Photonic Quantum Information Processing: a Concise Review

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Photonic quantum information processing: A concise review

Author Slussarenko, Sergei, Pryde, Geoff J

Published 2019

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DOI https://doi.org/10.1063/1.5115814

Copyright Statement © 2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Griffith Research Online https://research-repository.griffith.edu.au Photonic quantum information processing: A concise review

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

Sergei Slussarenko , and Geoff J. Pryde

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Paper published as part of the special topic on Quantum Computing Note: This paper is part of the Special Topic on Quantum Computing.

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

Photonic quantum information processing: A concise review

Cite as: Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 Submitted: 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.au b)Electronic mail: g.pryde@grifﬁth.edu.au

ABSTRACT Photons have been a flagship system for studying quantum mechanics, advancing quantum information science, and developing quantum technologies. Quantum entanglement, teleportation, quantum key distribution, and early quantum computing demonstrations were pioneered in this technology because photons represent a naturally mobile and low-noise system with quantum-limited detection readily available. 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 protocol realizations has raced ahead as both underlying technologies and theoretical schemes have continued to develop. Today, photonic quantum computing represents an exciting path to medium- and large-scale processing. It promises to put aside its reputation for requiring excessive resource overheads due to inefficient two-qubit gates. Instead, the ability to generate large numbers of photons—and the development of integrated platforms, improved sources and detectors, novel noise-tolerant theoretical approaches, and more—have solidified it as a leading contender for both quantum information processing and quantum networking. Our concise review provides a flyover of some key aspects of the field, with a focus on experiment. Apart from being a short and accessible introduction, its many references to in-depth articles and 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 CONTENTS I. INTRODUCTION I. INTRODUCTION ...... 1 A. Optical quantum computing A. Optical quantum computing...... 1 With the invention of the quantum computing (QC) concept, the B. Basics...... 2 development of suitable optical quantum technology became both an II. PHOTON TECHNOLOGY ...... 2 A. Detecting a photon...... 2 interesting approach to the problem and a necessity. On the one hand, B. Generating a photon...... 3 the advantages of using photons as information carriers seem to be C. Generating a photon deterministically ...... 4 obvious: photons are clean and decoherence-free quantum systems for D. Manipulating a photon ...... 5 which single-qubit operations can be easily performed with incredibly 1 E. Integrated quantum photonics ...... 6 high fidelity. On the other hand, quantum information handling with III. QUANTUM COMPUTING...... 7 photons as “flying qubits” is required for communication-based quan- A. Intermediate quantum computing ...... 7 tum information science tasks, such as networking quantum com- B. Cluster-state based computing ...... 9 puters and enabling distributed processing. IV. NETWORKING QUANTUM PROCESSORS...... 9 In terms of the traditional DiVincenzo criteria of a quantum V. CONCLUSION ...... 10 computer,2 five out of seven are essentially satisfied by choosing

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-1 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

photons. The remaining criteria are harder to satisfy because photons outlined by Reck et al.35 quite some time ago, with recent improve- do not easily interact, making deterministic two-qubit gates a chal- ments36 and expansions.37 In principle, Reck-type schemes could per- lenge. Among the additional technical considerations is photon loss, form universal processing with a single photon in many modes used which arises from currently imperfect detection and photon genera- to represent multiple qubits. Unfortunately, that encoding leads to tion techniques, and from scattering and absorption in optical compo- exponential scaling in the number of optical components, and thus nents comprising the computation circuits. Although photons are cannot be used to build a scalable quantum computer. Thus, the use of always flying, computing and networking tasks may need them to be multiple single photons is required for circuits with two-qubit gates delayed or stored, so an extra device—an optical quantum memory— and beyond. may sometimes be needed. Addressing each of these considerations It is natural, then, to implement one qubit per photon, with a requires additional resources, creating a notionally large optical QC dual-rail encoding. Two-qubit operations require the ability to apply a overhead that has sometimes led to negative perceptions of the pho- p phase shift rotation on one of the qubits depending on the state of tonic approach. the remaining qubit.38 These are trickier to implement than single Of course, there is intense research under way in the develop- qubit operations, since this is a nonlinear optical interaction, and such ment of deterministic optical (but matter-mediated) quantum gates,3–5 optical nonlinearities, at the single photon level, are extremely weak. which could take photonic quantum computing (PQC) in a new direc- An alternative is to mimic nonlinear operations with linear optics and tion. Meanwhile, the idea of linear optical quantum computing measurement, resulting in a probabilistic gate that provides the correct (LOQC) that relies on simple, but probabilistic, quantum operations operation after an appropriate postselection, or with an additional her- has increasing promise as it has continued development over the last alding signal. 20 years. The earlier history of the field is covered in previous Historically, a variety of approaches to efficient optical quantum 6–9 reviews that have appeared regularly in the literature. Here, we do computing were discussed and investigated, for example Ref. 39 and not provide a typical review—that is, we do not present a comprehen- references therein. However, the field of LOQC took off with the pro- sive encapsulation of all the achievements of the field during the past posal of Knill, Laflamme, and Milburn (KLM),40 who invented a scal- decade. Instead, we concentrate on the few technological, experimen- able photonic scheme that required linear optics components, single tal, and theoretical advances that we think play key roles on the path photon detection, and classical feed-forward only (the reader may toward a universal quantum computer operating with individual pho- enjoy reading Ref. 41 for comprehensive lecture notes and Ref. 7 for a tons and linear operations. On the technology side, we look at photon historical overview of KLM). The KLM scheme essentially works by detection and generation tools, and integrated waveguide technol- using nonclassical interference to generate a phase shift that is nonlin- ogy—and some new intermediate quantum computing demonstra- ear with respect to the photon number, conditioned on photons tions that these enable. On the conceptual side, we discuss a few appearing at certain heralding modes. These operations are then built promising ways toward a realistic universal linear optical quantum into nondeterministic logical gates. The gates are used in a repeat- 10 computer. We will concentrate on photonic quantum computing until-success mode, and the operation of a successful gate is teleported (PQC) that relies on qubits encoded in discrete variables, noting, how- onto the logical qubits. The use of a large number of concatenated ever, that quantum computing with continuous variables has now steps, and lots of ancilla photons, leads to essentially deterministic 11–14 become an important part of LOQC. But before that, we start gates. The KLM scheme theoretically allowed for a resource-efficient with a brief refresher on the basic conceptual elements and history of implementation of two- and multiqubit gates—unlike encoding a sin- PQC. gle photon across many modes, the resource scaling was not exponential in the number of qubits, but rather linear. Thus the KLM scheme B. Basics provides a pathway to build a universal quantum computer, albeit A qubit can be encoded as probability amplitudes corresponding with a large overhead of ancilla qubits (and their associated circuitry) to the photon occupation of two modes of some degree of freedom of to deal with the use of nondeterministic two-qubit gates. With the the optical field. This method is known as dual-rail encoding. The advent of a viable theoretical approach, photonic quantum computing most commonly used mode pairs are orthogonal polarizations or non- became the subject of extensive theoretical and experimental develop- overlapping propagation paths, but recently, other degrees of freedom ment. As well finding approaches that reduce the overhead due to such as transverse spatial,15–17 frequency mode,18–21 temporal bin- nondeterminism, making this scheme practical also requires high- ,22–25 and temporal mode-23,26–29 encoding are attracting attention. quality technological components to make, manipulate, and measure42 One-qubit operations—i.e., the shifting of single-photon population the photon qubits. We first turn our attention to these technology between the two modes that comprise the dual-rail qubit, and applica- considerations. tions of phase shifts between them—are easily and reliably imple- II. PHOTON TECHNOLOGY mented using interferometry in the degree of freedom of choice. A great advantage of optical quantum computing is that it does not have A. Detecting a photon to be confined to qubits: Many of the degrees of freedom listed provide A photon’s life in a quantum experiment starts with its genera- a natural way to encode multilevel qudits. Moreover, several degrees of tion and concludes with its detection. Both processes need to be effi- freedom of the same photon can be used simultaneously.30–34 (As we cient, and their performance and properties play essential roles in will discuss later, these tools provide a natural advantage for optics, PQC. In this section, we start from the end—with a look at single pho- allowing for simpler logical circuits even when working with qubits as ton detection43 technology. the basic logical elements.) A way to realize an arbitrary n-dimensional An ideal photon detector (PD) clicks every time a photon hits it unitary transformation on the mode space, with linear optics, has been and immediately restarts its operation. It does not produce false

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-2 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

positive signals when no real photons were detected (so-called “dark transition-edge sensors (TESs),66 can be also employed in experiments counts”) and it also tells exactly how many photons were detected in where photon number counting is essential. TES detectors work as the same spatiotemporal mode. Such ideal photon detectors do not bolometers with single-photon-level resolution: Absorption near the exist yet. Existing PDs are correspondingly characterized by detection superconducting transition changes the resistance of the device mono- efficiency gd, reset time sR (that sets the maximum detection rate), tonically with the photon number, which can be read out through an 67 detection time jitter sj, dark count rate Cd, and photon-number- integrating circuit. TESs have excellent PNR skills: In recent experi- resolving (PNR) capabilities. While a “perfect” PD is not actually ments, they were able to efficiently discriminate up to 20 photons in required for PQC,44 improving the PD performance to very high levels the same spatiotemporal mode.68 Atthesametime,theyhaveshown 69 is important for a realistic and scalable platform. to be able to reach gd 0:95 in the telecom wavelength range, with 70,71 Setting aside historical and exotic approaches, the PD of choice further developments leading to even higher gd 0:98, closely for optical quantum information science experiments has been the Si approaching the ideal gd ¼ 1. TESs can also be optimized to any avalanche photodiode (APD) operating in Geiger mode. These are rel- wavelength in the visible and IR ranges. A critical drawback of a TES atively fast (sR 100 ns), low-noise (typical Cd 100 counts per sec- detector is its slow operation, with microsecond reset times and ond) detectors. Unfortunately, their limited quantum efficiency, 50 ns time jitter. Efforts in improving TES time performance are 72 typically up to gd 65%, sets a practical limit on the number of pho- ongoing, with reset times as fast as sR 460 ns and time jitters of 73 tons that can be used simultaneously in an experiment. A probability down to sj ¼ 2:3ns (for 775 nm photons) having been demon- of detecting, say, ten photons with ten detectors is already less than strated. Still, these numbers are at least two orders of magnitude higher 2%, and things get exponentially worse with increasing photon num- than what might be considered practical for PQC, where clock cycles of ber. Si APDs do not possess PNR capabilities45 and their maximum Շ10 ns are likely required for the practical switching of flying photons. efficiency wavelength range is quite limited. In particular, it does not cover the telecommunications bands around 1310 and 1550 nm. The B. Generating a photon equivalent detector for 1550 nm, the InGaAs APD, suffers from lower Having exceptional detectors is not much use if one cannot effi- quantum efficiency and higher dark counts. ciently make high-quality photons on which to encode qubits. Inefficient detection was a significant limiting factor for PQC for Computing tasks in the near and long term requires the capabil- quite some time. Things started to turn for the better with the advent ity of simultaneously generating a large74 (N 10 1011)numberof 46,47 of superconducting nanowire single-photon detectors (SNSPDs). single photon states. The obvious way to achieve this is to have a large These provided something close to a direct substitute for the usual (N 10 1011) number of deterministic sources that can simulta- APDs: They have comparable (sR 40 ns) reset times, yet can achieve neously produce one and only one photon each at the push of a button detection efficiencies of up to gd 0:93 (Ref. 48) [and recently even (i.e., on a trigger event). Moreover, these photons must necessarily be: gd 0:95 (Ref. 49)] in the telecom wavelength range. SNSPDs work (a) efficiently collected so to be sent into the PQC processor and not by passing a current though a superconducting nanowire close to the lost (e.g., by absorption, scattering, diffraction, or mode mismatch dur- critical current—then, the energy absorbed from even a single photon ing the generation and fiber in-coupling process); (b) in a pure quan- can transition the device to normal resistivity. The subsequent voltage tum state and indistinguishable from one another; and (c) compatible spike is filtered and amplified, and registered as a detection. SNSPDs with the low-loss material and high-efficiency detection technology are a bit more complicated to operate than APDs, as they require cryo- from above. At present, sources that properly satisfy this list do not genic temperatures of 0.8 K–3 K (depending on the superconducting exist. However, truly deterministic, high-quality photon sources like material), but the massive enhancement in detection efficiency justifies this are being developed using diverse physical systems,45 such as the inconvenience. SNSPD performance can also be optimized to any trapped ions and atoms, color centers in diamonds, semiconductors, wavelength by selecting the appropriate material and designing a suit- quantum dots,75 and other, more exotic, methods (e.g., Refs. 76–78). able optical cavity that envelops the nanowire. They can also be Some of these rely on the use of a single emitter that, in principle, nat- designed to efficiently interface with fiber-optic inputs. In short, urally provides on-demand single-photon emission, while others— besides providing an enormous increase in detection efficiency, such as atomic ensemble79 and parametric nonlinear processes80— SNSPDs have enabled operation at telecom wavelength that benefits require heralding signals and switching to make them so. (The from previous development of optical materials and efficient photonic requirements for achieving deterministic operation in practice will be tools. This detector performance is also beneficial for quantum com- considered Subsection II C.) munication and other low-loss applications, e.g., Refs. 50–55. In the meantime, the key enabling technology for experimental Research on superconducting detectors is still ongoing, aimed at quantum optics, spontaneous parametric downconverson81–83 understanding detection mechanisms in different types of nanowire (SPDC), remains a practical way to generate high-quality single pho- materials,56–60 improving its performance in terms of reset times61 and tons nondeterminstically. Developments in this technology have effec- time jitter,62,63 and developing new methods of accurate detection effi- tively addressed the feature list (a)–(c) above. In this three-wave ciency measurements.64 Although intrinsic dark counts are low, mixing nonlinear vð2Þ process, a pump photon from a laser has a small SNSPDs are susceptible to picking up background thermal radiation probability to be converted into a pair of “daughter” (signal and idler) ~ ~ ~ from the input fiber’s room-temperature environment—this can be photons. The process must obey the momentum (kp ¼ ks þ ki,phase ~ overcome by spectral filtering. matching) and energy (xp ¼ xs þ xi) conservation laws, with kn and The key remaining limitation of this technology is the lack of xn, n ¼ p; s; i, being the wavevectors and angular frequencies of PNR capability. While schemes that turn SNSPDs into PNR detectors pump, signal and idler photons, respectively. SPDC is probabilistic, are being investigated,65 a different type of detector, based on but it can be used to produce “heralded” single photon (and more

