. Y. Choi, C. J. Chae, P. H. W. Leong, and B. J. Eggleton, “Active temporalmultiplexing of indistinguishable heralded single photons,” Nat. Commun. 7,10853 (2016).

182F. Kaneda and P. G. Kwiat, “High-efficiency single-photon generation vialarge-scale active time multiplexing,” preprint arXiv:1803.04803 (2018).

183M. Gimeno-Segovia, H. Cable, G. J. Mendoza, P. Shadbolt, J. W. Silverstone, J.Carolan, M. G. Thompson, J. L. O’Brien, and T. Rudolph, “Relative multiplex-ing for minimising switching in linear-optical quantum computing,” New J.Phys. 19, 063013 (2017).

184X. Zhang, Y. H. Lee, B. A. Bell, P. H. W. Leong, T. Rudolph, B. J. Eggleton,and C. Xiong, “Indistinguishable heralded single photon generation via rela-tive temporal multiplexing of two sources,” Opt. Express 25, 26067–26075(2017).

185F. Kaneda, F. Xu, J. Chapman, and P. G. Kwiat, “Quantum-memory-assistedmulti-photon generation for efficient quantum information processing,”Optica 4, 1034–1037 (2017).

186A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat.Photonics 3, 706 (2009).

187E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussieres, M.George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantummemory for entangled photons,” Nature 469, 512 (2011).

188D. J. Saunders, J. H. D. Munns, T. F. M. Champion, C. Qiu, K. T. Kaczmarek,E. Poem, P. M. Ledingham, I. A. Walmsley, and J. Nunn, “Cavity-enhancedroom-temperature broadband Raman memory,” Phys. Rev. Lett. 116, 090501(2016).

189K. T. Kaczmarek, P. M. Ledingham, B. Brecht, S. E. Thomas, G. S.Thekkadath, O. Lazo-Arjona, J. H. D. Munns, E. Poem, A. Feizpour, D. J.Saunders, J. Nunn, and I. A. Walmsley, “High-speed noise-free optical quan-tum memory,” Phys. Rev. A 97, 042316 (2018).

190A. Tiranov, P. C. Strassmann, J. Lavoie, N. Brunner, M. Huber, V. B. Verma,S. W. Nam, R. P. Mirin, A. E. Lita, F. Marsili, M. Afzelius, F. Bussieres, and N.Gisin, “Temporal multimode storage of entangled photon pairs,” Phys. Rev.Lett. 117, 240506 (2016).

191M. Rancic, M. P. Hedges, R. L. Ahlefeldt, and M. J. Sellars, “Coherence time ofover a second in a telecom-compatible quantum memory storage material,”Nat. Phys. 14, 50 (2017).

192G. T. Campbell, O. Pinel, M. Hosseini, T. C. Ralph, B. C. Buchler, and P. K.Lam, “Configurable unitary transformations and linear logic gates using quan-tum memories,” Phys. Rev. Lett. 113, 063601 (2014).

193V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat,“Storage and retrieval of vector beams of light in a multiple-degree-of-freedomquantum memory,” Nat. Commun. 6, 7706 (2015).

194T.-S. Yang, Z.-Q. Zhou, Y.-L. Hua, X. Liu, Z.-F. Li, P.-Y. Li, Y. Ma, C. Liu, P.-J.Liang, X. Li, Y.-X. Xiao, J. Hu, C.-F. Li, and G.-C. Guo, “Multiplexed storageand real-time manipulation based on a multiple degree-of-freedom quantummemory,” Nat. Commun. 9, 3407 (2018).

195M. Zhong, M. P. Hedges, R. L. Ahlefeldt, J. G. Bartholomew, S. E. Beavan, S.M. Wittig, J. J. Longdell, and M. J. Sellars, “Optically addressable nuclear spinsin a solid with a six-hour coherence time,” Nature 517, 177 (2015).

196M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum mem-ory for light,” Nature 465, 1052 (2010).

197Y.-W. Cho, G. T. Campbell, J. L. Everett, J. Bernu, D. B. Higginbottom, M. T.Cao, J. Geng, N. P. Robins, P. K. Lam, and B. C. Buchler, “Highly efficient

optical quantum memory with long coherence time in cold atoms,” Optica 3,100–107 (2016).

198Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S.-L. Zhu,“Efficient quantum memory for single-photon polarization qubits,” Nat.Photonics 13, 346–351 (2019).

199M. Hosseini, G. Campbell, B. M. Sparkes, P. K. Lam, and B. C. Buchler,“Unconditional room-temperature quantum memory,” Nat. Phys. 7, 794 (2011).

200P. Vernaz-Gris, K. Huang, M. Cao, A. S. Sheremet, and J. Laurat, “Highly-effi-cient quantum memory for polarization qubits in a spatially-multiplexed coldatomic ensemble,” Nat. Commun. 9, 363 (2018).

