## Abstract

Advances in photonic integrated circuits have recently enabled electrically reconfigurable optical systems that can implement universal linear optics transformations on spatial mode sets. This review paper covers progress in such “programmable nanophotonic processors” as well as emerging applications of the technology to problems including classical and quantum information processing and machine learning.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Photonic integrated circuits (PICs) have become increasingly important in classical communications applications over the past decades, including as transmitters and receivers in long-haul, metro, and data center interconnects. Many of the attributes that make PICs attractive for these applications—compactness, high bandwidth, and the ability to control large numbers of optical modes with high phase stability—also make them appealing for entirely new applications, such as hardware accelerators based on emerging classical and quantum computing concepts. However, these emerging applications come with highly demanding device and scaling requirements. For example, proposed optical matrix processors will likely require the control of at least hundreds of spatial modes to be useful as neural network hardware accelerators [1–3], optical quantum computing protocols may require similar numbers of optical modes for each logical quantum bit (qubit) [4–6], and quantum computing schemes based on atomic memories will also require high-performance control over large numbers of optical spatial modes [7–9]. In addition, many of these emerging applications will require new types of devices, such as extremely low-loss modulators, and may need to function at wavelengths outside the standard telecommunications band. Whereas these challenges may have appeared daunting a decade or two ago, rapid advances in PICs have recently enabled proof-of-concept demonstrations. Silicon-on-insulator (SOI), silicon nitride, and indium phosphide (InP) technology has, in many areas, led the way thanks in large part to the availability of mature fabrication processes and multi-project-wafer (MPW) services [10–12]. Recently, SOI PIC systems that can coherently control tens of optical modes have been demonstrated [1]. Crucially, it was shown that even though all individual photonic components are imperfect, nearly perfect mode transformations become possible in sufficiently large reconfigurable optical devices [13,14]—indicating that scaling optical systems while mitigating errors is feasible. Among reconfigurable optical systems, there has been much progress towards “universal linear optics” devices: photonic circuits that can be programmed to perform all possible linear optical transformations on a given set of input modes [15,16]. This paper will review progress towards such general-purpose “programmable nanophotonic processors” (PNPs) and emerging applications to problems including machine learning and quantum information processing. The PNPs considered here implement linear optical transformations by one-way propagation; we assume no resonators or other feedback loops, which are important for a number of applications, including RF filtering [17–20].

## 2. PROGRAMMABLE NANOPHOTONIC PROCESSORS

The most popular methods for constructing a programmable mode transformer from $N$ input to $N$ output modes break the problem up into a mesh of $2\times 2$ mode transformers consisting of Mach–Zehnder interferometers (MZIs) [13,15,21], as shown in Figs. 1(a) and 1(b). Each MZI consists of two 50% beam splitters and two phase shifters parameterized by $(\theta ,\varphi )$, as shown to the right of Fig. 1(b). In integrated photonics platforms, beam splitters are commonly realized by directional couplers that convert input modes ${a}_{1},{a}_{2}$ into output modes ${b}_{1,2}=\frac{1}{\sqrt{2}}({a}_{1,2}+i{a}_{2,1})$; note the $\pi /2$ phase in the cross terms guarantees unitarity of the directional coupler transformation. The MZI shown in the inset of Fig. 1(b) applies the $\mathrm{SU}(2)$ transformation,

For a universal unitary transformation, each of the
$N$ input modes must be coupled to each of the
$N$ output modes. Figure 1(a) shows an arrangement of MZIs
connecting $N=6$ modes. To allow connections between all
modes, one requires ${\mathrm{\Sigma}}_{n}=N(N-1)/2$ ($N$ choose 2) MZIs—15 MZIs for this
example. The triangular arrangement of Fig. 1(a) was first proposed by Reck
*et al.* [22]. Figure 1(b)
shows a more compact arrangement, described by Clements
*et al.* [21], that accomplishes the same $U(N)$ transformation; it also requires
15 MZIs for $N=6$ modes. Both the “Reck” and
“Clements” decomposition algorithms terminate with a matrix
that implements $U(N)$ up to a diagonal phase screen. The phase
screen can be implemented using phase shifters at each input mode, as
shown in Figs. 1(a) and
1(b). A cascaded binary tree
structure [23] that can implement
arbitrary unitary transformations has also been proposed.

The network shown in Fig. 1(c) was originally proposed by Miller as a method for realizing any linear transformation on a set of spatial modes [16]. This network uses a physical instantiation of the singular value decomposition, which is a factorization of any matrix ($M$) as $M=U\mathrm{\Sigma}{V}^{\u2020}$, where $U$ is an $m\times m$ unitary matrix; $\mathrm{\Sigma}$ is an $m\times n$ diagonal, rectangular matrix of nonnegative real numbers; and $V$ is an $n\times n$ unitary matrix. Here, two universal unitary circuits ($U,{V}^{\u2020}$) are connected by a column of single MZIs that are used as variable attenuators implementing $\mathrm{\Sigma}$. In the original implementation of the “Miller” network, each MZI is implemented using two internal phase shifters with the differential phase between the two phase shifters being one parameter and the global phase imparted by the two phase shifters as another parameter [13,16,23–25]. The “Miller” MZI configuration can be more compact than the standard configuration, since the overall unit cell length is reduced by the length of one phase shifter.

