Linear Optical Circuits Research Articles

Learning linear optical circuits with coherent states

We analyze the energy and training data requirements for supervised learning of an M-mode linear optical circuit by minimizing an empirical risk defined solely from the action of the circuit on coherent states. When the linear optical circuit acts non-trivially only on k < M unknown modes (i.e. a linear optical k-junta), we provide an energy-efficient, adaptive algorithm that identifies the junta set and learns the circuit. We compare two schemes for allocating a total energy, E, to the learning algorithm. In the first scheme, each of the T random training coherent states has energy E/T . In the second scheme, a single random MT-mode coherent state with energy E is partitioned into T training coherent states. The latter scheme exhibits a polynomial advantage in training data size sufficient for convergence of the empirical risk to the full risk due to concentration of measure on the (2MT−1) -sphere. Specifically, generalization bounds for both schemes are proven, which indicate that for ε-approximation of the full risk by the empirical risk with high probability, O(E2/3M2/3/ϵ2/3) training states are sufficient for the first scheme and O(E1/3M1/3/ϵ2/3) training states are sufficient for the second scheme.

Avoiding lateral mode leakage in thin film lithium niobate waveguides for the generation of spectrally pure photons at telecom wavelengths

Photonic integrated optical components, notably straight waveguides, serve as pivotal elements for on-chip generation and manipulation of quantum states of light. In this work, we focus on optimizing waveguides based on lithium niobate on insulator (LNOI) to generate photon pairs at telecom wavelengths using spontaneous parametric down-conversion (SPDC). Specifically, we investigate lateral leakage for all possible SPDC processes involving type 0, type I, and type II phase matching conditions in an X-cut lithium niobate waveguide and provide a recipe to avoid leakage loss for the interacting photons. Furthermore, focusing on type II phase matching, we engineer the waveguide in the single-mode regime such that it also satisfies group index matching for generating spectrally pure single photons with high purity (99.33%). We also address fabrication imperfections of the optimized design and find that the spectral purity of the generated photons is robust to fabrication errors. This work offers guidance for the suitable selection of morphological parameters to obtain lossless, single-mode LNOI waveguides for building linear optical circuits and photon pair generation at telecom wavelengths using desired phase-matching conditions.

Approximating outcome probabilities of linear optical circuits

Quasiprobability representations are important tools for analyzing a quantum system, such as a quantum state or a quantum circuit. In this work, we propose classical algorithms specialized for approximating outcome probabilities of a linear optical circuit using quasiprobability distributions. Notably, we can reduce the negativity bound of a circuit from exponential to at most polynomial for specific cases by modulating the shapes of quasiprobability distributions thanks to the symmetry of the linear optical transformation in the phase space. Consequently, our scheme provides an efficient estimation of outcome probabilities within an additive-error whose precision depends on the classicality of the input state. When the classicality is high enough, we reach a polynomial-time estimation algorithm of a probability within a multiplicative-error by an efficient sampling from a log-concave function. By choosing appropriate input states and measurements, our results provide plenty of quantum-inspired classical algorithms for approximating various matrix functions beating best-known results. Moreover, we give sufficient conditions for the classical simulability of Gaussian Boson sampling using our approximating algorithm for any (marginal) outcome probability under the poly-sparse condition.