201M. Rambach, A. Nikolova, T. J. Weinhold, and A. G. White, “Sub-megahertzlinewidth single photon source,” APL Photonics 1, 096101 (2016).

202M. Rambach, W. Y. S. Lau, S. Laibacher, V. Tamma, A. G. White, and T. J.Weinhold, “Hectometer revivals of quantum interference,” Phys. Rev. Lett.121, 093603 (2018).

203T. C. Ralph and A. P. Lund, “Nondeterministic noiseless linear amplificationof quantum systems,” in Proceedings of 9th International Conference onQuantum Communication, Measurement and Computing, edited by A.Lvovsky (2009), pp. 155–160.

204J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J.Tura, R. Augusiak, L. Mancinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q.Gong, A. Ac�ın, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G.Thompson, “Multidimensional quantum entanglement with large-scale inte-grated optics,” Science 360, 285–291 (2018).

205A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based onspectrally engineered sum frequency generation,” Opt. Express 19,13770–13778 (2011).

206S. Slussarenko, B. Piccirillo, V. Chigrinov, L. Marrucci, and E. Santamato,“Liquid crystal spatial-mode converters for the orbital angular momentum oflight,” J. Opt. 15, 025406 (2013).

207D. V. Sychev, A. E. Ulanov, E. S. Tiunov, A. A. Pushkina, A. Kuzhamuratov,V. Novikov, and A. I. Lvovsky, “Entanglement and teleportation betweenpolarization and wave-like encodings of an optical qubit,” Nat. Commun. 9,3672 (2018).

208S. Kasture, F. Lenzini, B. Haylock, A. Boes, A. Mitchell, E. W. Streed, and M.Lobino, “Frequency conversion between uv and telecom wavelengths in a lith-ium niobate waveguide for quantum communication with yb þ trappedions,” J. Opt. 18, 104007 (2016).

209S. Tanzilli, A. Martin, F. Kaiser, M. P. De Micheli, O. Alibart, and D. B.Ostrowsky, “On the genesis and evolution of integrated quantum optics,”Laser Photonics Rev. 6, 115–143 (2012).

210L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, andR. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev.Lett. 105, 200503 (2010).

211A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo,N. Viggianiello, L. Innocenti, P. Mataloni, and F. Sciarrino, “Suppression lawof quantum states in a 3d photonic fast Fourier transform chip,” Nat.Commun. 7, 10469 (2016).

212D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H.Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, and J. L. O’Brien,“Fast path and polarization manipulation of telecom wavelength single pho-tons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601(2012).

213J. P. H€opker, M. Bartnick, E. Meyer-Scott, F. Thiele, S. Krapick, N. Montaut,M. Santandrea, H. Herrmann, S. Lengeling, R. Ricken, V. Quiring, T. Meier,A. Lita, V. Verma, T. Gerrits, S. W. Nam, C. Silberhorn, and T. J. Bartley,“Towards integrated superconducting detectors on lithium niobate wave-guides,” Proc. SPIE 10358, 1035809 (2017).

214I. Krasnokutska, J.-L. J. Tambasco, X. Li, and A. Peruzzo, “Ultra-low loss pho-tonic circuits in lithium niobate on insulator,” Opt. Express 26, 897–904(2018).

215I. Krasnokutska, J.-L. J. Tambasco, and A. Peruzzo, “Tunable large free spec-tral range microring resonators in lithium niobate on insulator,” Sci. Rep. 9,11086 (2019).

216J. W. Silverstone, D. Bonneau, J. L. O’Brien, and M. G. Thompson, “Siliconquantum photonics,” IEEE J. Sel. Top. Quantum Electron. 22, 390–402(2016).

Applied Physics Reviews REVIEW scitation.org/journal/are

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-15

VC Author(s) 2019

217S. Ferrari, C. Schuck, and W. Pernice, “Waveguide-integrated superconductingnanowire single-photon detectors,” Nanophotonics 7, 1725 (2019).

218J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli,R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. H€ofling, R. Sanjines, and A. Fiore,“Waveguide superconducting single-photon detectors for integrated quantumphotonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).

219W. H. P. Pernice, C. Schuck, O. Minaeva, M. Li, G. N. Goltsman, A. V.Sergienko, and H. X. Tang, “High-speed and high-efficiency travelling wavesingle-photon detectors embedded in nanophotonic circuits,” Nat. Commun.3, 1325 (2012).

220C. Schuck, W. H. P. Pernice, and H. X. Tang, “Waveguide integrated lownoise nbtin nanowire single-photon detectors with milli-Hz dark count rate,”Sci. Rep. 3, 1893 (2013).

221O. Kahl, S. Ferrari, V. Kovalyuk, G. N. Goltsman, A. Korneev, and W. H. P.Pernice, “Waveguide integrated superconducting single-photon detectors withhigh internal quantum efficiency at telecom wavelengths,” Sci. Rep. 5, 10941(2015).

222M. K. Akhlaghi, E. Schelew, and J. F. Young, “Waveguide integrated supercon-ducting single-photon detectors implemented as near-perfect absorbers ofcoherent radiation,” Nat. Commun. 6, 8233 (2015).