PNPs have been demonstrated in a number of material platforms, some of which are summarized in Fig. 2. The SOI platform offers an especially high index contrast of 3.4:1.5, which enables low-loss waveguide bends with radii as small as 2 μm [28]. The resulting high component densities are especially important for large PNPs, which already can have up to 88 MZIs connecting 26 optical [1] modes, as shown in Fig. 2(a), and applications are demanding much larger devices. Figure 2(b) shows a silicon photonics-based $U(4)$ PNP that was used for separating a multimode channel into individual single-mode waveguides. The $U(6)$ PNP was realized in germanium-doped glass with thermal modulators, illustrated in Fig. 2(c), and enabled the demonstration of linear optical quantum gates and boson sampling schemes [15]. Figure 2(d) shows a silicon photonics-based $U(4)$ PNP used to demonstrate a universal coupler [26].

Phase shifter technology in MZIs is of central importance, and a number of phase shifter technologies are being advanced. Lossless phase shifting mechanisms in silicon include the thermo-optic effect (3 dB bandwidth up to a few hundred kilohertz) [29], mechanical effects ($\sim \mathrm{MHz}$ bandwidth) [30,31], and electric-field-induced electro-optic effects ($\sim \mathrm{GHz}$ bandwidth) [32]. Recent work [33] has investigated the integration of III-V materials with silicon photonics for compact, low-power phase shifting based on metal-oxide semiconductor capacitors. The possibility of monolithically integrated silicon transistor control circuits [34] and photonic components bolsters the case for large-scale PNPs in silicon. Phase modulation mechanisms that introduce dynamic loss, such as the plasma dispersion effect, are not ideal for realizing PNPs since they complicate the description of the MZI unit cell and give rise to nonunitary transformations. A number of avenues exist to further increase component density. One example is to shrink the directional couplers. Inverse design methods are particularly promising for producing wavelength-scale devices [35,36].

## 3. PNP PROGRAMMING

Configuring or programming $N\times N$ mode transformations in a PNP involves precise tuning of approximately ${N}^{2}$ phases. This can be a nontrivial problem, especially when considering MZI inhomogeneity and the potential for cross talk between modulators (especially relevant for thermal modulators). MZI phases are set by applying voltages or currents to each phase shifter, labeled here as $(i,j)$ within the array. Figure 3(c) outlines the basic programming flow. Before considering possible routes towards programming an entire PNP, it is instructive to consider the behavior of a single, programmable MZI. Some single-MZI programming examples are shown in Table 1; here, we assume the differential phase between the two input modes to an MZI can be controlled and is described by some phase $\gamma $. Without an external phase shifter ($\varphi $), transformations are confined to the plane shown in Fig. 3(b). To access the full Poincaré sphere, an external phase shifter is required.

A number of programming protocols have been developed, and they can broadly be grouped into one of three categories: (1) element-by-element, with phase shifter settings for each MZI considered individually; (2) compiled, with phase shifter settings for each MZI resulting from a matrix decomposition algorithm [16,21,22]; or (3) optimized, with phase shifter settings for each MZI resulting from the execution of an optimization protocol acting on the phase shifters [1,16].

PNPs acting as a switching matrix are generally programmed using a category (1) protocol. PNPs implementing matrices or quantum gates [2,15] (which can be specified as unitary matrices) are generally programmed using a category (2) protocol. A matrix is provided as input to a decomposition algorithm, which then returns the phase shifter settings required to realize the matrix transformation. PNPs used as black boxes that unscramble light [26] or scatter light to implement a specific output intensity pattern [1] are programmed using a category (3) protocol where the phase shifter settings are prescribed by an optimization algorithm.

To evaluate the accuracy of a program in a PNP, it is useful to characterize the unitary transformation it implements. Fortunately, efficient techniques exist [37] that use laser light to determine the amplitude elements (${|{U}_{i,j}|}^{2}$) and interferometry to determine the phase arguments ($\mathrm{arg}({U}_{i,j})$) up to an unobservable input and output phase screen. Circuit fidelity is a metric that quantifies the “closeness” between two unitary matrices and is given by ${F}_{C}=\mathrm{Tr}({|{U}^{\u2020}{U}_{T}|}^{2})$, where $\mathrm{Tr}(\text{\hspace{0.17em}})$ is the trace operator, $U$ is the measured unitary, and ${U}_{T}$ is a target unitary.

After fabrication, the initial state of the PNP is unknown due to static phase disorder within the waveguides. This effect has been studied in the context of silicon photonics and is parameterized by the static “phase coherence length” [38]; in silicon, this parameter is typically on the order of a few millimeters. To correct for this initial phase disorder, a PNP can be calibrated. There are several known algorithms for calibration, including self-configuring protocols [23] and progressive algorithms [39]. The ability to monitor the power at each MZI in a PNP enables dynamic, local measurements of the state of the system (at the cost of electronic control complexity); contactless integrated photonic probe (CLIPP) detectors avoid excess insertion loss by detecting light via bandgap defect states [26].