Generalized Mach Zehnder Interferometers Integrated on Si3N4 Waveguide Platform

We demonstrate 3 × 3 and 4 × 4 Generalized Mach Zehnder Interferometers (GMZI) integrated on the Si <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> waveguide platform and present their experimental performance as switching and variable power splitting modules. The GMZI structures have been theoretically modelled by means of a transfer matrix model, evaluating also the GMZI spectral response in the case of fabrication errors that may lead to variations in GMZI arm lengths, MMI phase shifts and MMI splitting ratios. Both 3 × 3 and 4 × 4 GMZI structures were fabricated as integrated Si <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -based circuits and were experimentally characterized with the experimental results being in close agreement with the theoretical predictions. They both reveal a resonant spectral behavior with an FSR of 1.9 nm and low average insertion losses per port that were measured to be 1 ± 0.2 and 2.4 ± 0.4 dB. The two GMZI structures were utilized as 1:3 and 1:4 switching elements, respectively, with average crosstalk levels being always greater than 10 dB for all possible input-output port combinations. Finally, their credentials to serve within linear optical circuit implementations were evaluated through their experimental characterization in variable power splitting applications, showing that the power level at a single output port can take any value between 0 and 100% of the input signal.

A Quadratic Speedup in the Optimization of Noisy Quantum Optical Circuits

Linear optical quantum circuits with photon number resolving (PNR) detectors are used for both Gaussian Boson Sampling (GBS) and for the preparation of non-Gaussian states such as Gottesman-Kitaev-Preskill (GKP), cat and NOON states. They are crucial in many schemes of quantum computing and quantum metrology. Classically optimizing circuits with PNR detectors is challenging due to their exponentially large Hilbert space, and quadratically more challenging in the presence of decoherence as state vectors are replaced by density matrices. To tackle this problem, we introduce a family of algorithms that calculate detection probabilities, conditional states (as well as their gradients with respect to circuit parametrizations) with a complexity that is comparable to the noiseless case. As a consequence we can simulate and optimize circuits with twice the number of modes as we could before, using the same resources. More precisely, for an M-mode noisy circuit with detected modes D and undetected modes U, the complexity of our algorithm is O(M2&amp;#x220F;i&amp;#x2208;UCi2&amp;#x220F;i&amp;#x2208;DCi), rather than O(M2&amp;#x220F;i&amp;#x2208;D&amp;#x222A;UCi2), where Ci is the Fock cutoff of mode i. As a particular case, our approach offers a full quadratic speedup for calculating detection probabilities, as in that case all modes are detected. Finally, these algorithms are implemented and ready to use in the open-source photonic optimization library MrMustard.

Photonic interface between subcarrier wave and dual-rail encodings

Quantum key distribution with multimode subcarrier wave encoding is propitious for being robust against environmental disturbance. For application in long-distance quantum communications, this encoding has to be compatible with entanglement-assisted quantum repeaters that are commonly designed to work with dual-rail encodings. We propose and demonstrate an interface between subcarrier wave and dual-rail encodings with a fidelity of $0.95\ifmmode\pm\else\textpm\fi{}0.03$ using a linear optical circuit. The developed scheme opens the way for future heterogeneous quantum networks that combine subcarrier wave quantum key distribution with trusted and nontrusted nodes.

Large Baseline Optical Imaging Assisted by Single Photons and Linear Quantum Optics.

In this Letter, we show that by combining quantum metrology and networking tools, it is possible to extend the baseline of an interferometric optical telescope and thus improve diffraction-limited imaging of point source positions. The quantum interferometer is based on single-photon sources, linear optical circuits, and efficient photon number counters. Surprisingly, with thermal (stellar) sources of low photon number per mode and high transmission losses across the baseline, the detected photon probability distribution still retains a large amount of Fisher information about the source position, allowing for a significant improvement in the resolution of positioning point sources, on the order of 10 μas. Our proposal can be implemented with current technology. In particular, our proposal does not require experimental optical quantum memories.