223J. M€unzberg, A. Vetter, F. Beutel, W. Hartmann, S. Ferrari, W. H. P. Pernice,and C. Rockstuhl, “Superconducting nanowire single-photon detector imple-mented in a 2d photonic crystal cavity,” Optica 5, 658–665 (2018).

224B. Calkins, P. L. Mennea, A. E. Lita, B. J. Metcalf, W. S. Kolthammer, A.Lamas-Linares, J. B. Spring, P. C. Humphreys, R. P. Mirin, J. C. Gates, P. G. R.Smith, I. A. Walmsley, T. Gerrits, and S. W. Nam, “High quantum-efficiencyphoton-number-resolving detector for photonic on-chip information proc-essing,” Opt. Express 21, 22657–22670 (2013).

225J. P. H€opker, T. Gerrits, A. Lita, S. Krapick, H. Herrmann, R. Ricken, V.Quiring, R. Mirin, S. W. Nam, C. Silberhorn, and T. J. Bartley, “Integratedtransition edge sensors on titanium in-diffused lithium niobate waveguides,”APL Photonics 4, 056103 (2019).

226T. Zhong, X. Hu, F. N. C. Wong, K. K. Berggren, T. D. Roberts, and P. Battle,“High-quality fiber-optic polarization entanglement distribution at 1.3 lm tel-ecom wavelength,” Opt. Lett. 35, 1392–1394 (2010).

227A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficientsingle-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett.106, 013603 (2011).

228G. Harder, V. Ansari, B. Brecht, T. Dirmeier, C. Marquardt, and C.Silberhorn, “An optimized photon pair source for quantum circuits,” Opt.Express 21, 13975–13985 (2013).

229N. Montaut, L. Sansoni, E. Meyer-Scott, R. Ricken, V. Quiring, H. Herrmann,and C. Silberhorn, “High-efficiency plug-and-play source of heralded singlephotons,” Phys. Rev. Appl. 8, 024021 (2017).

230S. Atzeni, A. S. Rab, G. Corrielli, E. Polino, M. Valeri, P. Mataloni, N.Spagnolo, A. Crespi, F. Sciarrino, and R. Osellame, “Integrated sources ofentangled photons at the telecom wavelength in femtosecond-laser-writtencircuits,” Optica 5, 311–314 (2018).

231E. Meyer-Scott, N. Prasannan, C. Eigner, V. Quiring, J. M. Donohue, S.Barkhofen, and C. Silberhorn, “High-performance source of spectrally pure,polarization entangled photon pairs based on hybrid integrated-bulk optics,”Opt. Express 26, 32475–32490 (2018).

232F. Lenzini, A. N. Poddubny, J. Titchener, P. Fisher, A. Boes, S. Kasture, B.Haylock, M. Villa, A. Mitchell, A. S. Solntsev, A. A. Sukhorukov, and M.Lobino, “Direct characterization of a nonlinear photonic circuit’s wave func-tion with laser light,” Light Sci. Appl. 7, 17143 (2018).

233J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson,A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscalesilicon waveguides,” Opt. Express 14, 12388–12393 (2006).

234M. Davanco, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W.M. J. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-bandheralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett.100, 261104 (2012).

235K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A.Walmsley, “Photon pair-state preparation with tailored spectral properties by

spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15,14870–14886 (2007).

236B. J. Smith, P. Mahou, O. Cohen, J. S. Lundeen, and I. A. Walmsley, “Photonpair generation in birefringent optical fibers,” Opt. Express 17, 23589–23602(2009).

237J. B. Spring, P. S. Salter, B. J. Metcalf, P. C. Humphreys, M. Moore, N.Thomas-Peter, M. Barbieri, X.-M. Jin, N. K. Langford, W. S. Kolthammer, M.J. Booth, and I. A. Walmsley, “On-chip low loss heralded source of pure singlephotons,” Opt. Express 21, 13522–13532 (2013).

238J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M.Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D.Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, “On-chip quantuminterference between silicon photon-pair sources,” Nat. Photonics 8, 104(2013).

239J. B. Spring, P. L. Mennea, B. J. Metcalf, P. C. Humphreys, J. C. Gates, H. L.Rogers, C. S€oller, B. J. Smith, W. S. Kolthammer, P. G. R. Smith, and I. A.Walmsley, “Chip-based array of near-identical, pure, heralded single-photonsources,” Optica 4, 90–96 (2017).

240N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M.Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spec-trally filtered correlated photons for large-scale quantum photonic systems,”Phys. Rev. X 4, 041047 (2014).

241D. Grassani, A. Simbula, S. Pirotta, M. Galli, M. Menotti, N. C. Harris, T.Baehr-Jones, M. Hochberg, C. Galland, M. Liscidini, and D. Bajoni, “Energycorrelations of photon pairs generated by a silicon microring resonator probedby stimulated four wave mixing,” Sci. Rep. 6, 23564 (2016).