## 4. APPLICATIONS

We now discuss a subset of recent PNP applications: self-configuration and mode mixing, quantum transport and quantum gates, and machine learning.

#### A. Self-Configuration

As mentioned above, accurate configuration of the many degrees of freedom (phase settings) in the PNP can pose a challenge, especially when accounting for inhomogeneity in constituent devices. In 2013, Miller proposed a self-configuring solution for one particular PNP function: the coherent addition of light from $N$ spatial input modes into one spatial output mode by canceling the fields in the remaining $N-1$ output modes [16]. This concept is illustrated in Figs. 4(a) and 4(b), where the phase shifters of MZIs A–D are consecutively tuned to cancel the photocurrents on the corresponding output detectors. An important advantage in this approach is that each MZI can function without global knowledge of the other MZIs or photodetectors, and this independent self-configuration promises that such coherent, nearly lossless mode adders could be very fast. The coherent field adder only works if the optical modes are locally phase stable; for example, it would be impossible to add single-photon excitations (which have no fixed relative phase) over the input modes. Instead, arbitrary linear optical mode converters require an $N\times N$ mesh. Using an extension of his previous work, Miller proposed such a self-configuring $N\times N$ mesh that uses detectors on each MZI [26]. Using SOI PIC platforms, a $4\times 4$ universal PNP with power monitoring taps was demonstrated in 2016 [27] [see Fig. 2(d)]. A $4\times 4$ dynamically self-configuring mode adder was demonstrated in 2017 [26]. As shown in Fig. 4(c), the authors used a 980 nm laser to generate a dynamic input state to the $4\times 4$ mesh and used CLIPP detectors to actively track and undo mode mixing. As they scale in numbers of modes, self-configuring circuits could enable a range of applications [40], from spatial multiplexing/demultiplexing—for example in multimode fiber communications—to beam tracking and quantum circuits. The “Clements” architecture cannot be self-configured in this way, though a scheme has been proposed to allow progressive configuration of such networks [24].

#### B. Quantum Information Processing

Photons are appealing as a carrier of quantum information due to their ability to propagate with low loss over long distances, phase stability, and their amenability to control even at room temperature in PICs [41]. Perhaps the greatest challenge lies in producing controlled interactions between photonic quantum states: deterministic two-photon gates require many ancillae photons together with measurement and fast active feedforward [9,42,43], or atom-mediated interactions [44,45] translated to PIC-compatible platforms [46–49]. As both approaches require phase-stable control of large numbers of optical modes with high precision, programmable PNPs are emerging as important platforms. In contrast to custom-built static PICs, PNPs also provide a platform for rapid prototyping of photonic quantum information processing protocols, including quantum computing protocols [15], quantum transport [1], and quantum simulation [6,50]. In the following, we briefly discuss some of these demonstrations.

### 1. Quantum Transport

A number of interesting problems, from coherent effects in biological processes [51] to quantum computing [52,53] and quantum search [54], involve the transport of quantum particles along chains of coupled quantum systems. One experimental approach relies on a photonic quantum walk along discrete lattice sites, which can be represented as the waveguides of the PNP. While nonlinear interactions between photons give rise to particularly rich phenomena and applications, even linear quantum walks of single or multiple photons have a number of applications [1,50,55,56] and have been proposed to be computationally hard on classical computers for large-enough problems [57].

Figure 5(a) shows the
topology of a 26-mode, fan-out PNP implemented in the silicon
photonic platform; this PNP consists of 88 programmable MZIs and
176 phase shifters and supports embeddings of universal unitary
circuits up to $U(9)$. An input state of photons enters
from the left and undergoes a quantum walk on a 1D chain as it
passes in time along the right. By programming the splitting
ratios of the sites (via the internal phase shift), it is possible
to explore discrete-site quantum transport on a number of graphs.
In a recent experiment, Harris *et al.*
[1] explored a single
photonic quantum walker under static and dynamic phase disorder.
Each of the MZIs were set to implement 50:50 splitting ratios, but
the external phase shifters were programmed to have either a
static phase variation [illustrated in Fig. 5(c)], a dynamically changing
phase [illustrated in Fig. 5(d)], or any combination of static and dynamic phase
variations. In this configuration, the PNP implements a balanced
coin quantum walk on a discrete-time, 1D graph. A sufficiently
large static-only phase variation can confine photons to a local
vicinity (as in Anderson localization), whereas a strong dynamic
phase variation causes a ballistic diffusion in time (due to
dephasing between the sites). An optimal trade-off between static
and dynamic disorder (which rises with effective system
temperature) had been predicted to facilitate environment-assisted
exciton transport in photosynthetic complexes [51]. In this regime, dynamic
disorder prevents a particle from becoming “stuck”
in one site. The programmability of the PNP made it possible to
carefully study this quantum transport across 64,400 unique
settings of static and dynamic disorder, and demonstrate this
environment-assisted quantum transport experimentally.