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-3 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

complex multiphoton84–87) states, where the presence of a photon is for spectral filtering. GVM at specific donwconversion wavelength sets heralded by the detection of its twin. Alternatively, SPDC can produce is attained by selecting an appropriate nonlinear material—KTP photon pairs that are naturally entangled in polarization,88 transverse (potassium titanyl phosphate) proved to be suitable for degenerate spatial modes,15,89 or frequency.90,91 With modest effort, it is possible downconversion in the telecom region. Using GVM, a number of fre- to produce photon pairs with entanglement in a time-bin encoding,22 quency uncorrelated,115,116 nondegenerate117 and degenerate indistin- or even in multiple degrees of freedom simultaneously.92 guishable106,118–120 pure photon-pair sources at telecom wavelength SPDC can be a simple and cost-effective way to get single pho- have been demonstrated. Combined with optimized mode matching tons and (entangled) photon pairs but, in its original and simplest with the optical fiber121 and high efficiency detection technology in form, it is far from an ideal photon source for PQC. Ongoing techno- the telelcom wavelength range, GVM allowed realization of pulsed tel- logical development is changing that. Among the immense variety of ecom photon-pair sources that are simultaneously pure, highly effi- SPDC-based sources that have been developed and reviewed over past cient and (if desired) entangled in a chosen degree of years,45,93–95 we concentrate here on some advances that directly serve freedom.52,53,55,122 Further tailoring of the crystal’s nonlinearity pro- realistic PQC. file123–125 provides photons that are fully uncorrelated in their spec- A typical SPDC output from a simple, critically phase-matched, trum,126–128 completely removing the need for lossy spectral filtering. bulk-crystal source88 is not compatible with efficient coupling into Investigation of the performance and limitations of periodically poled single-mode optical fiber, because its transverse spatial profile is far SPDC sources continues129–132 and even tools for complete SPDC 133 from a Gaussian mode, resulting in coupling loss. Also, the twin pho- optimization are now available. tons are intrinsically entangled in frequency. This means that detection These developments have provided an enormous leap forward of one photon—to herald the presence of another—without resolving for SPDC technology, helping it to get close to satisfying many of its wavelength degrades the purity of the heralded photon.96 The spec- the criteria (a)–(c) for ideal photon generation. Heralding efficien- 55,134 tral filtering necessary to remove this entanglement adds even more cies jumped to above 0.8. The entangled state quality is harder loss to the source. Moreover, traditional SPDC photon wavelengths sit to survey, because of the variety of figures of merit that are used. around 800 nm, due to the standard use of Si APDs and compatible Focussing on a couple of standard ones, quantum state purities with readily available pump laser wavelengths. At these wavelengths, over 0:997 (Ref. 135) have been observed, and entangled state 136 the material loss (e.g., in fibers) is significant, and detection efficiency qualities—equivalent to the fidelity with a maximally entangled 52,135,137–139 is limited. A typical experiment involving more than one photon pair state—above 0.99 have been achieved in the lab. These would have heralding efficiency (probability of a heralded photon to high-performance sources have also allowed realization of impor- 52,140 successfully travel from a source to a detector and produce a click97)of tant experiments in entanglement verification and quantum 55 Շ10 15%, although some experiments report 30% (Ref. 98). metrology. Under these conditions, setting up several photon-pair sources allowed However, these advances relate to what happens when a photon the creation of complex photonic states of up to ten photons,99 but the pair is generated—the pair generation process itself is probabilistic. In low collective detection rates, and achievable state quality, limited the Sec. II C, we consider how SPDC or other technologies may be used to long-term prospects of these sources. provide deterministic single photon generation. A significant step forward was the application of quasiphase matching100 (QPM), via periodically poled nonlinear crystals. This C. Generating a photon deterministically expanded the range of possible phasematching wavelengths and emis- Photon-pair sources from SPDC and related processes—like 101–104 sion geometries and enabled collinear, beam-like downconver- spontaneous four-wave mixing (SFWM)141–144—are not only nonde- sion in the telecom wavelength range. With both photons emitted into terministic but generally operate at low generation probabilities. In an almost-single, almost-identical, almost-Gaussian spatial mode, the order to keep the single photon state quality high, pump powers have mode-matching loss and fiber propagation loss could be kept very low, to be kept low; otherwise, multiple photon pairs will be generated at leading to high heralding efficiencies. Using type-II phase matching the same time.145 This limits practical photon-pair generation proba- meant that degenerate photons could be deterministically separated bility n, for SPDC and similar processes, to n Շ 1%. Directly combin- with polarization optics. With the addition of interferometric schemes ing an array of n such sources (that will together produce n to generate polarization entanglement,105,106 QPM SPDC sources simultaneous pairs with probability nn) to generate a larger quantum could deliver entangled photon pairs with either continuous state is essentially not a viable option for a scalable photonic quantum wave107,108 or pulsed laser pumps.109,110 computer. There remained the need to remove the residual spectral entan- A more feasible alternative is to employ a deterministic photon glement in downconverted photons. This was recently solved by source. In recent years, photon-on-demand sources based on quantum applying the concept of group velocity matching (GVM).96,111,112 By dots,146 both free-space75,147,148 and integrated149–151 into optical carefully engineering the relative group velocities of the pump, signal, waveguides, have demonstrated a significant increase in brightness, and idler photons, and adjusting the pump laser bandwidth and SPDC enabling new quantum computation experimental demonstrations.152 crystal length, the joint spectrum of the daughter photons can be con- (It is worth noting that although quantum dots are usually assumed to trolled. It can be arranged that the signal photon is in a single spectral provide single photons on demand, quantum dots can also generate mode, and the idler photon is in a single spectral mode, to high fidelity. entangled photon pairs24,153–157 and superpositions of photon number (Note that the photons do not need to be in the same spectral mode as states.158) Although quantum dots159 can couple to optical cavities one another.) This technique provides photon pairs that are inherently with very high efficiency,148,160,161 a currently outstanding problem is uncorrelated in their spectrum,113,114 and reduces or removes the need coupling light efficiently into single mode optical fibers, with present

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-4 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

coupling efficiencies162 Շ33% (Ref. 163). Moreover, each quantum that impinges on a single nonlinear crystal and thus without increasing dot is usually spectrally different from others due to structural and the amount of high-photon-number noise from multiple-pair genera- environmental inhomogeneities, so the photons emitted by two dots tion events. (In principle, the network can also filter out multiple-pair are distinguishable from each other. PQC relies on nonclassical inter- generation events if photon-number resolving detectors are used.) ference, and the lack of indistinguishability makes it complicated to This concept,169,170 experimentally demonstrated in 2011 (Ref. 171), increase the number of photons used simultaneously in an experiment. has moved significantly toward practicality since then,172–174 in part One way to fix this is to tune the emission spectrum of different quan- because of the use of fiber- and waveguide-based integrated platforms tum dots to make indistinguishable.164,165 Alternatively, a single quan- to help scaling. tum dot can be used to generate all the required photons. For this, a Another method, that does not require multiple separate sources, pulsed output stream of photons from the dot is demultiplexed into is to use time169,175,176 (or frequency177,178) multiplexing of a single different spatial channels via a free-space166,167 or integrated168 active source.169,175,176 In the time multiplexing approach, shown in Fig. optical network. The multiplexed photons are then each delayed by 1(b), a heralded photon pair is generated in a random time bin, but appropriate amounts, so as to be output simultaneously from the the timing is recorded through detection of the heralding signal. The source setup. heralded photons are sent into an active temporal delay network and Similar active optical circuits can also, in principle, turn probabil- switched so as to exit the network at a fixed, although lower, repetition istic sources such as SPDC into deterministic ones. To realize this, an rate. The number n of time bins that is used to output one single pho- array of sources is used—see Fig. 1(a). Detecting the heralding signals ton plays the role of n sources in a spatial multiplexing scheme. Thus, from such an array will label which source has successfully generated a the improvement in generation probability scales with the size of the photon pair. Then, the corresponding heralded photon can be actively delay network, but is affected negatively by the loss in optical compo- rerouted through an optical network toward the output, while other nents in it. This multiplexing idea has been recently implemented in a photons, if generated, would be discarded by the same network. Using number of experiments, demonstrating multiplexing with large- n sources, this way theoretically boosts the generation efficiency to scale,179 or large-scale and low-loss180 networks, or with devices that n 181 nmulti ¼ 1 ð1 nÞ , ideally, without increasing the pump power produce indistinguishable output photons. The experimental demonstration that includes all of these features182 produced single photons in the output fiber with a probability of 0:6, and these photons displayed a nonclassical interference visibility 0:9. A more in-depth look at near-deterministic sources can be found in Ref. 93. Interesting preliminary work has also been done toward combining these kinds of techniques to simultaneously generate more than one single photon at a time. The multiplexing approach can be applied to more than one probabilistic source to generate states with one pho- tonineachofN > 1modes.183,184 An alternative method is to use an optical quantum memory to synchronize several probabilistic sources.185 Although quantum memory might be as simple as a switchable optical delay (in a free-space, fiber, or waveguide loop, for example), there is also extensive theoretical and experimental development of memories based on matter systems,186 with recent achievements including but not limited to broadband,187,188 high-speed,189 multimode,79,190 telecom-compatible,51,191 or configurable192 memories, capable of storing vector-,193 vortex-,194 or entangled-51,190 qubits, and storage with long coherence times195 and high storage efficiency196–198 and fidelity.199,200 Over the span of slightly more than a decade, photon detection and the probabilistic generation of high-quality photons have under- gone transformational advances, and the development of deterministic FIG. 1. Schematic representation of two types of triggered photon sources based sources is well under way, with no in-principle barriers to their realiza- on SPDC. (These concepts can be adapted to nondeterministic sources based on tion. (There are also other interesting advances, such as spectrally nar- other technologies.) (a) Multiplexing scheme that combines multiple (here, four) rowband sources201 for metrology and fundamental physics probabilistic photon sources to provided an increased brightness. Detectors (upper applications202 that we do not cover here.) arms) are used to herald the production of a photon by one of the sources, which is then switched into the output mode by some logic (e.g., a field-programmable gate array) and switch array. Since there are multiple sources in parallel, this scheme D. Manipulating a photon increases such as probability of having a photon in an appropriate time bin, without Thus, before proceeding to Sec. II E, we briefly turn our attention increasing the probability of having more that one photon in that bin. With enough to technologies for manipulating photons for PQC. Precise and accu- sources in parallel and with low loss, the arrangement can approach an ideal, deter- rate control of photon’s polarization, path, or time-bin stat has always ministic single photon source. (b) Triggered source that uses only one probabilistic 203 source and an active delay network. The network rearranges the generated pho- been the strength of PQC. Recently, this has been extended to per- tons in time, so that they are output at a stable, although lower, repetition rate. This forming reconfigurable mode transformations in integrated quantum scheme also provides a deterministic source, in principle. optics.204 Modern electro-optic elements, such as Pockels cells or

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-5 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

integrated electro-optic modulators, allow fast polarization switching rate calculated in pairs per second per mode per unit of pump power) sufficient perform rigorous Bell tests with locality and freedom of and high heralding efficiency. This is advantageous compared to bulk choice loopholes closed52,140 or spatial mode switching for source mul- SPDC, where the spatial mode configuration for high brightness is dif- tiplexing purposes.166–168 Efficient tools for manipulation of more ferent from the one that provides high heralding efficiency.121 Using exotic degrees of freedom, such as frequency-time205 or transverse spa- integrated technology, efficient sources in the telecom wavelength tial modes,206 are also being developed, including the techniques that range,226 including ones with GVM,227,228 have also been realized, transfer information from one degree of freedom to another, such as leading to the development of a fully packaged, banana-sized,229 and polarization to spatial transverse mode,16 discrete variable to continu- highly efficient photon-pair source and similar sources in a variety of ous variable,207 frequency conversion,208 and so on. material platforms.230,231 Techniques have been demonstrated for direct and practical characterization of nonlinear operations (like E. Integrated quantum photonics SPDC) in integrated quantum photonics.232 Integrated optics has also shown the capability of using more than one degree of freedom of a While introducing the relevant advances in photon detection and 230 generation technology, we mostly limited ourselves to the “bulk” photon. A number of materials for integrated optical components have optics environment, with separate optical components sitting on a ð2Þ tabletop. As the scale of PQC demonstrations grows to larger numbers no v nonlinearity, making them unsuitable for SPDC-based photon-pair sources. In this case, a practical alternative is SFWM. It is a of photons and gates, the importance of technological scalability and ð3Þ miniaturization becomes increasingly apparent. Integrated waveguides v nonlinear parametric process where two pump photons (degener- and optical chips offer an obvious path to implementing circuits at ate or otherwise) are converted into two daughter photons (degenerate or otherwise), conserving energy and momentum. Historically investi- scale, i.e., with huge numbers of components packaged compactly. 141–144 Thus, these technologies are now playing a significant role in the gated in optical fibers due to the isotropic nature of amorphous field.209 Several characteristics are important for a waveguide platform: silica, this method is now commonly adopted in those integrated plat- ð3Þ 233,234 The achievable density of optical components; low propagation and forms where v nonlinearities dominate. A GVM-like approach coupling losses; and the ability and speed of active reconfiguration, for for controlling the joint spectrum of daughter photons was also subse- quently generalized to SFWM235 and implemented experimentally in example. It is also desirable to integrate sources and detectors onto the 236 237 optical chip. fiber and on a chip. The scalability of the integrated optics Different materials offer their own strengths and advantages approach allows one to fabricate arrays of nearly identical photon 238,239 for realizing a practical integrated quantum photonics platform. sources that are now actively used in PQC experiments in sili- 204 Femtosecond-laser-written waveguides (typically in a glass) sup- con. On the more technical side, a number of SFWM obstacles, port polarization qubits210 and are not restricted to a 2D geometry, including the challenge of strongly filtering out the strong pump field allowing realization of complex couplings in 3D interferometric from the generated photon field, have been overcome in recent 240,241 networks.211 Lithium niobate, a material that is already well estab- years. The interested reader can find more information on inte- lished in classical integrated photonics, is an efficient and flexible grated probabilistic sources in Ref. 93 and on recent advances in GVM platform for photon sources and fast switchable electro-optical bulk and waveguided sources in Ref. 94. components operating at the gigahertz rates. Both ion-indiffused The rapid development of quantum integrated photonics is per- and high-index-contrast etched waveguides are being developed haps most obvious in the growth in the scale, complexity and perfor- and employed.150,168,212–215 Silicon-based optical chips offer high mance of optical circuits for one- and multiqubit operations. The first 242 210,243 component density, low loss, the ability or potential to integrate optical chips with path- and polarization- qubit encoding did every necessary component, and compatibility with existing not immedaitely surpass the performance previously achieved with 244,245 foundry processes.216 An enormous range of other material plat- bulk optics (in repeating the factoring of 15 by a compiled Shor’s forms are also under consideration. algorithm, for example Ref. 246) but emphatically demonstrated the On the integrated detection front, a lot of work has been done217 promise of the integrated approach. Subsequent devices, and the appli- in embedding SNSPDs into optical chips since the first demonstration cations they implemented, started to increase in complexity really in 2011 (Ref. 218). This ongoing effort has already provided fast and quickly.247 This included increasing the number of interferometers on efficient,219 low-noise,220 fast and low-noise,221 or low-noise, efficient achip(Fig. 2) and adding slow or fast active phase212,248–250 and spec- and fast222 (and even faster223) detection at telecom wavelength. tral215,251 controls in various waveguide platforms. These capabilities Significant effort is being put into turning waveguided SNSPDs into have led to a realization of fully reconfigurable optical processors for waveguided PNR detectors, see for example Ref. 213 and references an increasing number of optical modes.252 It has been observed that, therein, and Ref. 65. Similar developments are happening on the TES for the moment at least, the number of components on integrated integration side.224,225 quantum photonics chips is undergoing a Moore’s-law-like exponen- The situation is even more vivid regarding integrated photon tial growth with time.253 sources. QPM-based downconversion, which now plays the key role in A challenge of integrated platforms is optical loss caused by heralded photon and photon-pair generation, was in fact first demon- material absorption, waveguide roughness, and coupling onto and off strated in fiber101 and integrated waveguides.102–104 An important chip. These are actively investigated by a variety of techniques includ- advantage here is the transverse spatial confinement of the three ing: Improved materials (e.g., higher purity); moving to high-index- (pump, signal, and idler) propagating optical modes along the entire contrast platforms where devices can be smaller (e.g., Ref. 214), by length of the nonlinear material. This confinement allows construction integrating sources and detectors directly on chip. Modular architec- of a photon-pair source with both high brightness (absolute generation tures are also being investigated.254

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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 2018 AAAS. 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 local manipulation of the state by linear optical circuits. Triangular networks of MZIs perform arbitrary local projective measurements. The photons are coupled off the chip into ﬁbers by means of grating couplers, and are detected by two SNSPDs. See Ref. 204 for details.