242A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-siliconwaveguide quantum circuits,” Science 320, 646–649 (2008).

243A. Crespi, R. Ramponi, R. Osellame, L. Sansoni, I. Bongioanni, F. Sciarrino, G.Vallone, and P. Mataloni, “Integrated photonic quantum gates for polariza-tion qubits,” Nat. Commun. 2, 566 (2011).

244C.-Y. Lu, D. E. Browne, T. Yang, and J.-W. Pan, “Demonstration of a com-piled version of Shor’s quantum factoring algorithm using photonic qubits,”Phys. Rev. Lett. 99, 250504 (2007).

245B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A.Gilchrist, and A. G. White, “Experimental demonstration of a compiled ver-sion of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett. 99,250505 (2007).

246A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algo-rithm on a photonic chip,” Science 325, 1221–1221 (2009).

247P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F.Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulatingand measuring entanglement and mixture with a reconfigurable photoniccircuit,” Nat. Photonics 6, 45 (2011).

248J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation ofmultiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3,346 (2009).

249B. J. Smith, D. Kundys, N. Thomas-Peter, P. G. R. Smith, and I. A. Walmsley,“Phase-controlled integrated photonic quantum circuits,” Opt. Express 17,13516–13525 (2009).

250F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’Ambrosio, P. Mataloni,F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame,“Thermally reconfigurable quantum photonic circuits at telecom wavelengthby femtosecond laser micromachining,” Light Sci. Appl. 4, e354 (2015).

251J. Notaros, J. Mower, M. Heuck, C. Lupo, N. C. Harris, G. R. Steinbrecher, D.Bunandar, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund,“Programmable dispersion on a photonic integrated circuit for classical andquantum applications,” Opt. Express 25, 21275–21285 (2017).

252J. Carolan, C. Harrold, C. Sparrow, E. Mart�ın-L�opez, N. J. Russell, J. W.Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall,M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A.Laing, “Universal linear optics,” Science 349, 711–716 (2015).

253M. G. Thompson, http://2016.qcrypt.net/wp-content/uploads/2015/11/ forpresentation at QCrypt 2016; accessed 11 June 2019.

254P. L. Mennea, W. R. Clements, D. H. Smith, J. C. Gates, B. J. Metcalf, R. H. S.Bannerman, R. Burgwal, J. J. Renema, W. S. Kolthammer, I. A. Walmsley, andP. G. R. Smith, “Modular linear optical circuits,” Optica 5, 1087–1090 (2018).

Applied Physics Reviews REVIEW scitation.org/journal/are

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-16

VC Author(s) 2019

255R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev.Lett. 86, 5188–5191 (2001).

256M. A. Nielsen, “Optical quantum computation using cluster states,” Phys.Rev. Lett. 93, 040503 (2004).

257Cluster states are graphs where the nodes are qubits and the edges are entan-gling links between the qubits, satisfying particular constraints.255

258R. B. Patel, J. Ho, F. Ferreyrol, T. C. Ralph, and G. J. Pryde, “A quantumFredkin gate,” Sci. Adv. 2, e1501531 (2016).

259T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White, “Linear opticalcontrolled-not gate in the coincidence basis,” Phys. Rev. A 65, 062324 (2002).

260T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logicoperations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).

261J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning,“Demonstration of an all-optical quantum controlled-not gate,” Nature 426,264 (2003).

262T. B. Pittman, M. J. Fitch, B. C. Jacobs, and J. D. Franson, “Experimentalcontrolled-not logic gate for single photons in the coincidence basis,” Phys.Rev. A 68, 032316 (2003).

263J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C.Ralph, and A. G. White, “Quantum process tomography of a controlled-notgate,” Phys. Rev. Lett. 93, 080502 (2004).

264P. P. Rohde, G. J. Pryde, J. L. O’Brien, and T. C. Ralph, “Quantum-gate char-acterization in an extended Hilbert space,” Phys. Rev. A 72, 032306 (2005).

265R. Okamoto, J. L. O’Brien, H. F. Hofmann, and S. Takeuchi, “Realization of aknill-laflamme-milburn controlled-not photonic quantum circuit combiningeffective optical nonlinearities,” PNAS 108, 10067–10071 (2011).

266C. H. Bennett, G. Brassard, C. Cr�epeau, R. Jozsa, A. Perez, and W. K.Wootters, “Teleporting an unknown quantum state via dual classical andEinstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).

267W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B.Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” PNAS107, 20869–20874 (2010).

268M. A. Nielsen and I. L. Chuang, Quantum Computation and QuantumInformation: 10th Anniversary Edition, 10th ed. (Cambridge University Press,New York, NY, USA, 2011).

269B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I.Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum com-puter,” Nat. Chem. 2, 106 (2010).