### 2. Quantum Gates

Universal quantum computers follow two predominant frameworks: the
circuit model [58], where
single qubit and multiqubit gates are performed sequentially on
qubits, and the cluster state model [59,60],
where a large entangled resource state is first created, and then
single qubit gates are performed, which encode the computation. In
linear optics photonic quantum computing, two-qubit processes are
realized probabilistically. It is therefore critical that the
successful operation of a gate be “heralded” by
ancillary photons. Carolan *et al.* [15] used a six-mode PNP alongside
an off-chip multiphoton source to implement a variety of heralded
gates in both the circuit and cluster state model.
Figures 6(a) and
6(b) show the symbol and
photonic circuit for a heralded controlled-NOT (CNOT) operation,
which uses two path-encoded computational photons and two
ancillary photons. Given a detection in the ancillary modes, the
CNOT logic is guaranteed to have taken place on the computational
photons [see Fig. 6(c)]. Technologically, the low coupling loss of
0.4 dB between silica waveguides and input/output fibers
was key to enabling multiphoton experiments of up to six photons.
While SOI PNPs have so far been limited to coupling losses of
3 dB, losses as low as 0.4 dB have been demonstrated
in silicon photonics [61],
pointing the way towards large-scale SOI PNPs suitable for
multiphoton quantum information.

#### C. Machine Learning

Artificial neural network (ANN) algorithms have dramatically improved natural language processing, image recognition, object detection, and more [62]. ANNs rely heavily on matrix-vector products and require frequent memory access during training and inference. Recent work has focused on developing tailored electronics architecture for ANNs that take advantage of the limited requirements on computational precision, large matrix sparsity, and other features to achieve improved computational rates and energy efficiency [63–68]. However, the computational speed and power efficiency achieved with these hardware architectures are still bound by underlying transistor device physics, including switching energies and electronic clock rates—two quantities that are closely linked.

Some machine learning algorithms, including neural networks, appear suited for analog computing architectures, including analog complementary metal-oxide semiconductor (CMOS) circuits [69], memristor arrays [70,71], photonic networks [2], and magnetic devices [72]. Photonic methods may simultaneously enable low latency, high energy efficiency, and high throughput [2]. While bulk-optical implementations of optical neural networks (ONNs) have been suggested in the past [73], it has only recently become possible to implement large-scale, phase-stable, and programmable linear transformations. Recent work has focused on implementing hybrid optical-electronic systems that implement spike processing [74] and reservoir computing [75–77]. Augmented with optical nonlinearities, PNPs promise high-speed and low-power implementations of neural networks fully in the optical domain.

As shown by Shen and Harris *et al.* [2], it is possible to directly map the
mathematical description of a multilayer perceptron, the most basic
form of deep neural network, onto arrays of PNPs connected by
nonlinear optical components. In each layer of a multilayer
perceptron, a matrix-vector product is evaluated, and then each entry
of the resultant output vector is passed through a nonlinear
“activation function.” A schematic representation of an
ONN is shown in Fig. 7(a), and a zoom into a single layer is shown in
Fig. 7(b). Matrix-vector
products are evaluated using optical interference units in the
“Miller” encoding [(PNPs implementing arbitrary,
nonunitary matrices as shown in Fig. 1(c)] [16],
and activation functions are realized with an optical nonlinearity
unit (ONU). Vectors are encoded in the intensity and phase
distribution of optical signals incident at the left of the ONN. These
optical signals propagate through the set of layers comprising the ONN
and are finally converted into electrical current using detectors,
shown at the right of Fig. 7(a). An ONU could be implemented using saturable absorbers
[78,79] or devices that exhibit bistability [80–82]; both kinds of
nonlinear optical devices have been demonstrated in integrated
photonic systems, but challenges remain in realizing an array of such
nonlinear devices in a single system.

Existing neural network training algorithms, such as backpropagation [83,84], executed on electronic computers can be used to determine the set of matrices to be programmed into the ONN. After training, a set of weights in each layer that minimizes an error metric is determined. These weight matrices can be decomposed into PNP phase shifter settings at each layer. After programming, the ONN can be used as an inference machine—classifying vectors that are not part of the training data set.

This adaptation of deep neural networks to integrated photonics was tested on a simple vowel recognition problem [2]. A two-layer neural network with four neurons per layer and a saturable absorber nonlinear activation function was trained on a 64-bit computer against a set of four-dimensional input vectors that represent recordings of people speaking one of four vowels. The data set contained 360 vectors; 180 were used for training and 180 were exclusively used for testing. After training, the ONN was able to correctly classify 138/180 spoken vowels (compared to 165/180 for a 64-bit digital computer). Advances in PNP programming fidelity and improved readout (including optical fiber packaging techniques) may reduce the performance gap between the ONN and the digitally simulated one.

## 5. DISCUSSION

PNPs are already finding applications in proof-of-concept demonstrations including classical computing systems [1–3], quantum computing systems [15], self-calibrating mode mixers [26], and matrix processors [2,15,27]. For real-world applications, it is still necessary to address some important challenges, including (1) the development of more compact, low-power phase shifters with ultralow loss and—for many applications—programmability at rates of MHz and higher; (2) operation outside the near-infrared spectrum, especially at shorter wavelengths; (3) precise electronic control over tens of thousands of phase shifters; and (4) more compact ultralow loss passive components, which may be developed by computational design [35,85].