III. QUANTUM COMPUTING development of individual quantum gates of increasing complexity in The advent of the KLM scheme40 in 2001, with its proof of the the circuit model; the implementation small-scale quantum algorithms scalability of optical processing, inspired a worldwide push toward a and nonuniversal circuits or clusters for them; the development of universal quantum computer with photons. Of course, a full-scale simplifying and supporting techniques within the circuit and cluster- error-corrected version could not be built at that time, and indeed a state models; and the advent of algorithms for sampling problems universal quantum computer remains a challenging quest today in any based on the fundamental properties of bosons. Intermediate quantum quantum system. The KLM scheme led to the development and computing research is helpful for optimization of the general schemes improvement of a variety of photonic encodings, schemes for quan- for PQC, and for developing and testing of individual components of a tum gates, and protocol and algorithm demonstrations.6–9 Circuit- future quantum computer. based approaches, having evolved from KLM, continue to be an active area of theoretical and experimental research as a path toward A. Intermediate quantum computing intermediate-scale and universal quantum processors. A significant development for PQC was the realization that the Photons can be readily and accurately manipulated at the single- qubit level—very high fidelity one-qubit gates can be constructed1 cluster-state model of quantum computing (also known as one-way 252 quantum computing)255 was well-suited to photon qubits.256 This is because of excellent optical mode control. Initially, particular attention primarily because large cluster states257 can, in principle, be built effi- fell on the controlled-NOT (CNOT) gate to complete a universal gate set ciently using entangled photon sources and teleportation gates of the in the quantum circuit model. Theoretical proposals for nondeterminis- 259,260 kind used in the KLM scheme. It is also important that photon mea- tic CNOT gates, demonstrating the basic measurement-induced surements are easy to perform reliably, and because cluster state nonlinearity concept of KLM, were quickly followed by experimental 261,262 263,264 schemes can be made tolerant to photon loss, the primary source of CNOT demonstrations and characterizations. These were noise in an optical environment. For these reasons, cluster state expanded to include heralded KLM-style265 and teleportation266-based39 schemes are widely viewed as offering a realistic path to scalable PQC. schemes.267 A number of proof of principle algorithms followed the early As the development of universal PQC has continued, a number demonstrations of photonic gates.6 While using CNOTs to build arbi- of intermediate goals have emerged, providing short- to medium-term trary unitary circuits is, of course, a working theoretical method, it is far targets and a path toward full-scale devices. These include: The from optimal. This is because, for example, the decomposition of a

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three-qubit gate, such as the Toffoli gate, into one- and two-qubit opera- This general approach was used to experimentally realize arbi- tions may require a large number of such gates.268 An alternative would trary controlled-single-qubit unitaries, a CNOT gate,274 and three- be to look for ways of implementing gates that can operate on a larger qubit gates—namely the Toffoli271 and Fredkin (controlled-SWAP, number of qubits directly. see Fig. 3)258 gates. It was also employed in experimentally implement- An interesting and important class of arbitrary-scale quantum ing a number of quantum computing tasks, such as solving systems of logic is the family of controlled-Unitary (CU) gates. In these, a (possi- two linear equations275 (this was also done without entanglement- bly multiqubit) unitary operation acts or not—depending on the state based gates276) factoring 21 by a version of Shor’s algorithm (Ref. of a control qubit—on the target qubits. CU gates are important in 277), measuring state overlaps and state purity,258 and eigenstate wit- various computational tasks, for example the phase estimation algo- nessing for simple quantum algorithms.270 Entanglement-based gates rithm that underlies Shor’s algorithm244,245 and in quantum chemis- are now also used in larger quantum circuits, including the ones real- try.269,270 A key realization is that implementing the unitary operation ized in an integrated platform.270,278 U alone may be possible or even easy, but adding the control opera- The use of various photonic quantum gate architectures has tion—i.e., conditional action—is difficult. allowed realization of a variety of intermediate scale simulations, A general scheme for adding a control operation to an arbitrary implemented in bulk and integrated optics platforms. Among these279 unitary transformation was proposed in 2009 (Ref. 271). In this are spin chain simulation,280 calculating molecular ground-state ener- method, given the unitary to be controlled, the Hilbert space dimen- gies,281 Hamiltonian learning282 and eigenstates witnessing,270 and sionality of the incoming target qubits is first doubled by using some complex state transformations such as Fourier211,283 or Kravchuk284 auxiliary degree of freedom of the corresponding photons. Half of the transforms. modes of each target qubit pass through the unitary, while the remain- A highly topical intermediate photonic quantum computing task ing half bypass it. Then, the control qubit state is used to route the tar- is that of BosonSampling,285–292 which is an example of sampling-type get qubits to either pass the unitary or bypass it, via the corresponding computational problems more generally.293 BosonSampling is a non- modes. After that, the modes are recombined, so the Hilbert space is universal protocol for which there is strong theoretical evidence that a shrunk to its original dimensionality. This effectively creates a CU gate. quantum advantage can be observed. Consider n single photons input (The scheme can be simplified even further, by substituting Hilbert- into m n optical modes, which are subjected to a random unitary space-expanding gates with photon sources that generate entanglement operation on the mode space. It is classically computationally hard to in the auxiliary degree of freedom. The term “entanglement-based” is obtain samples from the probability distribution representing where usually used in the literature to describe these types of gates, which are the photons appear at the output. By contrast, photons (and other not completely general due to the need to generate the initial entangle- bosons) traversing a unitary on the mode space perform this calcula- ment, but can be useful at circuit inputs.) This overall method is partic- tion naturally. Interestingly, the same quantum-classical performance ularly suitable for optical quantum computing, because high divide exists even if the photons are allowed to arrive at random inputs dimensional systems, multiple degrees of freedom, and means of trans- of the circuit.294 It is thought that better-than-classical BosonSampling fering information between them are readily available. Moreover, theo- performance may be achieved with 50–100 photons, promoting the retical studies also highlighted that adding control to arbitrary unitary idea that this system could well provide the first rigorous experimental gates is generally impossible for matter-based qubits,272,273 so the demonstration of a quantum computational advantage. Nevertheless, method demonstrates a benefit of using fields to quantum compute. 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 Commons Attribution 4.0 NonCommercial License. The Fredkin (or controlled-SWAP) gate uses the method of adding control to an arbitrary untiary operation. Entangled photons are produced 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 encoded on 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 after each 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 plates encode 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 conceptual shown theoretically that missing links and nodes (e.g., due to fusion and experimental aspects of the topic in more detail. failures or optical loss) in the constructed cluster state need not be Intermediate quantum computing is likely to lack fully fledged problematic. As long as their prevalence is below a certain threshold, error correction. Thus, photon loss and noise in PQC will need to be percolation theory can be used to reshape the entangled state and per- controlled by other methods. One prominent approach being investi- form universal computation.317,318 The percolation operation corre- gated for NISQ297 (noisy intermediate-scale quantum devices) is sponds, roughly, to a classically efficient relabeling of the cluster. machine learning (ML). ML provides a method to work with quantum Furthermore, error correction for fault-tolerant quantum operations protocols operating in an environment of unknown or uncharacter- seems achievable, especially given modest loss thresholds.310,319–321 ized noise, or where the full ab initio modeling of the protocol is In principle, cluster states can be generated and processed (via intractable,298 and can be applied to PQC and other systems.299 The adaptive measurement) on the fly, without the need to store photons flip side to ML helping quantum computation by controlling noise is in an optical quantum memory. This is known as ballistic cluster state the hope that quantum computers can enhance ML for other applica- computing.322 In this scheme, an array of sources and simple circuits 297 tions, possibly even in the NISQ regime. Other relevant proposals produce entangled photons at each time step—these photons are for ML quantum applications include long-distance quantum commu- entangled together to produce a 3D cluster where each layer represents 300 301,302 nication or metrology. Experimental demonstrations of ML a generation step in time. It has been shown that the depth of the clus- application to quantum information science have recently started to ter that needs to exist at any time is only of the order of a few tens of 282,303–305 appear, too. photons.323 In this case, given a suitably small number of faults in the cluster, the computation can proceed indefinitely in principle, with the B. Cluster-state based computing source array continuing to make new cluster layers at each time step In conventional PQC, uncorrelated input qubits are processed by and detectors measuring a layer at each step. a complicated quantum circuit of one-, two-, three-, and many-qubit Ongoing theoretical and experimental research on photonic clus- gates (which in turn can be decomposed to one- and two-qubit gates). ters and ballistic schemes is also addressing many technical details Here, generating many uncorrelated photonic qubits is considered the (e.g., optimal cluster geometry, error correction schemes, and sources “easy” part of the problem, and the logical circuit does the “hard”306 designed for cluster generation). However, it is emerging that photonic task of performing the computation. An alternative approach is one- cluster schemes, and closely related ideas, are extremely plausible 320 way (or cluster-state) quantum computing.255,256,307 In one-way com- approaches for realizing universal quantum computers. puting, a hard-to-make, highly entangled multiphoton state is sent In summary, in the span of less than two decades, photonic into an easy-to-implement processing circuit that consists only of quantum information science has matured immensely. New photon single-qubit operations, measurements, and classical feed-forward.6,308 generation and detection technologies have enormously improved the The key idea is that, in the absence of deterministic two-photon opera- efficiency and quality of photonic quantum states. Integrated circuits tions, the cluster state can be built up offline using nondeterministic grew from a simple demonstration of a beam splitter to massively mul- interactions, and then the computation progresses via those determin- timode reconfigurable circuits. The number of photons simultaneously istic single-qubit operations for which optics is especially suited. used in experiments has grown, from 2 to 4 up to 12 (Ref. 291). Follow-up development showed how to create cluster states more Overall, experimental PQC is steadily moving toward the major goal efficiently,309 leading to significantly reduced resource requirements of universal quantum computing and theoretical PQC is steadily pro- (characterized as Bell-pairs per effective two-qubit gate, a metric of gressing toward more resource-efficient and noise-tolerant schemes. PQC overhead; smaller $ better) compared to many other optical In parallel, nonuniversal quantum computation schemes such as schemes. The one-way computing approach is also more tolerant to BosonSampling are also rapidly scaling up toward the demonstration losses, compared to KLM.310 Since the first experimental demonstra- of the true quantum computational advantage over classical tion of the essentials of one-way quantum computing,311 considerable computers. steps have been made toward making larger cluster states,291,312 dem- IV. NETWORKING QUANTUM PROCESSORS onstrating larger computing networks,313 and improving the feed- forward performance.314 A type of one-way-based computing, where PQC is strongly interlinked with other optical quantum informa- the computer cannot determine the input data and performs the com- tion tasks. On the one hand, quantum phase estimation algorithms, putation blindly but correctly, has also been demonstrated.315 used in e.g., Shor’s algorithm and a number of intermediate quantum Developments in the theory of optical one-way computing have driven computing schemes (as in Ref. 281), are also useful in quantum- 324–327 increasingly realistic schemes for large-scale photonic quantum enhanced metrology. On the other hand, quantum communica- computing. tion is essential for building a distributed quantum processor from Indeed, recent theoretical developments suggest that cluster- interlinked quantum computers. Flying fast, photons (or other optical based quantum computation may be a more realistic approach toward states) are the obvious way to transmit quantum information. Thus, the future photonic quantum computer than gate-based models. photonic quantum interconnects can naturally be tasked with interfac- There are a number of key advantages to a cluster-state approach. One ing remote systems and, perhaps, local processing cores. Optical con- concerns the way that clusters are built, through progressive nondeter- nections make sense regardless of the quantum system chosen for ministic fusion operations309,316 that seek to merge two smaller processing, but using photonic processing means that the interconver- entangled states into a larger one. The key point is that the failure of sion between a stationary and a flying qubit can be skipped. (Indeed, the nondeterministic operation slightly reduces the sizes of the initial quantum teleportation—an entanglement-based protocol used in entangled states, but does not destroy them.309 In fact, it has been communication—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 to V. CONCLUSION adjust the spectral properties of photons between the communication Our short review has only touched briefly on other PQC ele- and the processor, and ways to do this are being investigated for a vari- ments including error correction in photonic schemes,355 optical ety of different interconversion wavelengths, and photon-carrying and 208,328–333 quantum memories, and algorithms and protocols. There is also a generating architectures. broad range of related research that is beyond our immediate scope, Creating verified communication links capable of sharing and including other qubit or qudit encodings—such as single-rail,6,356 par- transmitting entanglement is essential for networking quantum ity state,6,357 continuous-variable,13,358–360 and hybrid207,361—as well computers, and also quantum secure communication, small as other source and detector technologies. Some of these techniques communication-based processing tasks (quantum communication are also promising in terms of resource use and scalability. Instead, we complexity334,335), and quantum networks for distributed metrol- 336 have covered technologies and methods that are the main focus of the ogy. A major step in entanglement verification and distribution experimental development of photonic (Fock-state) quantum infor- was the experimental implementation of loophole-free Bell tests, 52,140 337 mation processing in the medium term, and provided a firm founda- executed with photonic and matter qubits. Besides definition for the development of large-scale devices. tively showing local realistic explanations of entanglement are not There is significant promise for the long term. Improvements in viable, these tests confirmed that entanglement can now be rigor- cluster-state schemes designed specifically for photonics are providing ously verified in a loophole-free manner, opening the road to the a reduction in the overhead (from nondeterminism) and in error unconditionally secure device-independent protocols (e.g., Ref. thresholds—especially for loss. In conjunction, exceptional quality 338). A remaining challenge is enabling these protocols in the sources, detectors, and gates—and large-scale integrated platforms— presence of very high loss in a communication channel used to dis- are providing the hardware advances required to build processors tributetheentanglement.AsinPQC,lossisthepredominant comprising very many elements. Intermediate tasks like source of added noise that degrades entanglement. BosonSampling provide a path to demonstrating a true quantum com- One can neglect the loss by postselecting only on successful puting advantage sooner rather than later. Photonics continues to be detection events; however, such experiments do not offer device- the dominant platform for connecting processors separated by dis- independent security or a quantum advantage in metrology. tance and for remote entanglement sharing, in general. Unfortunately, the no-cloning theorem forbids creation of identi- There remain other potentially transformational technologies for cal backup copies of unknown quantum states to be used if a pho- photonic processing. We have only touched briefly on nonlinear inter- ton is lost. A state-independent attempt to amplify a qubit or qudit actions at the single-photon level—mediated by atoms, for example. (i.e., to boost the photon number to its original value) would inevi- Such schemes, applied at scale, could massively reduce the overhead of tably lead to the degradation of the state purity. Noiseless amplifi- “linear plus measurement” approaches. However, there remain signifi- cation can only be performed in probabilistic manner—consistent cant research and development required to capitalize on their promise. with noise reduction being a nonunitary process—and produces a In the meantime, or perhaps in their stead, the convergence of techno- wrong output upon failure. Fortunately, heralded amplification logical performance and theoretical requirements in photonic linear (also known as noiseless linear amplification, NLA) is possible: In optics is pointing to a bright future for photon processing. this probabilistic scheme, successful amplification events are heralded by an independent photon detection signal, allowing them to ACKNOWLEDGMENTS be sorted from the failed trials.203 Heralded amplification can be used to distribute entanglement in the presence of loss—even with This work was conducted by the ARC Centre of Excellence for the detection loophole closed, in principle. Since the first demon- Quantum Computation and Communication Technology under strations,339–341 NLA has been actively researched in both the Grant No. CE170100012. discrete-variable (photon) and continuous-variables communities. 342 343 It has shown the ability to amplify polarization, path, and REFERENCES 344 time-bin qubits and has been used to restore mode entangle- 1 339,343,345 N. Peters, J. Altepeter, E. Jeffrey, D. Branning, and P. Kwiat, “Precise creation, ment that was degraded due to loss and versions of the characterization, and manipulation of single optical qubits,” Quantum Inf. 346 scheme have been applied to quantum communication and Comput. 3, 503 (2003). cloning.347 There have been proposals and experiments related to 2D. P. DiVincenzo, “The physical implementation of quantum computation,” implementing NLAs with quantum logic gates.348,349 Fortschr. Phys. 48, 771–783 (2000). 3 Many other communication protocols for sharing high-quality T. G. Tiecke, J. D. Thompson, N. P. de Leon, L. R. Liu, V. Vuletic´, and M. D. entanglement in lossy environments (i.e., for realizing quantum Lukin, “Nanophotonic quantum phase switch with a single atom,” Nature 350 508, 241 (2014). repeaters) are based on entanglement swapping. Recent advances 4D. Tiarks, S. Schmidt, G. Rempe, and S. Durr,€ “Optical p phase shift created have used entanglement swapping for sharing entanglement with the with a single-photon pulse,” Sci. Adv. 2, e1600036 (2016). detection loophole closed, even over high-loss channels.337,351 Other 5N. K. Langford, S. Ramelow, R. Prevedel, W. J. Munro, G. J. Milburn, and A. potential tools include quantum nondemoliton measurements of the Zeilinger, “Efficient quantum computing using coherent photon conversion,” 352,353 Nature 478, 360 (2011). photon number and, of course, a variety of error-correction- 6 code protocols (e.g., Ref. 354). Ultimately, entanglement-based net- T. Ralph and G. Pryde, “Optical quantum computation,” Prog. Opt. 54, 209–269 (2010). works will likely also require local processing (i.e., small quantum 7P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. computers) for distilling entanglement and quantum memory for syn- Milburn, “Linear optical quantum computing with photonic qubits,” Rev. chronizing operations. Mod. Phys. 79, 135–174 (2007).