270R. Santagati, J. Wang, A. A. Gentile, S. Paesani, N. Wiebe, J. R. McClean, S.Morley-Short, P. J. Shadbolt, D. Bonneau, J. W. Silverstone, D. P. Tew, X.Zhou, J. L. O’Brien, and M. G. Thompson, “Witnessing eigenstates for quan-tum simulation of Hamiltonian spectra,” Sci. Adv. 4, eaap9646 (2018).

271B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J.Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifyingquantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5,134–140 (2009).

272M. Ara�ujo, A. Feix, F. Costa, and �C. Brukner, “Quantum circuits cannot con-trol unknown operations,” New J. Phys. 16, 093026 (2014).

273J. Thompson, K. Modi, V. Vedral, and M. Gu, “Quantum plug n’play:Modular computation in the quantum regime,” New J. Phys. 20, 013004(2018).

274X.-Q. Zhou, T. C. Ralph, P. Kalasuwan, M. Zhang, A. Peruzzo, B. P. Lanyon,and J. L. O’Brien, “Adding control to arbitrary unknown quantum oper-ations,” Nat. Commun. 2, 413 (2011).

275X.-D. Cai, C. Weedbrook, Z.-E. Su, M.-C. Chen, M. Gu, M.-J. Zhu, L. Li, N.-L.Liu, C.-Y. Lu, and J.-W. Pan, “Experimental quantum computing to solve sys-tems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).

276S. Barz, I. Kassal, M. Ringbauer, Y. O. Lipp, B. Dakic, A. Aspuru-Guzik, andP. Walther, “A two-qubit photonic quantum processor and its application tosolving systems of linear equations,” Sci. Rep. 4, 6115 (2014).

277E. Mart�ın-L�opez, A. Laing, T. Lawson, R. Alvarez, X.-Q. Zhou, and J. L.O’Brien, “Experimental realization of Shor’s quantum factoring algorithmusing qubit recycling,” Nat. Photonics 6, 773 (2012).

278X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D.Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson,

and J. C. F. Matthews, “Large-scale silicon quantum photonics implementingarbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).

279Earlier reviews provide references to samples of older demonstrations in thefield.

280I. Pitsios, L. Banchi, A. S. Rab, M. Bentivegna, D. Caprara, A. Crespi, N.Spagnolo, S. Bose, P. Mataloni, R. Osellame, and F. Sciarrino, “Photonic simu-lation of entanglement growth and engineering after a spin chain quench,”Nat. Commun. 8, 1569 (2017).

281A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A.Aspuru-Guzik, and J. L. O’Brien, “A variational eigenvalue solver on a pho-tonic quantum processor,” Nat. Commun. 5, 4213 (2014).

282J. Wang, S. Paesani, R. Santagati, S. Knauer, A. A. Gentile, N. Wiebe, M.Petruzzella, J. L. O’Brien, J. G. Rarity, A. Laing, and M. G. Thompson,“Experimental quantum Hamiltonian learning,” Nat. Phys. 13, 551 (2017).

283S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Gr€afe, R.Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A.Szameit, “Implementation of quantum and classical discrete fractional Fouriertransforms,” Nat. Commun. 7, 11027 (2016).

284M. Stobi�nska, A. Buraczewski, M. Moore, W. R. Clements, J. J. Renema, S. W.Nam, T. Gerrits, A. Lita, W. S. Kolthammer, A. Eckstein, and I. A. Walmsley,“Quantum interference enables constant-time quantum information proc-essing,” Sci. Adv. 5, eaau9674 (2019).

285S. Aaronson and A. Arkhipov, “The computational complexity of linearoptics,” Theory Comput. 9, 143–252 (2013).

286M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C.Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,”Science 339, 794–798 (2013).

287J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M.Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates,B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a pho-tonic chip,” Science 339, 798–801 (2013).

288M. Tillmann, B. Dakic, R. Heilmann, S. Nolte, A. Szameit, and P. Walther,“Experimental boson sampling,” Nat. Photonics 7, 540 (2013).

289N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S.Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galv~ao,and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat.Photonics 8, 615 (2014).

290J. Carolan, J. D. A. Meinecke, P. J. Shadbolt, N. J. Russell, N. Ismail, K.W€orhoff, T. Rudolph, M. G. Thompson, J. L. O’Brien, J. C. F. Matthews, andA. Laing, “On the experimental verification of quantum complexity in linearoptics,” Nat. Photonics 8, 621 (2014).

291H.-S. Zhong, Y. Li, W. Li, L.-C. Peng, Z.-E. Su, Y. Hu, Y.-M. He, X. Ding, W.Zhang, H. Li, L. Zhang, Z. Wang, L. You, X.-L. Wang, X. Jiang, L. Li, Y.-A.Chen, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “12-photon entanglement and scal-able scattershot boson sampling with optimal entangled-photon pairs fromparametric down-conversion,” Phys. Rev. Lett. 121, 250505 (2018).

292D. J. Brod, E. F. Galv~ao, A. Crespi, R. Osellame, N. Spagnolo, and F. Sciarrino,“Photonic implementation of boson sampling: A review,” Adv. Photonics 1,034001 (2019).