Very-large-scale integrated quantum graph photonics

Graphs have provided an expressive mathematical tool to model quantum-mechanical devices and systems. In particular, it has been recently discovered that graph theory can be used to describe and design quantum components, devices, setups and systems, based on the two-dimensional lattice of parametric nonlinear optical crystals and linear optical circuits, different to the standard quantum photonic framework. Realizing such graph-theoretical quantum photonic hardware, however, remains extremely challenging experimentally using conventional technologies. Here we demonstrate a graph-theoretical programmable quantum photonic device in very-large-scale integrated nanophotonic circuits. The device monolithically integrates about 2,500 components, constructing a synthetic lattice of nonlinear photon-pair waveguide sources and linear optical waveguide circuits, and it is fabricated on an eight-inch silicon-on-insulator wafer by complementary metal–oxide–semiconductor processes. We reconfigure the quantum device to realize and process complex-weighted graphs with different topologies and to implement different tasks associated with the perfect matching property of graphs. As two non-trivial examples, we show the generation of genuine multipartite multidimensional quantum entanglement with different entanglement structures, and the measurement of probability distributions proportional to the modulus-squared hafnian (permanent) of the graph’s adjacency matrices. This work realizes a prototype of graph-theoretical quantum photonic devices manufactured by very-large-scale integration technologies, featuring arbitrary programmability, high architectural modularity and massive manufacturing scalability.

High-dimensional spatial mode sorting and optical circuit design using multi-plane light conversion

Multi-plane light converters (MPLCs) are an emerging class of optical devices capable of converting a set of input spatial light modes to a new target set of output modes. This operation represents a linear optical transformation—a much sought after capability in photonics. MPLCs have potential applications in both the classical and quantum optics domains, in fields ranging from optical communications to optical computing and imaging. They consist of a series of diffractive optical elements (the “planes”), typically separated by a free space. The phase delays imparted by each plane are determined by the process of inverse-design, most often using an adjoint algorithm known as the wavefront matching method (WMM), which optimizes the correlation between the target and actual MPLC outputs. In this work, we investigate high mode capacity MPLCs to create arbitrary spatial mode sorters and linear optical circuits. We focus on designs possessing low numbers of phase planes to render these MPLCs experimentally feasible. To best control light in this scenario, we develop a new inverse-design algorithm, based on gradient ascent with a specifically tailored objective function, and show how, in the low-plane limit, it converges to MPLC designs with a substantially lower modal cross-talk and higher fidelity than those achievable using the WMM. We experimentally demonstrate several prototype few-plane high-dimensional spatial mode sorters, operating on up to 55 modes, capable of sorting photons based on their Zernike mode or orbital angular momentum state, or an arbitrarily randomized spatial mode basis. We discuss the advantages and drawbacks of these proof-of-principle prototypes and describe future improvements. Our work points to a bright future for high-dimensional MPLC-based technologies.

Spin-orbit maximally discordant mixed states

We introduce a proposal to prepare spin-obit maximally discordant mixed states by a linear optical circuit, with quantum bits (qubits) encoded in the polarization and transverse mode degrees of freedom of photons. In particular, we discuss how to prepare non-balanced spin-orbit entangled states, applying this technique to obtain maximally discordant mixed states. We present a simulation of the optical circuit by using the Jones matrix formalism. We performed a study of entanglement, classical and quantum correlations. The results show excellent agreement with the underlying theory and open a new experimental approach for addressing quantum correlations in optical setups.

Strong simulation of tracking single photons with which-way-detectors in linear optics

Which-way-detectors (WWDs) are path-entangled detectors characterizing mutual exclusivity between path information and interference visibility in wave-particle duality experiments. We show surprisingly that WWDs allow to utilize single photons distinguishable in time domain to realize linear optical circuits where tracking their paths is exponentially hard for strong simulation analogous to rectangular lattice based Ising models. Distinguishable photons have scalability advantages of generation and detection compared with indistinguishable photons by promising both theoretical and experimental improvements in linear optical computing including boson sampling. We calculate strong simulation complexities by using variable elimination (VE) method for undirected graphs related to tensor network contraction for quantum circuits and recursive Feynman path-integral (RFPI) method to reduce space complexity. Two designs include either a single photon touring m times or m single photons propagating sequentially through an optical circuit composed of n beam splitters and phase shifters entangled with n WWDs. VE method for tracking results in undirected graphs matching with (2 m − 1) × (n + 1) and m × (n + 1) lattice Ising models with computational complexities of and in time and and in space for single and multi-photon based designs, respectively. We exploit RFPI method for m ≫ n to reduce space complexities to polynomial levels with respect to n and log m. Probability amplitude of specific cases of multi-photon design is represented in terms of Ising partition function with purely imaginary weights to characterize sampling complexity. Open issues about sampling complexity and experimental implementation of multi-WWD set-ups are discussed.