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-10 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

8J. L. O’Brien, A. Furusawa, and J. Vucˇkovic´, “Photonic quantum tech- 31T. M. Graham, H. J. Bernstein, T.-C. Wei, M. Junge, and P. G. Kwiat, nologies,” Nat. Photonics 3, 687–695 (2009). “Superdense teleportation using hyperentangled photons,” Nat. Commun. 6, 9T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. 7185 (2015). O’Brien, “Quantum computers,” Nature 464, 45 (2010). 32X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. 10In this article, we use the term “photonic” to refer to schemes with counted Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of photons, i.e. to discrete-variable schemes such as those with qubit photon a single photon,” Nature 518, 516 (2015). 33 or qudit photon. M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, 11S. Lloyd and S. L. Braunstein, “Quantum computation over continuous varia- “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248 bles,” Phys. Rev. Lett. 82, 1784–1787 (1999). (2016). 34 12S. L. Braunstein and P. van Loock, “Quantum information with continuous X.-L. Wang, Y.-H. Luo, H.-L. Huang, M.-C. Chen, Z.-E. Su, C. Liu, C. Chen, variables,” Rev. Mod. Phys. 77, 513–577 (2005). W. Li, Y.-Q. Fang, X. Jiang, J. Zhang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, 13N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Nielsen, “Universal quantum computation with continuous-variable cluster Rev. Lett. 120, 260502 (2018). 35 states,” Phys. Rev. Lett. 97, 110501 (2006). M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realiza- 14F. Lenzini, J. Janousek, O. Thearle, M. Villa, B. Haylock, S. Kasture, L. Cui, H.- tion of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994). 36 P. Phan, D. V. Dao, H. Yonezawa, P. K. Lam, E. H. Huntington, and M. W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Lobino, “Integrated photonic platform for quantum information with contin- Walmsley, “Optimal design for universal multiport interferometers,” Optica 4 3, 1460–1465 (2016). uous variables,” Sci. Adv. , eaat9331 (2018). 37 15A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, N. Tischler, C. Rockstuhl, and K. Słowik, “Quantum optical realization of “Experimental high-dimensional two-photon entanglement and violations of arbitrary linear transformations allowing for loss and gain,” Phys. Rev. X 8, 021017 (2018). generalized bell inequalities,” Nat. Phys. 7, 677 (2011). 38 16V. D’Ambrosio, E. Nagali, C. H. Monken, S. Slussarenko, L. Marrucci, and F. G. J. Milburn, “Quantum optical Fredkin gate,” Phys. Rev. Lett. 62, 2124–2127 (1989). Sciarrino, “Deterministic qubit transfer between orbital and spin angular 39 momentum of single photons,” Opt. Lett. 37, 172–174 (2012). D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal 17M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: New quantum computation using teleportation and single-qubit operations,” Nature 402, 390 (1999). quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018). 40 18J. Roslund, R. M. de Araujo, S. Jiang, C. Fabre, and N. Treps, “Wavelength- E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum multiplexed quantum networks with ultrafast frequency combs,” Nat. computation with linear optics,” Nature 409, 46 (2001). 41C. R. Myers and R. Laflamme, “Linear optics quantum computation: An over- Photonics 8, 109 (2013). 19 view,” preprint arXiv:quant-ph/0512104 (2004). Y. Cai, J. Roslund, G. Ferrini, F. Arzani, X. Xu, C. Fabre, and N. Treps, 42H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control “Multimode entanglement in reconfigurable graph states using optical fre- (Cambridge University Press, 2009). quency combs,” Nat. Commun. 8, 15645 (2017). 43 20 Single-Photon Generation and Detection: Physics and Applications, M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhang, Experimental Methods in the Physical Sciences Vol. 45, edited by A. Migdall, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana,~ and R. S. V. Polyakov, J. Fan, and J. C. Bienfang (Academic Press, 2013). Morandotti, “On-chip generation of high-dimensional entangled quantum 44M. Silva, M. Rotteler,€ and C. Zalka, “Thresholds for linear optics quantum states and their coherent control,” Nature 546, 622 (2017). computing with photon loss at the detectors,” Phys. Rev. A 72, 032307 21J. M. Lukens and P. Lougovski, “Frequency-encoded photonic qubits for scal- (2005). able quantum information processing,” Optica 4, 8–16 (2017). 45 M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: 22J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time Single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011). entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 46G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. 82, 2594–2597 (1999). 23 Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, P. C. Humphreys, B. J. Metcalf, J. B. Spring, M. Moore, X.-M. Jin, M. Barbieri, “Picosecond superconducting single-photon optical detector,” Appl. Phys. W. S. Kolthammer, and I. A. Walmsley, “Linear optical quantum computing Lett. 79, 705–707 (2001). in a single spatial mode,” Phys. Rev. Lett. 111, 150501 (2013). 47 24 K. M. Rosfjord, J. K. W. Yang, E. A. Dauler, A. J. Kerman, V. Anant, B. M. H. Jayakumar, A. Predojevic´,T. Kauten, T. Huber, G. S. Solomon, and G. Voronov, G. N. Gol’tsman, and K. K. Berggren, “Nanowire single-photon Weihs, “Time-bin entangled photons from a quantum dot,” Nat. Commun. 5, detector with an integrated optical cavity and anti-reflection coating,” Opt. 4251 (2014). 25 Express 14, 527–534 (2006). F. Samara, A. Martin, C. Autebert, M. Karpov, T. J. Kippenberg, H. Zbinden, 48F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. and R. Thew, “High-rate photon pairs and sequential time-bin entanglement Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting with si3n4 microring resonators,” Opt. Express 27, 19309–19318 (2019). 7 26 single infrared photons with 93% system efficiency,” Nat. Photonics , D. Kielpinski, J. F. Corney, and H. M. Wiseman, “Quantum optical waveform 210–214 (2013). conversion,” Phys. Rev. Lett. 106, 130501 (2011). 49D. V. Reddy, R. R. Nerem, A. E. Lita, S. W. Nam, R. P. Mirin, and V. B. 27 B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal Verma, “Exceeding 95% system efficiency within the telecom c-band in super- modes: A complete framework for quantum information science,” Phys. conducting nanowire single photon detectors,” in CLEO (OSA, 2019), p. Rev. X 5, 041017 (2015). FF1A.3. 28 V. Averchenko, D. Sych, G. Schunk, U. Vogl, C. Marquardt, and G. Leuchs, 50F. Bussie`res, C. Clausen, A. Tiranov, B. Korzh, V. B. Verma, S. W. Nam, F. “Temporal shaping of single photons enabled by entanglement,” Phys. Rev. A Marsili, A. Ferrier, P. Goldner, H. Herrmann, C. Silberhorn, W. Sohler, M. 96, 043822 (2017). Afzelius, and N. Gisin, “Quantum teleportation from a telecom-wavelength 29 V. Ansari, E. Roccia, M. Santandrea, M. Doostdar, C. Eigner, L. Padberg, I. photon to a solid-state quantum memory,” Nat. Photonics 8, 775 (2014). Gianani, M. Sbroscia, J. M. Donohue, L. Mancino, M. Barbieri, and C. 51E. Saglamyurek, J. Jin, V. B. Verma, M. D. Shaw, F. Marsili, S. W. Nam, D. Silberhorn, “Heralded generation of high-purity ultrashort single photons in Oblak, and W. Tittel, “Quantum storage of entangled telecom-wavelength programmable temporal shapes,” Opt. Express 26, 2764–2774 (2018). photons in an erbium-doped optical fibre,” Nat. Photonics 9, 83 (2015). 30W.-B. Gao, C.-Y. Lu, X.-C. Yao, P. Xu, O. Guhne,€ A. Goebel, Y.-A. Chen, 52L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. C.-Z. Peng, Z.-B. Chen, and J.-W. Pan, “Experimental demonstration of a J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. hyper-entangled ten-qubit Schrodinger€ cat state,” Nat. Phys. 6, 331–335 D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. (2010). Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A.

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-11 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

Stern, C. Abellan, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. 71D. Fukuda, G. Fujii, T. Numata, K. Amemiya, A. Yoshizawa, H. Tsuchida, H. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, “Strong loophole- Fujino, H. Ishii, T. Itatani, S. Inoue, and T. Zama, “Titanium-based transition- free test of local realism,” Phys. Rev. Lett. 115, 250402 (2015). edge photon number resolving detector with 98% detection efficiency with 53M. M. Weston, H. M. Chrzanowski, S. Wollmann, A. Boston, J. Ho, L. K. index-matched small-gap fiber coupling,” Opt. Express 19, 870–875 (2011). Shalm, V. B. Verma, M. S. Allman, S. W. Nam, R. B. Patel, S. Slussarenko, 72B. Calkins, A. E. Lita, A. E. Fox, and S. Woo Nam, “Faster recovery time of a and G. J. Pryde, “Efficient and pure femtosecond-pulse-length source of hot-electron transition-edge sensor by use of normal metal heat-sinks,” Appl. polarization-entangled photons,” Opt. Express 24, 10869–10879 (2016). Phys. Lett. 99, 241114 (2011). 54 R. Valivarthi, M. G. Puigibert, Q. Zhou, G. H. Aguilar, V. B. Verma, F. 73A. Lamas-Linares, B. Calkins, N. A. Tomlin, T. Gerrits, A. E. Lita, J. Beyer, R. Marsili, M. D. Shaw, S. W. Nam, D. Oblak, and W. Tittel, “Quantum telepor- P. Mirin, and S. Woo Nam, “Nanosecond-scale timing jitter for single photon tation across a metropolitan fibre network,” Nat. Photonics 10, 676 (2016). detection in transition edge sensors,” Appl. Phys. Lett. 102, 231117 (2013). 55 S. Slussarenko, M. M. Weston, H. M. Chrzanowski, L. K. Shalm, V. B. Verma, 74Y. Li, P. C. Humphreys, G. J. Mendoza, and S. C. Benjamin, “Resource costs S. W. Nam, and G. J. Pryde, “Unconditional violation of the shot-noise limit for fault-tolerant linear optical quantum computing,” Phys. Rev. X 5, 041007 in photonic quantum metrology,” Nat. Photonics 11, 700–703 (2017). (2015). 56 J. J. Renema, R. Gaudio, Q. Wang, Z. Zhou, A. Gaggero, F. Mattioli, R. Leoni, 75P. Senellart, G. Solomon, and A. White, “High-performance semiconductor D. Sahin, M. J. A. de Dood, A. Fiore, and M. P. van Exter, “Experimental test quantum-dot single-photon sources,” Nat. Nanotechnol. 12, 1026 (2017). of theories of the detection mechanism in a nanowire superconducting single 76X. He, N. F. Hartmann, X. Ma, Y. Kim, R. Ihly, J. L. Blackburn, W. Gao, J. photon detector,” Phys. Rev. Lett. 112, 117604 (2014). 57 Kono, Y. Yomogida, A. Hirano, T. Tanaka, H. Kataura, H. Htoon, and S. K. A. Engel, J. J. Renema, K. Il’in, and A. Semenov, “Detection mechanism of Doorn, “Tunable room-temperature single-photon emission at telecom wave- superconducting nanowire single-photon detectors,” Supercond. Sci. Technol. lengths from sp3 defects in carbon nanotubes,” Nat. Photonics 11, 577 28, 114003 (2015). 58 (2017). R. Gaudio, J. J. Renema, Z. Zhou, V. B. Verma, A. E. Lita, J. Shainline, M. J. 77T. Vogl, G. Campbell, B. C. Buchler, Y. Lu, and P. K. Lam, “Fabrication and Stevens, R. P. Mirin, S. W. Nam, M. P. van Exter, M. J. A. de Dood, and A. deterministic transfer of high-quality quantum emitters in hexagonal boron Fiore, “Experimental investigation of the detection mechanism in wsi nano- nitride,” ACS Photonics 5, 2305–2312 (2018). wire superconducting single photon detectors,” Appl. Phys. Lett. 109, 031101 78T. T. Tran, D. Wang, Z.-Q. Xu, A. Yang, M. Toth, T. W. Odom, and I. (2016). Aharonovich, “Deterministic coupling of quantum emitters in 2d materials to 59F. Marsili, M. J. Stevens, A. Kozorezov, V. B. Verma, C. Lambert, J. A. Stern, plasmonic nanocavity arrays,” Nano Lett. 17, 2634–2639 (2017). R. D. Horansky, S. Dyer, S. Duff, D. P. Pappas, A. E. Lita, M. D. Shaw, R. P. 79 K. R. Ferguson, S. E. Beavan, J. J. Longdell, and M. J. Sellars, “Generation of Mirin, and S. W. Nam, “Hotspot relaxation dynamics in a current-carrying light with multimode time-delayed entanglement using storage in a solid- superconductor,” Phys. Rev. B 93, 094518 (2016). state spin-wave quantum memory,” Phys. Rev. Lett. 117, 020501 (2016). 60J. J. Renema, R. Gaudio, Q. Wang, A. Gaggero, F. Mattioli, R. Leoni, M. P. 80F. Dell’Anno, S. D. Siena, and F. Illuminati, “Multiphoton quantum optics van Exter, A. Fiore, and M. J. A. de Dood, “Probing the hotspot interaction and quantum state engineering,” Phys. Rep. 428, 53–168 (2006). length in nbn nanowire superconducting single photon detectors,” Appl. 81W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise Phys. Lett. 110, 233103 (2017). 61 in parametric processes. I,” Phys. Rev. 124, 1646–1654 (1961). A. J. Kerman, D. Rosenberg, R. J. Molnar, and E. A. Dauler, “Readout of 82D. N. Klyshko, “Scattering of light in a medium with nonlinear polarizability,” superconducting nanowire single-photon detectors at high count rates,” Sov. Phys. JETP 28, 522 (1969). J. Appl. Phys. 113, 144511 (2013). 83 62 D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in paramet- I. Esmaeil Zadeh, J. W. N. Los, R. B. M. Gourgues, V. Steinmetz, G. Bulgarini, ric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970). S. M. Dobrovolskiy, V. Zwiller, and S. N. Dorenbos, “Single-photon detectors 84C. Wagenknecht, C.-M. Li, A. Reingruber, X.-H. Bao, A. Goebel, Y.-A. Chen, combining high efficiency, high detection rates, and ultra-high timing resolution,” APL Photonics 2, 111301 (2017). Q. Zhang, K. Chen, and J.-W. Pan, “Experimental demonstration of a her- 63 alded entanglement source,” Nat. Photonics 4, 549 (2010). B. A. Korzh, Q.-Y. Zhao, S. Frasca, J. P. Allmaras, T. M. Autry, E. A. Bersin, 85 M. Colangelo, G. M. Crouch, A. E. Dane, T. Gerrits, F. Marsili, G. Moody, E. S. Barz, G. Cronenberg, A. Zeilinger, and P. Walther, “Heralded generation of entangled photon pairs,” Nat. Photonics 4, 553 (2010). Ramirez, J. D. Rezac, M. J. Stevens, E. E. Wollman, D. Zhu, P. D. Hale, K. L. 86 Silverman, R. P. Mirin, S. W. Nam, M. D. Shaw, and K. K. Berggren, D. R. Hamel, L. K. Shalm, H. Hubel,€ A. J. Miller, F. Marsili, V. B. Verma, R. “Demonstrating sub-3 ps temporal resolution in a superconducting nanowire P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801 (2014). single-photon detector,” preprint arXiv:1804.06839 (2018). 87 64J. Tiedau, E. Meyer-Scott, T. Nitsche, S. Barkhofen, T. J. Bartley, and C. S. Krapick, B. Brecht, H. Herrmann, V. Quiring, and C. Silberhorn, “On-chip generation of photon-triplet states,” Opt. Express 24, 2836–2849 (2016). Silberhorn, “A high dynamic range optical detector for measuring single pho- 88 tons and bright light,” Opt. Express 27, 1–15 (2019). P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. 65F. Mattioli, Z. Zhou, A. Gaggero, R. Gaudio, S. Jahanmirinejad, D. Sahin, F. Shih, “New high-intensity source of polarization-entangled photon pairs,” Marsili, R. Leoni, and A. Fiore, “Photon-number-resolving superconducting Phys. Rev. Lett. 75, 4337–4341 (1995). 89 nanowire detectors,” Supercond. Sci. Technol. 28, 104001 (2015). A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, “Entanglement of the angular 66B. Cabrera, R. M. Clarke, P. Colling, A. J. Miller, S. Nam, and R. W. Romani, momentum states of photons,” Nature 412, 313–315 (2001). 90 “Detection of single infrared, optical, and ultraviolet photons using supercon- V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, “Generating ducting transition edge sensors,” Appl. Phys. Lett. 73, 735–737 (1998). entangled two-photon states with coincident frequencies,” Phys. Rev. Lett. 88, 67I. A. Burenkov, A. K. Sharma, T. Gerrits, G. Harder, T. J. Bartley, C. 183602 (2002). 91 Silberhorn, E. A. Goldschmidt, and S. V. Polyakov, “Full statistical mode O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. K€artner, reconstruction of a light field via a photon-number-resolved measurement,” “Two-photon coincident-frequency entanglement via extended phase Phys. Rev. A 95, 053806 (2017). matching,” Phys. Rev. Lett. 94, 083601 (2005). 92 68G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of “Single-mode parametric-down-conversion states with 50 photons as a source hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005). 93 for mesoscopic quantum optics,” Phys. Rev. Lett. 116, 143601 (2016). L. Caspani, C. Xiong, B. J. Eggleton, D. Bajoni, M. Liscidini, M. Galli, R. 69A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-pho- Morandotti, and D. J. Moss, “Integrated sources of photon quantum states tons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008). based on nonlinear optics,” Light Sci. Appl. 6, e17100 (2017). 70A. E. Lita, B. Calkins, L. A. Pellouchoud, A. J. Miller, and S. W. Nam, 94V. Ansari, J. M. Donohue, B. Brecht, and C. Silberhorn, “Tailoring nonlinear “Superconducting transition-edge sensors optimized for high-efficiency pho- processes for quantum optics with pulsed temporal-mode encodings,” Optica ton-number resolving detectors,” Proc. SPIE 7681, 76810D (2010). 5, 534–550 (2018).