While there are many challenges towards scaling PNPs, significant progress is being made on multiple fronts. Optoelectronic systems with over 1000 active elements and the circuits that control them have been monolithically integrated in CMOS processes [86]; MEMS and NEMS switches show promise for low-power switch arrays [27,31]; and a growing range of materials are becoming available, including SOI, silicon nitride, and InP. These developments point to a new era in photonics design and applications in which high-volume manufacturing will make general purpose PNPs containing an abundance of components cost-effective over custom-designed PICs in many applications. As field programmable gate arrays (FPGAs) have enabled a new paradigm for electronics, PNPs, or “optical FPGAs,” will enable unforeseen applications and advances for optical processing.

## Funding

Air Force Office of Scientific Research (AFOSR) (FA8750-14-2-0120, FA9550-13-1-0027, FA9550-14-1-0052); Air Force Research Laboratory (AFRL) Program (FA8750-14-2-0120, FA8750-16-2-0141); Office of the Secretary of Defense (OSD); Applied Research for Advanced Science and Technology (ARAP) Quantum Science and Engineering Program (QSEP) program.

## Acknowledgment

D. E. would like to acknowledge support from AFRL. M. L. F., A. M. S., C. C. T., and P. M. A. would like to acknowledge support of this work from OSD. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of Air Force Research Laboratory.

## REFERENCES

**1. **N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport
simulations in a programmable nanophotonic
processor,” Nat. Photonics **11**, 447–452
(2017). [CrossRef]

**2. **Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with
coherent nanophotonic circuits,” Nat.
Photonics **11**,
441–446
(2017). [CrossRef]

**3. **A. N. Tait, T. F. de Lima, E. Zhou, A. X. Wu, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Neuromorphic photonic
networks using silicon photonic weight banks,”
Sci. Rep. **7**, 7430
(2017). [CrossRef]

**4. **M. Pant, H. Krovi, D. Englund, and S. Guha, “Rate-distance tradeoff
and resource costs for all-optical quantum
repeaters,” Phys. Rev. A **95**, 012304 (2017). [CrossRef]

**5. **K. Kieling, T. Rudolph, and J. Eisert, “Percolation,
renormalization, and quantum computing with nondeterministic
gates,” Phys. Rev. Lett. **99**, 130501 (2007). [CrossRef]

**6. **A. Aspuru-Guzik and P. Walther, “Photonic quantum
simulators,” Nat. Phys. **8**, 285–291
(2012). [CrossRef]

**7. **Y. Li, P. C. Humphreys, G. J. Mendoza, and S. C. Benjamin, “Resource costs for
fault-tolerant linear optical quantum
computing,” Phys. Rev. X **5**, 041007 (2015). [CrossRef]

**8. **K. Nemoto, M. Trupke, S. J. Devitt, A. M. Stephens, B. Scharfenberger, K. Buczak, T. Nöbauer, M. S. Everitt, J. Schmiedmayer, and W. J. Munro, “Photonic architecture
for scalable quantum information processing in
diamond,” Phys. Rev. X **4**, 031022 (2014). [CrossRef]

**9. **M. Pant, D. Towsley, D. Englund, and S. Guha, “Percolation thresholds
for photonic quantum computing,”
arXiv:1701.03775 (2017).

**10. **T. Baehr-Jones, R. Ding, A. Ayazi, T. Pinguet, M. Streshinsky, N. Harris, J. Li, L. He, M. Gould, Y. Zhang, A. Eu-Jin Lim, T.-Y. Liow, S. Hwee-Gee Teo, G.-Q. Lo, and M. Hochberg, “A
25 Gb/s silicon photonics
platform,” arXiv:1203.0767
(2012).

**11. **P. P. Absil, P. De Heyn, H. Chen, P. Verheyen, G. Lepage, M. Pantouvaki, J. De Coster, A. Khanna, Y. Drissi, D. Van Thourhout, and J. Van Campenhout, “Imec iSiPP25G silicon
photonics: a robust CMOS-based photonics technology
platform,” Proc. SPIE **9367**, 93670V
(2015). [CrossRef]

**12. **K. Wörhoff, R. G. Heideman, A. Leinse, and M. Hoekman, “TriPleX: a versatile
dielectric photonic platform,” Adv.
Opt. Technol. **4**,
189–207
(2015). [CrossRef]

**13. **D. A. B. Miller, “Perfect optics with
imperfect components,” Optica **2**, 747–750
(2015). [CrossRef]

**14. **J. Mower, N. C. Harris, G. R. Steinbrecher, Y. Lahini, and D. Englund, “High-fidelity quantum
state evolution in imperfect photonic integrated
circuits,” Phys. Rev. A **92**, 032322 (2015). [CrossRef]

**15. **J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear
optics,” Science **349**, 711–716
(2015). [CrossRef]

**16. **D. A. B. Miller, “Self-configuring
universal linear optical component (invited),”
Photon. Res. **1**,
1–15 (2013). [CrossRef]