293A. P. Lund, M. J. Bremner, and T. C. Ralph, “Quantum sampling problems,Bosonsampling and quantum supremacy,” npj Quantum Inf. 3, 15 (2017).

294A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O’Brien, and T. C.Ralph, “Boson sampling from a Gaussian state,” Phys. Rev. Lett. 113, 100502(2014).

295S. Rahimi-Keshari, T. C. Ralph, and C. M. Caves, “Sufficient conditions forefficient classical simulation of quantum optics,” Phys. Rev. X 6, 021039(2016).

296A. Neville, C. Sparrow, R. Clifford, E. Johnston, P. M. Birchall, A. Montanaro,and A. Laing, “Classical boson sampling algorithms with superior perfor-mance to near-term experiments,” Nat. Phys. 13, 1153 (2017).

297J. Preskill, “Quantum Computing in the NISQ era and beyond,” Quantum 2,79 (2018).

298M. Y. Niu, S. Boixo, V. N. Smelyanskiy, and H. Neven, “Universal quantumcontrol through deep reinforcement learning,” npj Quantum Inf. 5, 33 (2019).

299S. Mavadia, V. Frey, J. Sastrawan, S. Dona, and M. J. Biercuk, “Prediction andreal-time compensation of qubit decoherence via machine learning,” Nat.Commun. 8, 14106 (2017).

Applied Physics Reviews REVIEW scitation.org/journal/are

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-17

VC Author(s) 2019

300J. Walln€ofer, A. A. Melnikov, W. D�ur, and H. J. Briegel, “Machine learningfor long-distance quantum communication,” preprint arXiv:1904.10797(2019).

301A. Hentschel and B. C. Sanders, “Efficient algorithm for optimizing adaptivequantum metrology processes,” Phys. Rev. Lett. 107, 233601 (2011).

302A. Lumino, E. Polino, A. S. Rab, G. Milani, N. Spagnolo, N. Wiebe, and F.Sciarrino, “Experimental phase estimation enhanced by machine learning,”Phys. Rev. Appl. 10, 044033 (2018).

303X.-D. Cai, D. Wu, Z.-E. Su, M.-C. Chen, X.-L. Wang, L. Li, N.-L. Liu, C.-Y.Lu, and J.-W. Pan, “Entanglement-based machine learning on a quantumcomputer,” Phys. Rev. Lett. 114, 110504 (2015).

304J. Gao, L.-F. Qiao, Z.-Q. Jiao, Y.-C. Ma, C.-Q. Hu, R.-J. Ren, A.-L. Yang, H.Tang, M.-H. Yung, and X.-M. Jin, “Experimental machine learning of quan-tum states,” Phys. Rev. Lett. 120, 240501 (2018).

305A. Pepper, N. Tischler, and G. J. Pryde, “Experimental realization of a quan-tum autoencoder: The compression of qutrits via machine learning,” Phys.Rev. Lett. 122, 060501 (2019).

306The terms “hard” and “easy” are rather qualitative, but they give an idea ofthe motivation for this approach.

307R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quan-tum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).

308H. J. Briegel, D. E. Browne, W. D€ur, R. Raussendorf, and M. Van den Nest,“Measurement-based quantum computation,” Nat. Phys. 5, 19 (2009).

309D. E. Browne and T. Rudolph, “Resource-efficient linear optical quantumcomputation,” Phys. Rev. Lett. 95, 010501 (2005).

310M. Varnava, D. E. Browne, and T. Rudolph, “How good must single photonsources and detectors be for efficient linear optical quantum computation?”Phys. Rev. Lett. 100, 060502 (2008).

311P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M.Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,”Nature 434, 169 (2005).

312C.-Y. Lu, X.-Q. Zhou, O. G€uhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel,T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graphstates,” Nat. Phys. 3, 91 (2007).

313C. Greganti, M.-C. Roehsner, S. Barz, T. Morimae, and P. Walther,“Demonstration of measurement-only blind quantum computing,” New J.Phys. 18, 013020 (2016).

314R. Prevedel, P. Walther, F. Tiefenbacher, P. Bohi, R. Kaltenbaek, T. Jennewein,and A. Zeilinger, “High-speed linear optics quantum computing using activefeed-forward,” Nature 445, 65–69 (2007).

315S. Barz, E. Kashefi, A. Broadbent, J. F. Fitzsimons, A. Zeilinger, and P.Walther, “Demonstration of blind quantum computing,” Science 335,303–308 (2012).

316F. Ewert and P. van Loock, “3/4-efficient bell measurement with passive linearoptics and unentangled ancillae,” Phys. Rev. Lett. 113, 140403 (2014).

317K. Kieling, T. Rudolph, and J. Eisert, “Percolation, renormalization, and quan-tum computing with nondeterministic gates,” Phys. Rev. Lett. 99, 130501(2007).