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-12 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

95F. Flamini, N. Spagnolo, and F. Sciarrino, “Photonic quantum information 117F. Kaneda, K. Garay-Palmett, A. B. U’Ren, and P. G. Kwiat, “Heralded single- processing: A review,” Rep. Prog. Phys. 82, 016001 (2019). photon source utilizing highly nondegenerate, spectrally factorable spontane- 96W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability ous parametric downconversion,” Opt. Express 24, 10733–10747 (2016). in type-II down-conversion with a broadband pump,” Phys. Rev. A 56, 1627 118T. Gerrits, M. J. Stevens, B. Baek, B. Calkins, A. Lita, S. Glancy, E. Knill, S. W. (1997). Nam, R. P. Mirin, R. H. Hadfield, R. S. Bennink, W. P. Grice, S. Dorenbos, T. 97D. N. Klyshko, “Use of two-photon light for absolute calibration of photoelec- Zijlstra, T. Klapwijk, and V. Zwiller, “Generation of degenerate, factorizable, tric detectors,” Sov. J. Quantum Electron. 10, 1112–1116 (1980). pulsed squeezed light at telecom wavelengths,” Opt. Express 19, 24434–24447 98X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y. (2011). 119 Lu, Y.-A. Chen, and J.-W. Pan, “Observation of eight-photon entanglement,” N. Bruno, A. Martin, T. Guerreiro, B. Sanguinetti, and R. T. Thew, “Pulsed Nat. Photonics 6, 225–228 (2012). source of spectrally uncorrelated and indistinguishable photons at telecom 99X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.- wavelengths,” Opt. Express 22, 17246–17253 (2014). 120 E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.- C. Greganti, P. Schiansky, I. A. Calafell, L. M. Procopio, L. A. Rozema, and P. A. Chen, C.-Y. Lu, and J.-W. Pan, “Experimental ten-photon entanglement,” Walther, “Tuning single-photon sources for telecom multi-photon Phys. Rev. Lett. 117, 210502 (2016). experiments,” Opt. Express 26, 3286–3302 (2018). 121 100D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Phys. 8, 180– 198 R. S. Bennink, “Optimal collinear Gaussian beams for spontaneous parametric (2007). down-conversion,” Phys. Rev. A 81, 053805 (2010). 122 101G. Bonfrate, V. Pruneri, P. G. Kazansky, P. Tapster, and J. G. Rarity, R.-B. Jin, R. Shimizu, K. Wakui, M. Fujiwara, T. Yamashita, S. Miki, H. Terai, “Parametric fluorescence in periodically poled silica fibers,” Appl. Phys. Lett. Z. Wang, and M. Sasaki, “Pulsed sagnac polarization-entangled photon source 75, 2356–2358 (1999). with a ppktp crystal at telecom wavelength,” Opt. Express 22, 11498–11507 102 (2014). S.Tanzilli,H.D.Riedmatten,W.Tittel,H.Zbinden,P.Baldi,M.D.Micheli, 123 D. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using peri- A. M. Branczyk, A. Fedrizzi, T. M. Stace, T. C. Ralph, and A. G. White, odically poled lithium niobate waveguide,” Electron. Lett. 37(2), 26–28 “Engineered optical nonlinearity for quantum light sources,” Opt. Express 19, 55–65 (2011). (2001). 124 103K. Sanaka, K. Kawahara, and T. Kuga, “New high-efficiency source of photon B. P. Dixon, J. H. Shapiro, and F. N. C. Wong, “Spectral engineering by pairs for engineering quantum entanglement,” Phys. Rev. Lett. 86, 5620–5623 Gaussian phase-matching for quantum photonics,” Opt. Express 21, 5879–5890 (2013). (2001). 125 104 K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated pho- F. Graffitti, D. Kundys, D. T. Reid, A. M. Branczyk, and A. Fedrizzi, “Pure tons in controlled spatial modes by downconversion in nonlinear wave- down-conversion photons through sub-coherence-length domain engineering,” Quantum Sci. Technol. 2, 035001 (2017). guides,” Opt. Lett. 26, 1367–1369 (2001). 126 105 C. Chen, C. Bo, M. Y. Niu, F. Xu, Z. Zhang, J. H. Shapiro, and F. N. C. Wong, A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A “Efficient generation and characterization of spectrally factorable biphotons,” wavelength-tunable fiber-coupled source of narrowband entangled photons,” Opt. Express 25, 7300–7312 (2017). Opt. Express 15, 15377–15386 (2007). 127F. Graffitti, P. Barrow, M. Proietti, D. Kundys, and A. Fedrizzi, “Independent 106P. G. Evans, R. S. Bennink, W. P. Grice, T. S. Humble, and J. Schaake, “Bright high-purity photons created in domain-engineered crystals,” Optica 5, source of spectrally uncorrelated polarization-entangled photons with nearly 514–517 (2018). single-mode emission,” Phys. Rev. Lett. 105, 253601 (2010). 128C. Chen, J. E. Heyes, K.-H. Hong, M. Y. Niu, A. E. Lita, T. Gerrits, S. W. Nam, 107T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of J. H. Shapiro, and F. N. C. Wong, “Indistinguishable single-mode photons polarization-entangled photons using a polarization Sagnac interferometer,” from spectrally engineered biphotons,” Opt. Express 27, 11626–11634 (2019). Phys. Rev. A 73, 012316 (2006). 129 E. Meyer-Scott, N. Montaut, J. Tiedau, L. Sansoni, H. Herrmann, T. J. Bartley, 108F. N. C. Wong, J. H. Shapiro, and T. Kim, “Efficient generation of and C. Silberhorn, “Limits on the heralding efficiencies and spectral purities of 16 polarization-entangled photons in a nonlinear crystal,” Laser Phys. , spectrally filtered single photons from photon-pair sources,” Phys. Rev. A 95, 1517–1524 (2006). 109 061803 (2017). O. Kuzucu and F. N. C. Wong, “Pulsed Sagnac source of narrow-band polari- 130F. Laudenbach, R.-B. Jin, C. Greganti, M. Hentschel, P. Walther, and H. zation-entangled photons,” Phys. Rev. A 77, 032314 (2008). Hubel,€ “Numerical investigation of photon-pair generation in periodically 110T. Scheidl, F. Tiefenbacher, R. Prevedel, F. Steinlechner, R. Ursin, and A. poled mTiOxo4 (m ¼ K, rb, cs; x ¼ p, as),” Phys. Rev. Appl. 8, 024035 (2017). Zeilinger, “Crossed-crystal scheme for femtosecond-pulsed entangled photon 131K. Zielnicki, K. Garay-Palmett, D. Cruz-Delgado, H. Cruz-Ramirez, M. F. generation in periodically poled potassium titanyl phosphate,” Phys. Rev. A O’Boyle, B. Fang, V. O. Lorenz, A. B. U’Ren, and P. G. Kwiat, “Joint spectral 89, 042324 (2014). 111 characterization of photon-pair sources,” J. Mod. Opt. 65, 1141–1160 (2018). T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for sponta- 132F. Graffitti, J. Kelly-Massicotte, A. Fedrizzi, and A. M. Branczyk, “Design con- neous parametric down-conversion driven by a narrow pump pulse,” Phys. siderations for high-purity heralded single-photon sources,” Phys. Rev. A 98, Rev. A 56, 1534 (1997). 112 053811 (2018). F. Konig€ and F. N. C. Wong, “Extended phase matching of second-harmonic 133L. K. Shalm, K. Garay, J. Palfree, J. Migdall, A. U’Ren, and S. W. Nam, http:// generation in periodically poled ktiopo4 with zero group-velocity mismatch,” spdcalc.org for “Spontaneous parametric downconversion calculator.” Appl. Phys. Lett. 84, 1644–1646 (2004). 134J. Renema, V. Shchesnovich, and R. Garcia-Patron, “Classical simulability of 113 W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and noisy boson sampling,” preprint arXiv:1809.01953 (2019). space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 135N. Tischler, F. Ghafari, T. J. Baker, S. Slussarenko, R. B. Patel, M. M. Weston, (2001). S. Wollmann, L. K. Shalm, V. B. Verma, S. W. Nam, H. C. Nguyen, H. M. 114 A. U’Ren, C. Silberhorn, K. Banaszek, I. Walmsley, R. Erdmann, W. Grice, and Wiseman, and G. J. Pryde, “Conclusive experimental demonstration of one- M. Raymer, “Generation of pure-state single-photon wavepackets by condi- way Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 121, 100401 (2018). ppffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi tional preparation based on spontaneous parametiric downconversion,” Laser 136We use F ¼½Tr qr q2 as the definition of the fidelity between the states Phys. 15, 146–161 (2005). q and r. 115 P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. 137R. Rangarajan, M. Goggin, and P. Kwiat, “Optimizing type-i polarization- Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single pho- entangled photons,” Opt. Express 17, 18920–18933 (2009). tons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008). 138P. Bierhorst, E. Knill, S. Glancy, Y. Zhang, A. Mink, S. Jordan, A. Rommal, Y.- 116R.-B. Jin, R. Shimizu, K. Wakui, H. Benichi, and M. Sasaki, “Widely tunable K. Liu, B. Christensen, S. W. Nam, M. J. Stevens, and L. K. Shalm, single photon source with high purity at telecom wavelength,” Opt. Express “Experimentally generated randomness certified by the impossibility of super- 21, 10659–10666 (2013). luminal signals,” Nature 556, 223–226 (2018).