**17. **D. Pérez, I. Gasulla, and J. Capmany, “Programmable
multifunctional integrated nanophotonics,”
Nanophotonics **7**,
1351–1371
(2018). [CrossRef]

**18. **D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon
photonics signal processor core,” Nat.
Commun. **8**, 636
(2017). [CrossRef]

**19. **D. Pérez, I. Gasulla, J. Capmany, and R. A. Soref, “Reconfigurable lattice
mesh designs for programmable photonic
processors,” Opt. Express **24**, 12093–12106
(2016). [CrossRef]

**20. **L. Zhuang, C. G. Roeloffzen, M. Hoekman, K.-J. Boller, and A. J. Lowery, “Programmable photonic
signal processor chip for radiofrequency
applications,” Optica **2**, 854–859
(2015). [CrossRef]

**21. **W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. Steven Kolthammer, and I. A. Walmsley, “Optimal design for
universal multiport interferometers,”
Optica **3**,
1460–1465
(2016). [CrossRef]

**22. **M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental
realization of any discrete unitary operator,”
Phys. Rev. Lett. **73**,
58–61 (1994). [CrossRef]

**23. **D. A. B. Miller, “Self-aligning universal
beam coupler,” Opt. Express **21**, 6360–6370
(2013). [CrossRef]

**24. **D. A. B. Miller, “Setting up meshes of
interferometers—reversed local light interference
method,” Opt. Express **25**, 29233–29248
(2017). [CrossRef]

**25. **D. A. Miller, “How complicated must an
optical component be?” J. Opt. Soc. Am.
A **30**,
238–251
(2013). [CrossRef]

**26. **A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. B. Miller, A. Melloni, and F. Morichetti, “Unscrambling
light—automatically undoing strong mixing between
modes,” Light Sci. Appl. **6**, e17110 (2017). [CrossRef]

**27. **A. Ribeiro, A. Ruocco, L. Vanacker, and W. Bogaerts, “Demonstration of a
4 × 4-port universal linear
circuit,” Optica **3**, 1348–1357
(2016). [CrossRef]

**28. **M. R. Watts, “Adiabatic microring
resonators,” Opt. Lett. **35**, 3231–3233
(2010). [CrossRef]

**29. **N. C. Harris, Y. Ma, J. Mower, T. Baehr-Jones, D. Englund, M. Hochberg, and C. Galland, “Efficient, compact and
low loss thermo-optic phase shifter in
silicon,” Opt. Express **22**, 10487–10493
(2014). [CrossRef]

**30. **M. Poot and H. X. Tang, “Broadband
nanoelectromechanical phase shifting of light on a
chip,” Appl. Phys. Lett. **104**, 061101
(2014). [CrossRef]

**31. **S. Han, T. J. Seok, N. Quack, B.-W. Yoo, and M. C. Wu, “Large-scale silicon
photonic switches with movable directional
couplers,” Optica **2**, 370–375
(2015). [CrossRef]

**32. **E. Timurdogan, C. V. Poulton, M. J. Byrd, and M. R. Watts, “Electric field-induced
second-order nonlinear optical effects in silicon
waveguides,” Nat. Photonics **11**, 200–206
(2017). [CrossRef]

**33. **M. Takenaka, J.-H. Han, J.-K. Park, F. Boeuf, J. Fujikata, S. Takahashi, and S. Takagi, “High-efficiency,
low-loss optical phase modulator based on III-V/Si hybrid MOS
capacitor,” in *Optical Fiber
Communication Conference* (Optical Society of
America, 2018), paper Tu3K.3.

**34. **C. Sun, M. T. Wade, Y. Lee, J. S. Orcutt, L. Alloatti, M. S. Georgas, A. S. Waterman, J. M. Shainline, R. R. Avizienis, S. Lin, B. R. Moss, R. Kumar, F. Pavanello, A. H. Atabaki, H. M. Cook, A. J. Ou, J. C. Leu, Y.-H. Chen, K. Asanović, R. J. Ram, M. Popović, and V. M. Stojanović, “Single-chip
microprocessor that communicates directly using
light,” Nature **528**, 534–538
(2015). [CrossRef]

**35. **A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, “Inverse design and
demonstration of a compact and broadband on-chip wavelength
demultiplexer,” Nat. Photonics **9**, 374–377
(2015). [CrossRef]

**36. **Y. Zhang, S. Yang, A. E.-J. Lim, G.-Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss
Y-junction for submicron silicon waveguide,”
Opt. Express **21**,
1310–1316
(2013). [CrossRef]

**37. **S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, and A. G. White, “Direct characterization
of linear-optical networks,” Opt.
Express **21**,
13450–13458
(2013). [CrossRef]

**38. **Y. Yang, Y. Ma, H. Guan, Y. Liu, S. Danziger, S. Ocheltree, K. Bergman, T. Baehr-Jones, and M. Hochberg, “Phase coherence length
in silicon photonic platform,” Opt.
Express **23**,
16890–16902
(2015). [CrossRef]

**39. **N. C. Harris, “Programmable nanophotonics for
quantum information processing and artificial
intelligence,” Ph.D. thesis
(Massachusetts Institute of
Technology, 2017).