318M. Pant, D. Towsley, D. Englund, and S. Guha, “Percolation thresholds forphotonic quantum computing,” Nat. Commun. 10, 1070 (2019).

319M. Varnava, D. E. Browne, and T. Rudolph, “Loss tolerance in one-way quan-tum computation via counterfactual error correction,” Phys. Rev. Lett. 97,120501 (2006).

320T. Rudolph, “Why I am optimistic about the silicon-photonic route to quan-tum computing,” APL Photonics 2, 030901 (2017).

321S. Morley-Short, M. Gimeno-Segovia, T. Rudolph, and H. Cable, “Loss-toler-ant teleportation on large stabilizer states,” Quantum Sci. Technol. 4, 025014(2019).

322M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From three-photon greenberger-horne-zeilinger states to ballistic universal quantumcomputation,” Phys. Rev. Lett. 115, 020502 (2015).

323S. Morley-Short, S. Bartolucci, M. Gimeno-Segovia, P. Shadbolt, H. Cable, andT. Rudolph, “Physical-depth architectural requirements for generating univer-sal photonic cluster states,” Quantum Sci. Technol. 3, 015005 (2017).

324H. Wiseman, D. Berry, S. Bartlett, B. Higgins, and G. Pryde, “Adaptive mea-surements in the optical quantum information laboratory,” IEEE J. Sel. Top.Quantum Electron. 15, 1661–1672 (2009).

325B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde,“Entanglement-free Heisenberg-limited phase estimation,” Nature 450,393–396 (2007).

326A. A. Berni, T. Gehring, B. M. Nielsen, V. H€andchen, M. G. A. Paris, and U.L. Andersen, “Ab initio quantum-enhanced optical phase estimation usingreal-time feedback control,” Nat. Photonics 9, 577 (2015).

327S. Daryanoosh, S. Slussarenko, D. W. Berry, H. M. Wiseman, and G. J. Pryde,“Experimental optical phase measurement approaching the exact Heisenberglimit,” Nat. Commun. 9, 4606 (2018).

328M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum trans-duction of telecommunications-band single photons from a quantum dot byfrequency upconversion,” Nat. Photonics 4, 786 (2010).

329R. Ikuta, Y. Kusaka, T. Kitano, H. Kato, T. Yamamoto, M. Koashi, and N.Imoto, “Wide-band quantum interface for visible-to-telecommunicationwavelength conversion,” Nat. Commun. 2, 537 (2011).

330K. De Greve, L. Yu, P. L. McMahon, J. S. Pelc, C. M. Natarajan, N. Y. Kim, E.Abe, S. Maier, C. Schneider, M. Kamp, S. H. ofling, R. H. Hadfield, A. Forchel,M. M. Fejer, and Y. Yamamoto, “Quantum-dot spin-photon entanglement viafrequency downconversion to telecom wavelength,” Nature 491, 421 (2012).

331B. A. Bell, J. He, C. Xiong, and B. J. Eggleton, “Frequency conversion in siliconin the single photon regime,” Opt. Express 24, 5235–5242 (2016).

332H. R€utz, K.-H. Luo, H. Suche, and C. Silberhorn, “Towards a quantum inter-face between telecommunication and uv wavelengths: Design and classicalperformance,” Appl. Phys. B 122, 13 (2016).

333A. Dr�eau, A. Tchebotareva, A. E. Mahdaoui, C. Bonato, and R. Hanson,“Quantum frequency conversion of single photons from a nitrogen-vacancycenter in diamond to telecommunication wavelengths,” Phys. Rev. Appl. 9,064031 (2018).

334H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communi-cation complexity,” Rev. Mod. Phys. 82, 665–698 (2010).

335K. Wei, N. Tischler, S.-R. Zhao, Y.-H. Li, J. M. Arrazola, Y. Liu, W. Zhang, H.Li, L. You, Z. Wang, Y.-A. Chen, B. C. Sanders, Q. Zhang, G. J. Pryde, F. Xu,and J.-W. Pan, “Experimental quantum switching for exponentially superiorquantum communication complexity,” Phys. Rev. Lett. 122, 120504 (2019).

336D. Gottesman, T. Jennewein, and S. Croke, “Longer-baseline telescopes usingquantum repeaters,” Phys. Rev. Lett. 109, 070503 (2012).

337B. Hensen, H. Bernien, A. E. Dreau, A. Reiserer, N. Kalb, M. S. Blok, J.Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellan, W. Amaya, V.Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner,T. H. Taminiau, and R. Hanson, “Loophole-free bell inequality violation usingelectron spins separated by 1.3 kilometres,” Nature 526, 682–686 (2015).

338A. Ac�ın, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,”Phys. Rev. Lett. 98, 230501 (2007).

339G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heraldednoiseless linear amplification and distillation of entanglement,” Nat.Photonics 4, 316 (2010).

340F. Ferreyrol, M. Barbieri, R. Blandino, S. Fossier, R. Tualle-Brouri, and P.Grangier, “Implementation of a nondeterministic optical noiseless amplifier,”Phys. Rev. Lett. 104, 123603 (2010).