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-13 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

139A. Lohrmann, A. Villar, A. Stolk, and A. Ling, “High fidelity field stop collec- 158J. C. Loredo, C. Anton, B. Reznychenko, P. Hilaire, A. Harouri, C. Millet, H. tion for polarization-entangled photon pair sources,” Appl. Phys. Lett. 113, Ollivier, N. Somaschi, L. De Santis, A. Lema^ıtre, I. Sagnes, L. Lanco, A. 171109 (2018). Auffe`ves, O. Krebs, and P. Senellart, “Generation of non-classical light in a 140M. Giustina, M. A. M. Versteegh, S. Wengerowsky, J. Handsteiner, A. photon-number superposition,” Nat. Photonics (2019). Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-A˚ . Larsson, C. Abellan, W. 159We have focused on quantum dots as these are the leading contender to Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. SPDC/SFWM for photon sources in PQC. However, as mentioned earlier, a Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, variety of other single emitters are also suitable in principle. “Significant-loophole-free test of bell’s theorem with entangled photons,” 160D. Englund, A. Faraon, B. Zhang, Y. Yamamoto, and J. Vucˇkovic´, “Generation Phys. Rev. Lett. 115, 250401 (2015). and transfer of single photons on a photonic crystal chip,” Opt. Express 15, 141M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair 5550–5558 (2007). source for quantum communications,” IEEE Photonics Technol. Lett. 14, 161H. Wang, Y.-M. He, T.-H. Chung, H. Hu, Y. Yu, S. Chen, X. Ding, M.-C. 983–985 (2002). Chen, J. Qin, X. Yang, R.-Z. Liu, Z.-C. Duan, J.-P. Li, S. Gerhardt, K. Winkler, 142J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. J. Russell, J. Jurkat, L.-J. Wang, N. Gregersen, Y.-H. Huo, Q. Dai, S. Yu, S. Hofling,€ C.-Y. “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13, Lu, and J.-W. Pan, “Towards optimal single-photon sources from polarized 534–544 (2005). microcavities,” Nat. Photonics (2019). 143J. Fulconis, O. Alibart, W. J. Wadsworth, P. S. Russell, and J. G. Rarity, “High 162Here, we mean the ratio of the rate of photons coupled to the rate of trigger brightness single mode source of correlated photon pairs using a photonic pulses. crystal fiber,” Opt. Express 13, 7572–7582 (2005). 163H. Wang, Y. He, Y.-H. Li, Z.-E. Su, B. Li, H.-L. Huang, X. Ding, M.-C. Chen, 144J. Fan, A. Migdall, and L. J. Wang, “Efficient generation of correlated photon C. Liu, J. Qin, J.-P. Li, Y.-M. He, C. Schneider, M. Kamp, C.-Z. Peng, S. pairs in a microstructure fiber,” Opt. Lett. 30, 3368–3370 (2005). Hofling,€ C.-Y. Lu, and J.-W. Pan, “High-efficiency multiphoton boson 145J. Fulconis, O. Alibart, W. J. Wadsworth, and J. G. Rarity, “Quantum interfer- sampling,” Nat. Photonics 11, 361 (2017). ence with photon pairs using two micro-structured fibres,” New J. Phys. 9, 276 164R. B. Patel, A. J. Bennett, I. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J. (2007). Shields, “Two-photon interference of the emission from electrically tunable 146O. Gazzano and G. S. Solomon, “Toward optical quantum information proc- remote quantum dots,” Nat. Photonics 4, 632 (2010). essing with quantum dots coupled to microstructures [invited],” J. Opt. Soc. 165D. J. P. Ellis, A. J. Bennett, C. Dangel, J. P. Lee, J. P. Griffiths, T. A. Mitchell, Am. B 33, C160–C175 (2016). T.-K. Paraiso, P. Spencer, D. A. Ritchie, and A. J. Shields, “Independent indis- 147O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. tinguishable quantum light sources on a reconfigurable photonic integrated Sagnes, L. Lanco, A. Lema^ıtre, and P. Senellart, “Bright solid-state sources of circuit,” Appl. Phys. Lett. 112, 211104 (2018). indistinguishable single photons,” Nat. Commun. 4, 1425 (2013). 166H. Wang, W. Li, X. Jiang, Y.-M. He, Y.-H. Li, X. Ding, M.-C. Chen, J. Qin, C.- 148N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Z. Peng, C. Schneider, M. Kamp, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, J. P. Hornecker, S. L. Portalupi, T. Grange, C. Anton, J. Demory, C. Gomez, I. Dowling, S. Hofling,€ C.-Y. Lu, and J.-W. Pan, “Toward scalable boson sam- Sagnes, N. D. Lanzillotti-Kimura, A. Lema^ıtre, A. Auffeves, A. G. White, L. pling with photon loss,” Phys. Rev. Lett. 120, 230502 (2018). Lanco, and P. Senellart, “Near-optimal single-photon sources in the solid 167C. Anton, J. C. Loredo, G. Coppola, H. Ollivier, N. Viggianiello, A. Harouri, state,” Nat. Photonics 10, 340 (2016). N. Somaschi, A. Crespi, I. Sagnes, A. Lema^ıtre, L. Lanco, R. Osellame, F. 149J.-H. Kim, S. Aghaeimeibodi, C. J. K. Richardson, R. P. Leavitt, D. Englund, Sciarrino, and P. Senellart, “Interfacing scalable photonic platforms: Solid- and E. Waks, “Hybrid integration of solid-state quantum emitters on a silicon state based multi-photon interference in a reconfigurable glass chip,” preprint photonic chip,” Nano Lett. 17, 7394–7400 (2017). arXiv:1905.00936 (2019). 150S. Aghaeimeibodi, B. Desiatov, J.-H. Kim, C.-M. Lee, M. A. Buyukkaya, A. 168F. Lenzini, B. Haylock, J. C. Loredo, R. A. Abr~ahao, N. A. Zakaria, S. Kasture, Karasahin, C. J. K. Richardson, R. P. Leavitt, M. Loncˇar, and E. Waks, I. Sagnes, A. Lemaitre, H.-P. Phan, D. V. Dao, P. Senellart, M. P. Almeida, A. “Integration of quantum dots with lithium niobate photonics,” Appl. Phys. G. White, and M. Lobino, “Active demultiplexing of single photons from a Lett. 113, 221102 (2018). solid-state source,” Laser Photonics Rev. 11, 1600297 (2017). 151S. Dutta, T. Cai, M. A. Buyukkaya, S. Barik, S. Aghaeimeibodi, and E. Waks, 169A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon and “Coupling quantum emitters in wse2 monolayers to a metal-insulator-metal multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. waveguide,” Appl. Phys. Lett. 113, 191105 (2018). A 66, 053805 (2002). 152Y. He, X. Ding, Z.-E. Su, H.-L. Huang, J. Qin, C. Wang, S. Unsleber, C. Chen, 170J. H. Shapiro and F. N. Wong, “On-demand single-photon generation using a H. Wang, Y.-M. He, X.-L. Wang, W.-J. Zhang, S.-J. Chen, C. Schneider, M. modular array of parametric downconverters with electro-optic polarization Kamp, L.-X. You, Z. Wang, S. Hofling,€ C.-Y. Lu, and J.-W. Pan, “Time-bin- controls,” Opt. Lett. 32, 2698–2700 (2007). encoded boson sampling with a single-photon device,” Phys. Rev. Lett. 118, 171X-s Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental gen- 190501 (2017). eration of single photons via active multiplexing,” Phys. Rev. A 83, 043814 153A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lema^ıtre, I. Sagnes, J. (2011). Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon 172M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, pairs,” Nature 466, 217 (2010). M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of her- 154A. J. Bennett, M. A. Pooley, R. M. Stevenson, M. B. Ward, R. B. Patel, A. B. de alded single-photon sources,” Nat. Commun. 4, 2582 (2013). la Giroday, N. Skold,€ I. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J. Shields, 173T. Meany, L. A. Ngah, M. J. Collins, A. S. Clark, R. J. Williams, B. J. Eggleton, “Electric-field-induced coherent coupling of the exciton states in a single M. J. Steel, M. J. Withford, O. Alibart, and S. Tanzilli, “Hybrid photonic circuit quantum dot,” Nat. Phys. 6, 947 (2010). for multiplexed heralded single photons,” Laser Photonics Rev. 8, L42–L46 155D. Heinze, A. Zrenner, and S. Schumacher, “Polarization-entangled twin pho- (2014). tons from two-photon quantum-dot emission,” Phys. Rev. B 95, 245306 174R. J. A. Francis-Jones, R. A. Hoggarth, and P. J. Mosley, “All-fiber multiplexed (2017). source of high-purity single photons,” Optica 3, 1270–1273 (2016). 156M. Prilmuller,€ T. Huber, M. Muller,€ P. Michler, G. Weihs, and A. Predojevic´, 175T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Single photons on pseudode- “Hyperentanglement of photons emitted by a quantum dot,” Phys. Rev. Lett. mand from stored parametric down-conversion,” Phys. Rev. A 66, 042303 121, 110503 (2018). (2002). 157D. Huber, M. Reindl, S. F. Covre da Silva, C. Schimpf, J. Martın-Sanchez, H. 176E. Jeffrey, N. A. Peters, and P. G. Kwiat, “Towards a periodic deterministic Huang, G. Piredda, J. Edlinger, A. Rastelli, and R. Trotta, “Strain-tunable source of arbitrary single-photon states,” New J. Phys. 6, 100 (2004). GaAs quantum dot: A nearly dephasing-free source of entangled photon pairs 177M. Grimau Puigibert, G. H. Aguilar, Q. Zhou, F. Marsili, M. D. Shaw, V. B. on demand,” Phys. Rev. Lett. 121, 033902 (2018). Verma, S. W. Nam, D. Oblak, and W. Tittel, “Heralded single photons based

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-14 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

on spectral multiplexing and feed-forward control,” Phys. Rev. Lett. 119, optical quantum memory with long coherence time in cold atoms,” Optica 3, 083601 (2017). 100–107 (2016). 178C. Joshi, A. Farsi, S. Clemmen, S. Ramelow, and A. L. Gaeta, “Frequency mul- 198Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S.-L. Zhu, tiplexing for quasi-deterministic heralded single-photon sources,” Nat. “Efficient quantum memory for single-photon polarization qubits,” Nat. Commun. 9, 847 (2018). Photonics 13, 346–351 (2019). 179G. J. Mendoza, R. Santagati, J. Munns, E. Hemsley, M. Piekarek, E. Martın- 199M. Hosseini, G. Campbell, B. M. Sparkes, P. K. Lam, and B. C. Buchler, Lopez, G. D. Marshall, D. Bonneau, M. G. Thompson, and J. L. O’Brien, “Unconditional room-temperature quantum memory,” Nat. Phys. 7, 794 (2011). 200 “Active temporal and spatial multiplexing of photons,” Optica 3, 127–132 P. Vernaz-Gris, K. Huang, M. Cao, A. S. Sheremet, and J. Laurat, “Highly-effi- (2016). cient quantum memory for polarization qubits in a spatially-multiplexed cold 180F. Kaneda, B. G. Christensen, J. J. Wong, H. S. Park, K. T. McCusker, and P. atomic ensemble,” Nat. Commun. 9, 363 (2018). 201 G. Kwiat, “Time-multiplexed heralded single-photon source,” Optica 2, M. Rambach, A. Nikolova, T. J. Weinhold, and A. G. White, “Sub-megahertz 1010–1013 (2015). linewidth single photon source,” APL Photonics 1, 096101 (2016). 202 181C. Xiong, X. Zhang, Z. Liu, M. J. Collins, A. Mahendra, L. G. Helt, M. J. Steel, M. Rambach, W. Y. S. Lau, S. Laibacher, V. Tamma, A. G. White, and T. J. D. Y. Choi, C. J. Chae, P. H. W. Leong, and B. J. Eggleton, “Active temporal Weinhold, “Hectometer revivals of quantum interference,” Phys. Rev. Lett. multiplexing of indistinguishable heralded single photons,” Nat. Commun. 7, 121, 093603 (2018). 203 10853 (2016). T. C. Ralph and A. P. Lund, “Nondeterministic noiseless linear amplification 182F. Kaneda and P. G. Kwiat, “High-efficiency single-photon generation via of quantum systems,” in Proceedings of 9th International Conference on large-scale active time multiplexing,” preprint arXiv:1803.04803 (2018). Quantum Communication, Measurement and Computing, edited by A. 183 Lvovsky (2009), pp. 155–160. M. Gimeno-Segovia, H. Cable, G. J. Mendoza, P. Shadbolt, J. W. Silverstone, J. 204 Carolan, M. G. Thompson, J. L. O’Brien, and T. Rudolph, “Relative multiplex- J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. ing for minimising switching in linear-optical quantum computing,” New J. Tura, R. Augusiak, L. Mancˇinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Phys. 19, 063013 (2017). Gong, A. Acın, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. 184X. Zhang, Y. H. Lee, B. A. Bell, P. H. W. Leong, T. Rudolph, B. J. Eggleton, Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018). and C. Xiong, “Indistinguishable heralded single photon generation via rela- 205 tive temporal multiplexing of two sources,” Opt. Express 25, 26067–26075 A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on 19 (2017). spectrally engineered sum frequency generation,” Opt. Express , 185F. Kaneda, F. Xu, J. Chapman, and P. G. Kwiat, “Quantum-memory-assisted 13770–13778 (2011). 206S. Slussarenko, B. Piccirillo, V. Chigrinov, L. Marrucci, and E. Santamato, multi-photon generation for efficient quantum information processing,” “Liquid crystal spatial-mode converters for the orbital angular momentum of Optica 4, 1034–1037 (2017). 186 light,” J. Opt. 15, 025406 (2013). A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. 207D. V. Sychev, A. E. Ulanov, E. S. Tiunov, A. A. Pushkina, A. Kuzhamuratov, Photonics 3, 706 (2009). 187 V. Novikov, and A. I. Lvovsky, “Entanglement and teleportation between E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussie`res, M. polarization and wave-like encodings of an optical qubit,” Nat. Commun. 9, George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum 3672 (2018). memory for entangled photons,” Nature 469, 512 (2011). 208S. Kasture, F. Lenzini, B. Haylock, A. Boes, A. Mitchell, E. W. Streed, and M. 188D. J. Saunders, J. H. D. Munns, T. F. M. Champion, C. Qiu, K. T. Kaczmarek, Lobino, “Frequency conversion between uv and telecom wavelengths in a lith- E. Poem, P. M. Ledingham, I. A. Walmsley, and J. Nunn, “Cavity-enhanced ium niobate waveguide for quantum communication with yb þ trapped room-temperature broadband Raman memory,” Phys. Rev. Lett. 116, 090501 ions,” J. Opt. 18, 104007 (2016). (2016). 209 S. Tanzilli, A. Martin, F. Kaiser, M. P. De Micheli, O. Alibart, and D. B. 189K. T. Kaczmarek, P. M. Ledingham, B. Brecht, S. E. Thomas, G. S. Ostrowsky, “On the genesis and evolution of integrated quantum optics,” Thekkadath, O. Lazo-Arjona, J. H. D. Munns, E. Poem, A. Feizpour, D. J. Laser Photonics Rev. 6, 115–143 (2012). Saunders, J. Nunn, and I. A. Walmsley, “High-speed noise-free optical quan- 210L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and tum memory,” Phys. Rev. A 97, 042316 (2018). 190 R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. A. Tiranov, P. C. Strassmann, J. Lavoie, N. Brunner, M. Huber, V. B. Verma, Lett. 105, 200503 (2010). S. W. Nam, R. P. Mirin, A. E. Lita, F. Marsili, M. Afzelius, F. Bussie`res, and N. 211A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, Gisin, “Temporal multimode storage of entangled photon pairs,” Phys. Rev. N. Viggianiello, L. Innocenti, P. Mataloni, and F. Sciarrino, “Suppression law Lett. 117, 240506 (2016). 191 of quantum states in a 3d photonic fast Fourier transform chip,” Nat. M. Rancîc´, M. P. Hedges, R. L. Ahlefeldt, and M. J. Sellars, “Coherence time of Commun. 7, 10469 (2016). over a second in a telecom-compatible quantum memory storage material,” 212D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Nat. Phys. 14, 50 (2017). Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, and J. L. O’Brien, 192 G. T. Campbell, O. Pinel, M. Hosseini, T. C. Ralph, B. C. Buchler, and P. K. “Fast path and polarization manipulation of telecom wavelength single pho- Lam, “Configurable unitary transformations and linear logic gates using quan- tons in lithium niobate waveguide devices,” Phys. Rev. Lett. 108, 053601 tum memories,” Phys. Rev. Lett. 113, 063601 (2014). (2012). 193 V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, 213J. P. Hopker,€ M. Bartnick, E. Meyer-Scott, F. Thiele, S. Krapick, N. Montaut, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom M. Santandrea, H. Herrmann, S. Lengeling, R. Ricken, V. Quiring, T. Meier, quantum memory,” Nat. Commun. 6, 7706 (2015). A. Lita, V. Verma, T. Gerrits, S. W. Nam, C. Silberhorn, and T. J. Bartley, 194 T.-S. Yang, Z.-Q. Zhou, Y.-L. Hua, X. Liu, Z.-F. Li, P.-Y. Li, Y. Ma, C. Liu, P.-J. “Towards integrated superconducting detectors on lithium niobate wave- Liang, X. Li, Y.-X. Xiao, J. Hu, C.-F. Li, and G.-C. Guo, “Multiplexed storage guides,” Proc. SPIE 10358, 1035809 (2017). and real-time manipulation based on a multiple degree-of-freedom quantum 214I. Krasnokutska, J.-L. J. Tambasco, X. Li, and A. Peruzzo, “Ultra-low loss pho- memory,” Nat. Commun. 9, 3407 (2018). tonic circuits in lithium niobate on insulator,” Opt. Express 26, 897–904 195 M. Zhong, M. P. Hedges, R. L. Ahlefeldt, J. G. Bartholomew, S. E. Beavan, S. (2018). M. Wittig, J. J. Longdell, and M. J. Sellars, “Optically addressable nuclear spins 215I. Krasnokutska, J.-L. J. Tambasco, and A. Peruzzo, “Tunable large free spec- in a solid with a six-hour coherence time,” Nature 517, 177 (2015). tral range microring resonators in lithium niobate on insulator,” Sci. Rep. 9, 196M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum mem- 11086 (2019). ory for light,” Nature 465, 1052 (2010). 216J. W. Silverstone, D. Bonneau, J. L. O’Brien, and M. G. Thompson, “Silicon 197Y.-W. Cho, G. T. Campbell, J. L. Everett, J. Bernu, D. B. Higginbottom, M. T. quantum photonics,” IEEE J. Sel. Top. Quantum Electron. 22, 390–402 Cao, J. Geng, N. P. Robins, P. K. Lam, and B. C. Buchler, “Highly efficient (2016).