**40. **D. A. B. Miller, “Sorting out
light,” Science **347**, 1423–1424
(2015). [CrossRef]

**41. **J. W. Silverstone, D. Bonneau, J. L. O’Brien, and M. G. Thompson, “Silicon quantum
photonics,” IEEE J. Sel. Top. Quantum
Electron. **22**,
390–402
(2016). [CrossRef]

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

**43. **T. Rudolph, “Why I am optimistic
about the silicon-photonic route to quantum
computing,” APL Photon. **2**, 030901 (2017). [CrossRef]

**44. **B. Hacker, S. Welte, G. Rempe, and S. Ritter, “A photon–photon
quantum gate based on a single atom in an optical
resonator,” Nature **536**, 193–196
(2016). [CrossRef]

**45. **D. J. Brod and J. Combes, “Passive CPHASE gate via
cross-Kerr nonlinearities,” Phys. Rev.
Lett. **117**, 080502
(2016). [CrossRef]

**46. **S. Sun, H. Kim, Z. Luo, G. S. Solomon, and E. Waks, “A single-photon switch
and transistor enabled by a solid-state quantum
memory,” Science **361**, 57–60
(2018). [CrossRef]

**47. **A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond
nanophotonics platform for quantum optical
networks,” Science **354**, 847–850
(2016). [CrossRef]

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

**49. **S. L. Mouradian, T. Schröder, C. B. Poitras, L. Li, J. Goldstein, E. H. Chen, M. Walsh, J. Cardenas, M. L. Markham, D. J. Twitchen, M. Lipson, and D. Englund, “Scalable integration of
long-lived quantum memories into a photonic
circuit,” Phys. Rev. X **5**, 031009 (2015). [CrossRef]

**50. **C. Sparrow, E. Martn-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the
vibrational quantum dynamics of molecules using
photonics,” Nature **557**, 660–667
(2018). [CrossRef]

**51. **P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd, and A. Aspuru-Guzik, “Environment-assisted
quantum transport,” New J.
Phys. **11**, 033003
(2009). [CrossRef]

**52. **A. M. Childs, D. Gosset, and Z. Webb, “Universal computation
by multiparticle quantum walk,”
Science **339**,
791–794
(2013). [CrossRef]

**53. **Y. Lahini, G. R. Steinbrecher, A. D. Bookatz, and D. Englund, “Quantum logic using
correlated one-dimensional quantum walks,”
npj Quantum Inf. **4**, 2
(2018). [CrossRef]

**54. **S. Aaronson and A. Ambainis, “Quantum search of
spatial regions,” in *44th Annual IEEE
Symposium on Foundations of Computer Science*
(2003),
pp. 200–209.

**55. **A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of
correlated photons,” Science **329**, 1500–1503
(2010). [CrossRef]

**56. **A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization
of entangled photons in an integrated quantum
walk,” Nat. Photonics **7**, 322–328
(2013). [CrossRef]

**57. **S. Aaronson and A. Arkhipov, “The computational
complexity of linear optics,” in *43rd
Annual ACM Symposium on Theory of Computing (STOC)*
(ACM, 2011),
pp. 333–342.

**58. **M. A. Nielsen and I. Chuang, “Quantum computation and
quantum information,” Am. J.
Phys. **70**,
558–559
(2002). [CrossRef]

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

**60. **R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based
quantum computation on cluster states,”
Phys. Rev. A **68**,
022312 (2003). [CrossRef]

**61. **J. Notaros, F. Pavanello, M. T. Wade, C. M. Gentry, A. Atabaki, L. Alloatti, R. J. Ram, and M. A. Popović, “Ultra-efficient CMOS
fiber-to-chip grating couplers,” in
*Optical Fiber Communications Conference and Exhibition
(OFC)* (2016),
pp. 1–3.

**62. **Y. LeCun, Y. Bengio, and G. Hinton, “Deep
learning,” Nature **521**, 436–444
(2015). [CrossRef]

**63. **C.-S. Poon and K. Zhou, “Neuromorphic silicon
neurons and large-scale neural networks: challenges and
opportunities,” Front.
Neurosci. **5**, 108
(2011). [CrossRef]

**64. **A. Shafiee, A. Nag, N. Muralimanohar, R. Balasubramonian, J. P. Strachan, M. Hu, R. S. Williams, and V. Srikumar, “ISAAC: a convolutional
neural network accelerator with in-situ analog arithmetic in
crossbars,” in *ACM/IEEE 43rd Annual
International Symposium on Computer Architecture (ISCA)*
(2016),
pp. 14–26.