341A. Zavatta, J. Fiur�a�sek, and M. Bellini, “A high-fidelity noiseless amplifier forquantum light states,” Nat. Photonics 5, 52 (2010).

342S. Kocsis, G. Y. Xiang, T. C. Ralph, and G. J. Pryde, “Heralded noiseless ampli-fication of a photon polarization qubit,” Nat. Phys. 9, 23–28 (2013).

343F. Monteiro, E. Verbanis, V. C. Vivoli, A. Martin, N. Gisin, H. Zbinden, andR. T. Thew, “Heralded amplification of path entangled quantum states,”Quantum Sci. Technol. 2, 024008 (2017).

344N. Bruno, V. Pini, A. Martin, V. B. Verma, S. W. Nam, R. Mirin, A. Lita, F.Marsili, B. Korzh, F. Bussieres, N. Sangouard, H. Zbinden, N. Gisin, and R.Thew, “Heralded amplification of photonic qubits,” Opt. Express 24, 125–133(2016).

345A. E. Ulanov, I. A. Fedorov, A. A. Pushkina, Y. V. Kurochkin, T. C. Ralph,and A. I. Lvovsky, “Undoing the effect of loss on quantum entanglement,”Nat. Photonics 9, 764–768 (2015).

346H. M. Chrzanowski, N. Walk, S. M. Assad, J. Janousek, S. Hosseini, T. C.Ralph, T. Symul, and P. K. Lam, “Measurement-based noiseless linear amplifi-cation for quantum communication,” Nat. Photonics 8, 333 (2014).

Applied Physics Reviews REVIEW scitation.org/journal/are

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-18

VC Author(s) 2019

347J. Y. Haw, J. Zhao, J. Dias, S. M. Assad, M. Bradshaw, R. Blandino, T. Symul,T. C. Ralph, and P. K. Lam, “Surpassing the no-cloning limit with a heraldedhybrid linear amplifier for coherent states,” Nat. Commun. 7, 13222 (2016).

348N. A. McMahon, A. P. Lund, and T. C. Ralph, “Optimal architecture for anondeterministic noiseless linear amplifier,” Phys. Rev. A 89, 023846 (2014).

349J. Ho, A. Boston, M. Palsson, and G. Pryde, “Experimental noiseless linearamplification using weak measurements,” New J. Phys. 18, 093026 (2016).

350T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger, “Experimental nonlocal-ity proof of quantum teleportation and entanglement swapping,” Phys. Rev.Lett. 88, 017903 (2001).

351M. M. Weston, S. Slussarenko, H. M. Chrzanowski, S. Wollmann, L. K. Shalm,V. B. Verma, M. S. Allman, S. W. Nam, and G. J. Pryde, “Heralded quantumsteering over a high-loss channel,” Sci. Adv. 4, e1701230 (2018).

352G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, “Measuringa photonic qubit without destroying it,” Phys. Rev. Lett. 92, 190402 (2004).

353E. Meyer-Scott, D. McCloskey, K. Gołos, J. Z. Salvail, K. A. G. Fisher, D. R.Hamel, A. Cabello, K. J. Resch, and T. Jennewein, “Certifying the presence ofa photonic qubit by splitting it in two,” Phys. Rev. Lett. 116, 070501 (2016).

354A. Bolt, G. Duclos-Cianci, D. Poulin, and T. M. Stace, “Foliated quantumerror-correcting codes,” Phys. Rev. Lett. 117, 070501 (2016).

355J. M. Auger, H. Anwar, M. Gimeno-Segovia, T. M. Stace, and D. E. Browne,“Fault-tolerant quantum computation with nondeterministic entanglinggates,” Phys. Rev. A 97, 030301 (2018).

356A. P. Lund and T. C. Ralph, “Nondeterministic gates for photonic single-railquantum logic,” Phys. Rev. A 66, 032307 (2002).

357A. Gilchrist, A. J. F. Hayes, and T. C. Ralph, “Efficient parity-encoded opticalquantum computing,” Phys. Rev. A 75, 052328 (2007).

358N. C. Menicucci, “Fault-tolerant measurement-based quantum computingwith continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504(2014).

359S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S.Suzuki, J-i Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa,“Ultra-large-scale continuous-variable cluster states multiplexed in the timedomain,” Nat. Photonics 7, 982 (2013).

360M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multi-partite entanglement of 60 modes of a quantum optical frequency comb,”Phys. Rev. Lett. 112, 120505 (2014).

361D. Drahi, D. V. Sychev, K. K. Pirov, E. A. Sazhina, V. A. Novikov, I. A.Walmsley, and A. I. Lvovsky, “Quantum interface between single- and dual-rail optical qubits,” preprint arXiv:1905.08562 (2019).

Applied Physics Reviews REVIEW scitation.org/journal/are

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-19

VC Author(s) 2019