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-15 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

217S. Ferrari, C. Schuck, and W. Pernice, “Waveguide-integrated superconducting spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, nanowire single-photon detectors,” Nanophotonics 7, 1725 (2019). 14870–14886 (2007). 218J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, 236B. J. Smith, P. Mahou, O. Cohen, J. S. Lundeen, and I. A. Walmsley, “Photon R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hofling,€ R. Sanjines, and A. Fiore, pair generation in birefringent optical fibers,” Opt. Express 17, 23589–23602 “Waveguide superconducting single-photon detectors for integrated quantum (2009). 237 photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011). J. B. Spring, P. S. Salter, B. J. Metcalf, P. C. Humphreys, M. Moore, N. 219W. H. P. Pernice, C. Schuck, O. Minaeva, M. Li, G. N. Goltsman, A. V. Thomas-Peter, M. Barbieri, X.-M. Jin, N. K. Langford, W. S. Kolthammer, M. Sergienko, and H. X. Tang, “High-speed and high-efficiency travelling wave J. Booth, and I. A. Walmsley, “On-chip low loss heralded source of pure single single-photon detectors embedded in nanophotonic circuits,” Nat. Commun. photons,” Opt. Express 21, 13522–13532 (2013). 238 3, 1325 (2012). J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. 220C. Schuck, W. H. P. Pernice, and H. X. Tang, “Waveguide integrated low Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. noise nbtin nanowire single-photon detectors with milli-Hz dark count rate,” Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, “On-chip quantum Sci. Rep. 3, 1893 (2013). interference between silicon photon-pair sources,” Nat. Photonics 8, 104 221O. Kahl, S. Ferrari, V. Kovalyuk, G. N. Goltsman, A. Korneev, and W. H. P. (2013). 239 Pernice, “Waveguide integrated superconducting single-photon detectors with J. B. Spring, P. L. Mennea, B. J. Metcalf, P. C. Humphreys, J. C. Gates, H. L. high internal quantum efficiency at telecom wavelengths,” Sci. Rep. 5, 10941 Rogers, C. Soller,€ B. J. Smith, W. S. Kolthammer, P. G. R. Smith, and I. A. (2015). Walmsley, “Chip-based array of near-identical, pure, heralded single-photon 222M. K. Akhlaghi, E. Schelew, and J. F. Young, “Waveguide integrated supercon- sources,” Optica 4, 90–96 (2017). 240 ducting single-photon detectors implemented as near-perfect absorbers of N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. coherent radiation,” Nat. Commun. 6, 8233 (2015). Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spec- 223J. Munzberg,€ A. Vetter, F. Beutel, W. Hartmann, S. Ferrari, W. H. P. Pernice, trally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014). and C. Rockstuhl, “Superconducting nanowire single-photon detector imple- 241 mented in a 2d photonic crystal cavity,” Optica 5, 658–665 (2018). D. Grassani, A. Simbula, S. Pirotta, M. Galli, M. Menotti, N. C. Harris, T. 224B. Calkins, P. L. Mennea, A. E. Lita, B. J. Metcalf, W. S. Kolthammer, A. Baehr-Jones, M. Hochberg, C. Galland, M. Liscidini, and D. Bajoni, “Energy Lamas-Linares, J. B. Spring, P. C. Humphreys, R. P. Mirin, J. C. Gates, P. G. R. correlations of photon pairs generated by a silicon microring resonator probed by stimulated four wave mixing,” Sci. Rep. 6, 23564 (2016). Smith, I. A. Walmsley, T. Gerrits, and S. W. Nam, “High quantum-efficiency 242 photon-number-resolving detector for photonic on-chip information proc- A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008). essing,” Opt. Express 21, 22657–22670 (2013). 243 225 A. Crespi, R. Ramponi, R. Osellame, L. Sansoni, I. Bongioanni, F. Sciarrino, G. J. P. Hopker,€ T. Gerrits, A. Lita, S. Krapick, H. Herrmann, R. Ricken, V. Vallone, and P. Mataloni, “Integrated photonic quantum gates for polariza- Quiring, R. Mirin, S. W. Nam, C. Silberhorn, and T. J. Bartley, “Integrated tion qubits,” Nat. Commun. 2, 566 (2011). transition edge sensors on titanium in-diffused lithium niobate waveguides,” 244C.-Y. Lu, D. E. Browne, T. Yang, and J.-W. Pan, “Demonstration of a com- APL Photonics 4, 056103 (2019). piled version of Shor’s quantum factoring algorithm using photonic qubits,” 226T. Zhong, X. Hu, F. N. C. Wong, K. K. Berggren, T. D. Roberts, and P. Battle, Phys. Rev. Lett. 99, 250504 (2007). “High-quality fiber-optic polarization entanglement distribution at 1.3 lm tel- 245B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. ecom wavelength,” Opt. Lett. 35, 1392–1394 (2010). Gilchrist, and A. G. White, “Experimental demonstration of a compiled ver- 227A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient sion of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett. 99, single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 250505 (2007). 106, 013603 (2011). 246 A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algo- 228G. Harder, V. Ansari, B. Brecht, T. Dirmeier, C. Marquardt, and C. rithm on a photonic chip,” Science 325, 1221–1221 (2009). Silberhorn, “An optimized photon pair source for quantum circuits,” Opt. 247P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Express 21, 13975–13985 (2013). 229 Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating N. Montaut, L. Sansoni, E. Meyer-Scott, R. Ricken, V. Quiring, H. Herrmann, and measuring entanglement and mixture with a reconfigurable photonic and C. Silberhorn, “High-efficiency plug-and-play source of heralded single circuit,” Nat. Photonics 6, 45 (2011). photons,” Phys. Rev. Appl. 8, 024021 (2017). 248 230 J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of S. Atzeni, A. S. Rab, G. Corrielli, E. Polino, M. Valeri, P. Mataloni, N. multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, Spagnolo, A. Crespi, F. Sciarrino, and R. Osellame, “Integrated sources of 346 (2009). entangled photons at the telecom wavelength in femtosecond-laser-written 249B. J. Smith, D. Kundys, N. Thomas-Peter, P. G. R. Smith, and I. A. Walmsley, circuits,” Optica 5, 311–314 (2018). 231 “Phase-controlled integrated photonic quantum circuits,” Opt. Express 17, E. Meyer-Scott, N. Prasannan, C. Eigner, V. Quiring, J. M. Donohue, S. 13516–13525 (2009). Barkhofen, and C. Silberhorn, “High-performance source of spectrally pure, 250F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’Ambrosio, P. Mataloni, polarization entangled photon pairs based on hybrid integrated-bulk optics,” F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, Opt. Express 26, 32475–32490 (2018). “Thermally reconfigurable quantum photonic circuits at telecom wavelength 232 F. Lenzini, A. N. Poddubny, J. Titchener, P. Fisher, A. Boes, S. Kasture, B. by femtosecond laser micromachining,” Light Sci. Appl. 4, e354 (2015). Haylock, M. Villa, A. Mitchell, A. S. Solntsev, A. A. Sukhorukov, and M. 251J. Notaros, J. Mower, M. Heuck, C. Lupo, N. C. Harris, G. R. Steinbrecher, D. Lobino, “Direct characterization of a nonlinear photonic circuit’s wave func- Bunandar, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, tion with laser light,” Light Sci. Appl. 7, 17143 (2018). “Programmable dispersion on a photonic integrated circuit for classical and 233 J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, quantum applications,” Opt. Express 25, 21275–21285 (2017). A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale 252J. Carolan, C. Harrold, C. Sparrow, E. Martın-Lopez, N. J. Russell, J. W. silicon waveguides,” Opt. Express 14, 12388–12393 (2006). Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, 234 M. Davanço,J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. M. J. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-band Laing, “Universal linear optics,” Science 349, 711–716 (2015). heralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett. 253M. G. Thompson, http://2016.qcrypt.net/wp-content/uploads/2015/11/ for 100, 261104 (2012). presentation at QCrypt 2016; accessed 11 June 2019. 235K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel- 254P. L. Mennea, W. R. Clements, D. H. Smith, J. C. Gates, B. J. Metcalf, R. H. S. Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Bannerman, R. Burgwal, J. J. Renema, W. S. Kolthammer, I. A. Walmsley, and Walmsley, “Photon pair-state preparation with tailored spectral properties by P. G. R. Smith, “Modular linear optical circuits,” Optica 5, 1087–1090 (2018).

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-16 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

255R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing Lett. 86, 5188–5191 (2001). arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018). 256M. A. Nielsen, “Optical quantum computation using cluster states,” Phys. 279Earlier reviews provide references to samples of older demonstrations in the Rev. Lett. 93, 040503 (2004). field. 257Cluster states are graphs where the nodes are qubits and the edges are entan- 280I. Pitsios, L. Banchi, A. S. Rab, M. Bentivegna, D. Caprara, A. Crespi, N. gling links between the qubits, satisfying particular constraints.255 Spagnolo, S. Bose, P. Mataloni, R. Osellame, and F. Sciarrino, “Photonic simu- 258R. B. Patel, J. Ho, F. Ferreyrol, T. C. Ralph, and G. J. Pryde, “A quantum lation of entanglement growth and engineering after a spin chain quench,” Fredkin gate,” Sci. Adv. 2, e1501531 (2016). Nat. Commun. 8, 1569 (2017). 259T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White, “Linear optical 281A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. controlled-not gate in the coincidence basis,” Phys. Rev. A 65, 062324 (2002). Aspuru-Guzik, and J. L. O’Brien, “A variational eigenvalue solver on a pho- 260T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic tonic quantum processor,” Nat. Commun. 5, 4213 (2014). operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001). 282J. Wang, S. Paesani, R. Santagati, S. Knauer, A. A. Gentile, N. Wiebe, M. 261J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, Petruzzella, J. L. O’Brien, J. G. Rarity, A. Laing, and M. G. Thompson, “Demonstration of an all-optical quantum controlled-not gate,” Nature 426, “Experimental quantum Hamiltonian learning,” Nat. Phys. 13, 551 (2017). 264 (2003). 283S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Gr€afe, R. 262T. B. Pittman, M. J. Fitch, B. C. Jacobs, and J. D. Franson, “Experimental Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. controlled-not logic gate for single photons in the coincidence basis,” Phys. Szameit, “Implementation of quantum and classical discrete fractional Fourier Rev. A 68, 032316 (2003). transforms,” Nat. Commun. 7, 11027 (2016). 263J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. 284M. Stobinska, A. Buraczewski, M. Moore, W. R. Clements, J. J. Renema, S. W. Ralph, and A. G. White, “Quantum process tomography of a controlled-not Nam, T. Gerrits, A. Lita, W. S. Kolthammer, A. Eckstein, and I. A. Walmsley, gate,” Phys. Rev. Lett. 93, 080502 (2004). “Quantum interference enables constant-time quantum information proc- 264P. P. Rohde, G. J. Pryde, J. L. O’Brien, and T. C. Ralph, “Quantum-gate char- essing,” Sci. Adv. 5, eaau9674 (2019). acterization in an extended Hilbert space,” Phys. Rev. A 72, 032306 (2005). 285S. Aaronson and A. Arkhipov, “The computational complexity of linear 265R. Okamoto, J. L. O’Brien, H. F. Hofmann, and S. Takeuchi, “Realization of a optics,” Theory Comput. 9, 143–252 (2013). knill-laflamme-milburn controlled-not photonic quantum circuit combining 286M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. effective optical nonlinearities,” PNAS 108, 10067–10071 (2011). Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” 266C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Perez, and W. K. Science 339, 794–798 (2013). Wootters, “Teleporting an unknown quantum state via dual classical and 287J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993). Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, 267W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a pho- Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation- tonic chip,” Science 339, 798–801 (2013). based realization of an optical quantum two-qubit entangling gate,” PNAS 288M. Tillmann, B. Dakic´, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, 107, 20869–20874 (2010). “Experimental boson sampling,” Nat. Photonics 7, 540 (2013). 268M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum 289N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Information: 10th Anniversary Edition, 10th ed. (Cambridge University Press, Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galv~ao, New York, NY, USA, 2011). and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. 269B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Photonics 8, 615 (2014). Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru- 290J. Carolan, J. D. A. Meinecke, P. J. Shadbolt, N. J. Russell, N. Ismail, K. Guzik, and A. G. White, “Towards quantum chemistry on a quantum com- Worhoff,€ T. Rudolph, M. G. Thompson, J. L. O’Brien, J. C. F. Matthews, and puter,” Nat. Chem. 2, 106 (2010). A. Laing, “On the experimental verification of quantum complexity in linear 270R. Santagati, J. Wang, A. A. Gentile, S. Paesani, N. Wiebe, J. R. McClean, S. optics,” Nat. Photonics 8, 621 (2014). Morley-Short, P. J. Shadbolt, D. Bonneau, J. W. Silverstone, D. P. Tew, X. 291H.-S. Zhong, Y. Li, W. Li, L.-C. Peng, Z.-E. Su, Y. Hu, Y.-M. He, X. Ding, W. Zhou, J. L. O’Brien, and M. G. Thompson, “Witnessing eigenstates for quan- Zhang, H. Li, L. Zhang, Z. Wang, L. You, X.-L. Wang, X. Jiang, L. Li, Y.-A. tum simulation of Hamiltonian spectra,” Sci. Adv. 4, eaap9646 (2018). Chen, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “12-photon entanglement and scal- 271B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. able scattershot boson sampling with optimal entangled-photon pairs from Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying parametric down-conversion,” Phys. Rev. Lett. 121, 250505 (2018). quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 292D. J. Brod, E. F. Galv~ao, A. Crespi, R. Osellame, N. Spagnolo, and F. Sciarrino, 134–140 (2009). “Photonic implementation of boson sampling: A review,” Adv. Photonics 1, 272M. Araujo, A. Feix, F. Costa, and C. Brukner, “Quantum circuits cannot con- 034001 (2019). trol unknown operations,” New J. Phys. 16, 093026 (2014). 293A. P. Lund, M. J. Bremner, and T. C. Ralph, “Quantum sampling problems, 273J. Thompson, K. Modi, V. Vedral, and M. Gu, “Quantum plug n’play: Bosonsampling and quantum supremacy,” npj Quantum Inf. 3, 15 (2017). Modular computation in the quantum regime,” New J. Phys. 20, 013004 294A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O’Brien, and T. C. (2018). Ralph, “Boson sampling from a Gaussian state,” Phys. Rev. Lett. 113, 100502 274X.-Q. Zhou, T. C. Ralph, P. Kalasuwan, M. Zhang, A. Peruzzo, B. P. Lanyon, (2014). and J. L. O’Brien, “Adding control to arbitrary unknown quantum oper- 295S. Rahimi-Keshari, T. C. Ralph, and C. M. Caves, “Sufficient conditions for ations,” Nat. Commun. 2, 413 (2011). efficient classical simulation of quantum optics,” Phys. Rev. X 6, 021039 275X.-D. Cai, C. Weedbrook, Z.-E. Su, M.-C. Chen, M. Gu, M.-J. Zhu, L. Li, N.-L. (2016). Liu, C.-Y. Lu, and J.-W. Pan, “Experimental quantum computing to solve sys- 296A. Neville, C. Sparrow, R. Clifford, E. Johnston, P. M. Birchall, A. Montanaro, tems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013). and A. Laing, “Classical boson sampling algorithms with superior perfor- 276S. Barz, I. Kassal, M. Ringbauer, Y. O. Lipp, B. Dakic´, A. Aspuru-Guzik, and mance to near-term experiments,” Nat. Phys. 13, 1153 (2017). P. Walther, “A two-qubit photonic quantum processor and its application to 297J. Preskill, “Quantum Computing in the NISQ era and beyond,” Quantum 2, solving systems of linear equations,” Sci. Rep. 4, 6115 (2014). 79 (2018). 277E. Martın-Lopez, A. Laing, T. Lawson, R. Alvarez, X.-Q. Zhou, and J. L. 298M. Y. Niu, S. Boixo, V. N. Smelyanskiy, and H. Neven, “Universal quantum O’Brien, “Experimental realization of Shor’s quantum factoring algorithm control through deep reinforcement learning,” npj Quantum Inf. 5, 33 (2019). using qubit recycling,” Nat. Photonics 6, 773 (2012). 299S. Mavadia, V. Frey, J. Sastrawan, S. Dona, and M. J. Biercuk, “Prediction and 278X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. real-time compensation of qubit decoherence via machine learning,” Nat. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, Commun. 8, 14106 (2017).