**65. **J. Misra and I. Saha, “Artificial neural
networks in hardware: a survey of two decades of
progress,” Neurocomputing **74**, 239–255
(2010). [CrossRef]

**66. **D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. van den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of
Go with deep neural networks and tree search,”
Nature **529**,
484–489
(2016). [CrossRef]

**67. **Y. H. Chen, T. Krishna, J. S. Emer, and V. Sze, “Eyeriss: an
energy-efficient reconfigurable accelerator for deep convolutional
neural networks,” IEEE J. Solid-State
Circuits **52**,
127–138
(2017). [CrossRef]

**68. **A. Graves, G. Wayne, M. Reynolds, T. Harley, I. Danihelka, A. Grabska-Barwińska, S. G. Colmenarejo, E. Grefenstette, T. Ramalho, J. Agapiou, A. P. Badia, K. M. Hermann, Y. Zwols, G. Ostrovski, A. Cain, H. King, C. Summerfield, P. Blunsom, K. Kavukcuoglu, and D. Hassabis, “Hybrid computing using
a neural network with dynamic external memory,”
Nature **538**,
471–476
(2016). [CrossRef]

**69. **A. Biswas and A. P. Chandrakasan, “Conv-RAM: an
energy-efficient SRAM with embedded convolution computation for
low-power CNN-based machine learning
applications,” in *IEEE International
Solid-State Circuits Conference (ISSCC)* (2018),
pp. 488–490.

**70. **A. Pantazi, S. Woźniak, T. Tuma, and E. Eleftheriou, “All-memristive
neuromorphic computing with level-tuned
neurons,” Nanotechnology **27**, 355205 (2016). [CrossRef]

**71. **M. Hu, C. E. Graves, C. Li, Y. Li, N. Ge, E. Montgomery, N. Davila, H. Jiang, R. S. Williams, J. J. Yang, Q. Xia, and J. P. Strachan, “Memristor-based analog
computation and neural network classification with a dot product
engine,” Adv. Mater. **30**, 1705914
(2018). [CrossRef]

**72. **J. Torrejon, M. Riou, F. A. Araujo, S. Tsunegi, G. Khalsa, D. Querlioz, P. Bortolotti, V. Cros, K. Yakushiji, A. Fukushima, H. Kubota, S. Yuasa, M. D. Stiles, and J. Grollier, “Neuromorphic computing
with nanoscale spintronic oscillators,”
Nature **547**,
428–431
(2017). [CrossRef]

**73. **N. H. Farhat, D. Psaltis, A. Prata, and E. Paek, “Optical implementation
of the Hopfield model,” Appl.
Opt. **24**,
1469–1475
(1985). [CrossRef]

**74. **S. K. Esser, P. A. Merolla, J. V. Arthur, A. S. Cassidy, R. Appuswamy, A. Andreopoulos, D. J. Berg, J. L. McKinstry, T. Melano, D. R. Barch, C. di Nolfo, P. Datta, A. Amir, B. Taba, M. D. Flickner, and D. S. Modha, “Convolutional networks
for fast, energy-efficient neuromorphic
computing,” Proc. Natl. Acad. Sci.
USA **113**,
11441–11446
(2016). [CrossRef]

**75. **K. Vandoorne, P. Mechet, T. Van Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental
demonstration of reservoir computing on a silicon photonics
chip,” Nat. Commun. **5**, 3541 (2014). [CrossRef]

**76. **L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information
processing beyond Turing: an optoelectronic implementation of
reservoir computing,” Opt.
Express **20**,
3241–3249
(2012). [CrossRef]

**77. **Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic
reservoir computing,” Sci.
Rep. **2**, 287
(2012). [CrossRef]

**78. **A. C. Selden, “Pulse transmission
through a saturable absorber,” Br. J.
Appl. Phys. **18**,
743–748
(1967). [CrossRef]

**79. **Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a
saturable absorber in a mode-locked laser,”
Nano Res. **4**,
297–307
(2011). [CrossRef]

**80. **B. Xu and N. B. Ming, “Experimental
observations of bistability and instability in a two-dimensional
nonlinear optical superlattice,” Phys.
Rev. Lett. **71**,
3959–3962
(1993). [CrossRef]

**81. **E. Centeno and D. Felbacq, “Optical bistability in
finite-size nonlinear bidimensional photonic crystals doped by a
microcavity,” Phys. Rev. B **62**, R7683–R7686
(2000). [CrossRef]

**82. **K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule
all-optical switching using a photonic-crystal
nanocavity,” Nat. Photonics **4**, 477–483
(2010). [CrossRef]

**83. **D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning
representations by back-propagating errors,”
Nature **323**,
533–536
(1986). [CrossRef]

**84. **T. W. Hughes, M. Minkov, Y. Shi, and S. Fan, “Training of photonic
neural networks through in situ backpropagation and gradient
measurement,” Optica **5**, 864–871
(2018). [CrossRef]

**85. **B. Shen, P. Wang, R. Polson, and R. Menon, “An
integrated-nanophotonics polarization beamsplitter with
2.4 × 2.4 μm^{2}
footprint,” Nat. Photonics **9**, 378–382
(2015). [CrossRef]

**86. **S. Chung, H. Abediasl, and H. Hashemi, “15.4 A
1024-element scalable optical phased array in
0.18 μm SOI CMOS,” in
*IEEE International Solid-State Circuits Conference
(ISSCC)* (IEEE,
2017),
pp. 262–263.