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-17 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

300J. Wallnofer,€ A. A. Melnikov, W. Dur, and H. J. Briegel, “Machine learning 325B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, for long-distance quantum communication,” preprint arXiv:1904.10797 “Entanglement-free Heisenberg-limited phase estimation,” Nature 450, (2019). 393–396 (2007). 301A. Hentschel and B. C. Sanders, “Efficient algorithm for optimizing adaptive 326A. A. Berni, T. Gehring, B. M. Nielsen, V. H€andchen, M. G. A. Paris, and U. quantum metrology processes,” Phys. Rev. Lett. 107, 233601 (2011). L. Andersen, “Ab initio quantum-enhanced optical phase estimation using 302A. Lumino, E. Polino, A. S. Rab, G. Milani, N. Spagnolo, N. Wiebe, and F. real-time feedback control,” Nat. Photonics 9, 577 (2015). Sciarrino, “Experimental phase estimation enhanced by machine learning,” 327S. Daryanoosh, S. Slussarenko, D. W. Berry, H. M. Wiseman, and G. J. Pryde, Phys. Rev. Appl. 10, 044033 (2018). “Experimental optical phase measurement approaching the exact Heisenberg 303X.-D. Cai, D. Wu, Z.-E. Su, M.-C. Chen, X.-L. Wang, L. Li, N.-L. Liu, C.-Y. limit,” Nat. Commun. 9, 4606 (2018). Lu, and J.-W. Pan, “Entanglement-based machine learning on a quantum 328M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum trans- computer,” Phys. Rev. Lett. 114, 110504 (2015). duction of telecommunications-band single photons from a quantum dot by 304J. Gao, L.-F. Qiao, Z.-Q. Jiao, Y.-C. Ma, C.-Q. Hu, R.-J. Ren, A.-L. Yang, H. frequency upconversion,” Nat. Photonics 4, 786 (2010). Tang, M.-H. Yung, and X.-M. Jin, “Experimental machine learning of quan- 329R. Ikuta, Y. Kusaka, T. Kitano, H. Kato, T. Yamamoto, M. Koashi, and N. tum states,” Phys. Rev. Lett. 120, 240501 (2018). Imoto, “Wide-band quantum interface for visible-to-telecommunication 305A. Pepper, N. Tischler, and G. J. Pryde, “Experimental realization of a quan- wavelength conversion,” Nat. Commun. 2, 537 (2011). tum autoencoder: The compression of qutrits via machine learning,” Phys. 330K.DeGreve,L.Yu,P.L.McMahon,J.S.Pelc,C.M.Natarajan,N.Y.Kim,E. Rev. Lett. 122, 060501 (2019). Abe,S.Maier,C.Schneider,M.Kamp,S.H.ofling,R.H.Hadfield,A.Forchel, 306The terms “hard” and “easy” are rather qualitative, but they give an idea of M. M. Fejer, and Y. Yamamoto, “Quantum-dot spin-photon entanglement via the motivation for this approach. frequency downconversion to telecom wavelength,” Nature 491, 421 (2012). 307R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quan- 331B. A. Bell, J. He, C. Xiong, and B. J. Eggleton, “Frequency conversion in silicon tum computation on cluster states,” Phys. Rev. A 68, 022312 (2003). in the single photon regime,” Opt. Express 24, 5235–5242 (2016). 308H. J. Briegel, D. E. Browne, W. Dur,€ R. Raussendorf, and M. Van den Nest, 332H. Rutz,€ K.-H. Luo, H. Suche, and C. Silberhorn, “Towards a quantum inter- “Measurement-based quantum computation,” Nat. Phys. 5, 19 (2009). face between telecommunication and uv wavelengths: Design and classical 309D. E. Browne and T. Rudolph, “Resource-efficient linear optical quantum performance,” Appl. Phys. B 122, 13 (2016). computation,” Phys. Rev. Lett. 95, 010501 (2005). 333A. Dreau, A. Tchebotareva, A. E. Mahdaoui, C. Bonato, and R. Hanson, 310M. Varnava, D. E. Browne, and T. Rudolph, “How good must single photon “Quantum frequency conversion of single photons from a nitrogen-vacancy sources and detectors be for efficient linear optical quantum computation?” center in diamond to telecommunication wavelengths,” Phys. Rev. Appl. 9, Phys. Rev. Lett. 100, 060502 (2008). 064031 (2018). 311P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. 334H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communi- Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” cation complexity,” Rev. Mod. Phys. 82, 665–698 (2010). Nature 434, 169 (2005). 335K. Wei, N. Tischler, S.-R. Zhao, Y.-H. Li, J. M. Arrazola, Y. Liu, W. Zhang, H. 312C.-Y. Lu, X.-Q. Zhou, O. Guhne,€ W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, Li, L. You, Z. Wang, Y.-A. Chen, B. C. Sanders, Q. Zhang, G. J. Pryde, F. Xu, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph and J.-W. Pan, “Experimental quantum switching for exponentially superior states,” Nat. Phys. 3, 91 (2007). quantum communication complexity,” Phys. Rev. Lett. 122, 120504 (2019). 313C. Greganti, M.-C. Roehsner, S. Barz, T. Morimae, and P. Walther, 336D. Gottesman, T. Jennewein, and S. Croke, “Longer-baseline telescopes using “Demonstration of measurement-only blind quantum computing,” New J. quantum repeaters,” Phys. Rev. Lett. 109, 070503 (2012). Phys. 18, 013020 (2016). 337B. Hensen, H. Bernien, A. E. Dreau, A. Reiserer, N. Kalb, M. S. Blok, J. 314R. Prevedel, P. Walther, F. Tiefenbacher, P. Bohi, R. Kaltenbaek, T. Jennewein, Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellan, W. Amaya, V. and A. Zeilinger, “High-speed linear optics quantum computing using active Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, feed-forward,” Nature 445, 65–69 (2007). T. H. Taminiau, and R. Hanson, “Loophole-free bell inequality violation using 315S. Barz, E. Kashefi, A. Broadbent, J. F. Fitzsimons, A. Zeilinger, and P. electron spins separated by 1.3 kilometres,” Nature 526, 682–686 (2015). Walther, “Demonstration of blind quantum computing,” Science 335, 338A. Acın, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device- 303–308 (2012). independent security of quantum cryptography against collective attacks,” 316F. Ewert and P. van Loock, “3/4-efficient bell measurement with passive linear Phys. Rev. Lett. 98, 230501 (2007). optics and unentangled ancillae,” Phys. Rev. Lett. 113, 140403 (2014). 339G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded 317K. Kieling, T. Rudolph, and J. Eisert, “Percolation, renormalization, and quan- noiseless linear amplification and distillation of entanglement,” Nat. tum computing with nondeterministic gates,” Phys. Rev. Lett. 99, 130501 Photonics 4, 316 (2010). (2007). 340F. Ferreyrol, M. Barbieri, R. Blandino, S. Fossier, R. Tualle-Brouri, and P. 318M. Pant, D. Towsley, D. Englund, and S. Guha, “Percolation thresholds for Grangier, “Implementation of a nondeterministic optical noiseless amplifier,” photonic quantum computing,” Nat. Commun. 10, 1070 (2019). Phys. Rev. Lett. 104, 123603 (2010). 319M. Varnava, D. E. Browne, and T. Rudolph, “Loss tolerance in one-way quan- 341A. Zavatta, J. Fiurasek, and M. Bellini, “A high-fidelity noiseless amplifier for tum computation via counterfactual error correction,” Phys. Rev. Lett. 97, quantum light states,” Nat. Photonics 5, 52 (2010). 120501 (2006). 342S. Kocsis, G. Y. Xiang, T. C. Ralph, and G. J. Pryde, “Heralded noiseless ampli- 320T. Rudolph, “Why I am optimistic about the silicon-photonic route to quan- fication of a photon polarization qubit,” Nat. Phys. 9, 23–28 (2013). tum computing,” APL Photonics 2, 030901 (2017). 343F. Monteiro, E. Verbanis, V. C. Vivoli, A. Martin, N. Gisin, H. Zbinden, and 321S. Morley-Short, M. Gimeno-Segovia, T. Rudolph, and H. Cable, “Loss-toler- R. T. Thew, “Heralded amplification of path entangled quantum states,” ant teleportation on large stabilizer states,” Quantum Sci. Technol. 4, 025014 Quantum Sci. Technol. 2, 024008 (2017). (2019). 344N. Bruno, V. Pini, A. Martin, V. B. Verma, S. W. Nam, R. Mirin, A. Lita, F. 322M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From three- Marsili, B. Korzh, F. Bussie`res, N. Sangouard, H. Zbinden, N. Gisin, and R. photon greenberger-horne-zeilinger states to ballistic universal quantum Thew, “Heralded amplification of photonic qubits,” Opt. Express 24, 125–133 computation,” Phys. Rev. Lett. 115, 020502 (2015). (2016). 323S. Morley-Short, S. Bartolucci, M. Gimeno-Segovia, P. Shadbolt, H. Cable, and 345A. E. Ulanov, I. A. Fedorov, A. A. Pushkina, Y. V. Kurochkin, T. C. Ralph, T. Rudolph, “Physical-depth architectural requirements for generating univer- and A. I. Lvovsky, “Undoing the effect of loss on quantum entanglement,” sal photonic cluster states,” Quantum Sci. Technol. 3, 015005 (2017). Nat. Photonics 9, 764–768 (2015). 324H. Wiseman, D. Berry, S. Bartlett, B. Higgins, and G. Pryde, “Adaptive mea- 346H. M. Chrzanowski, N. Walk, S. M. Assad, J. Janousek, S. Hosseini, T. C. surements in the optical quantum information laboratory,” IEEE J. Sel. Top. Ralph, T. Symul, and P. K. Lam, “Measurement-based noiseless linear amplifi- Quantum Electron. 15, 1661–1672 (2009). cation for quantum communication,” Nat. Photonics 8, 333 (2014).

Appl. Phys. Rev. 6, 041303 (2019); doi: 10.1063/1.5115814 6, 041303-18 VC Author(s) 2019 Applied Physics Reviews REVIEW scitation.org/journal/are

347J. Y. Haw, J. Zhao, J. Dias, S. M. Assad, M. Bradshaw, R. Blandino, T. Symul, 355J. M. Auger, H. Anwar, M. Gimeno-Segovia, T. M. Stace, and D. E. Browne, T. C. Ralph, and P. K. Lam, “Surpassing the no-cloning limit with a heralded “Fault-tolerant quantum computation with nondeterministic entangling hybrid linear amplifier for coherent states,” Nat. Commun. 7, 13222 (2016). gates,” Phys. Rev. A 97, 030301 (2018). 348N. A. McMahon, A. P. Lund, and T. C. Ralph, “Optimal architecture for a 356A. P. Lund and T. C. Ralph, “Nondeterministic gates for photonic single-rail nondeterministic noiseless linear amplifier,” Phys. Rev. A 89, 023846 (2014). quantum logic,” Phys. Rev. A 66, 032307 (2002). 349J. Ho, A. Boston, M. Palsson, and G. Pryde, “Experimental noiseless linear 357A. Gilchrist, A. J. F. Hayes, and T. C. Ralph, “Efficient parity-encoded optical amplification using weak measurements,” New J. Phys. 18, 093026 (2016). quantum computing,” Phys. Rev. A 75, 052328 (2007). 350T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger, “Experimental nonlocal- 358N. C. Menicucci, “Fault-tolerant measurement-based quantum computing ity proof of quantum teleportation and entanglement swapping,” Phys. Rev. with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504 Lett. 88, 017903 (2001). (2014). 351M. M. Weston, S. Slussarenko, H. M. Chrzanowski, S. Wollmann, L. K. Shalm, 359S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. V. B. Verma, M. S. Allman, S. W. Nam, and G. J. Pryde, “Heralded quantum Suzuki, J-i Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, steering over a high-loss channel,” Sci. Adv. 4, e1701230 (2018). “Ultra-large-scale continuous-variable cluster states multiplexed in the time 352G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, “Measuring domain,” Nat. Photonics 7, 982 (2013). a photonic qubit without destroying it,” Phys. Rev. Lett. 92, 190402 (2004). 360M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multi- 353E. Meyer-Scott, D. McCloskey, K. Gołos, J. Z. Salvail, K. A. G. Fisher, D. R. partite entanglement of 60 modes of a quantum optical frequency comb,” Hamel, A. Cabello, K. J. Resch, and T. Jennewein, “Certifying the presence of Phys. Rev. Lett. 112, 120505 (2014). a photonic qubit by splitting it in two,” Phys. Rev. Lett. 116, 070501 (2016). 361D. Drahi, D. V. Sychev, K. K. Pirov, E. A. Sazhina, V. A. Novikov, I. A. 354A. Bolt, G. Duclos-Cianci, D. Poulin, and T. M. Stace, “Foliated quantum Walmsley, and A. I. Lvovsky, “Quantum interface between single- and dual- error-correcting codes,” Phys. Rev. Lett. 117, 070501 (2016). rail optical qubits,” preprint arXiv:1905.08562 (2019).

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