Time-domain Hong-Ou-Mandel interference of quasi-thermal fields and its application in linear optical circuit characterization.

We study temporal correlations of interfering quasi-thermal fields, obtained by scattering laser radiation on a rotating ground glass disk. We show that the Doppler effect causes oscillations in the temporal cross correlation function. Furthermore, we propose how to use Hong-Ou-Mandel interference of quasi-thermal fields in the time domain to characterize linear optical circuits.

WDM equipped universal linear optics for programmable neuromorphic photonic processors

Non-von-Neumann computing architectures and deep learning training models have sparked a new computational era where neurons are forming the main architectural backbone and vector, matrix and tensor multiplications comprise the basic mathematical toolbox. This paradigm shift has triggered a new race among hardware technology candidates; within this frame, the field of neuromorphic photonics promises to convolve the targeted algebraic portfolio along a computational circuitry with unique speed, parallelization, and energy efficiency advantages. Fueled by the inherent energy efficient analog matrix multiply operations of optics, the staggering advances of photonic integration and the enhanced multiplexing degrees offered by light, neuromorphic photonics has stamped the resurgence of optical computing brining a unique perspective in low-energy and ultra-fast linear algebra functions. However, the field of neuromorphic photonics has relied so far on two basic architectural schemes, i.e., coherent linear optical circuits and incoherent WDM approaches, where wavelengths have still not been exploited as a new mathematical dimension. In this paper, we present a radically new approach for promoting the synergy of WDM with universal linear optics and demonstrate a new, high-fidelity crossbar-based neuromorphic photonic platform, able to support matmul with multidimensional operands. Going a step further, we introduce the concept of programmable input and weight banks, supporting in situ reconfigurability, forming in this way the first WDM-equipped universal linear optical operator and demonstrating different operational modes like matrix-by-matrix and vector-by-tensor multiplication. The benefits of our platform are highlighted in a fully convolutional neural network layout that is responsible for parity identification in the MNIST handwritten digit dataset, with physical layer simulations revealing an accuracy of ∼94%, degraded by only 2% compared to respective results obtained when executed entirely by software. Finally, our in-depth analysis provides the guidelines for neuromorphic photonic processor performance improvement, revealing along the way that 4 bit quantization is sufficient for inputs, whereas the weights can be implemented with as low as 2 bits of precision, offering substantial benefits in terms of driving circuitry complexity and energy savings.

Classical Simulation of Boson Sampling Based on Graph Structure.

Boson sampling is a fundamentally and practically important task that can be used to demonstrate quantum supremacy using noisy intermediate-scale quantum devices. In this Letter, we present classical sampling algorithms for single-photon and Gaussian input states that take advantage of a graph structure of a linear-optical circuit. The algorithms' complexity grows as so-called treewidth, which is closely related to the connectivity of a given linear-optical circuit. Using the algorithms, we study approximated simulations for local Haar-random linear-optical circuits. For equally spaced initial sources, we show that, when the circuit depth is less than the quadratic in the lattice spacing, the efficient simulation is possible with an exponentially small error. Notably, right after this depth, photons start to interfere each other and the algorithms' complexity becomes subexponential in the number of sources, implying that there is a sharp transition of its complexity. Finally, when a circuit is sufficiently deep enough for photons to typically propagate to all modes, the complexity becomes exponential as generic sampling algorithms. We numerically implement a likelihood test with a recent Gaussian boson sampling experiment and show that the treewidth-based algorithm with a limited treewidth renders a larger likelihood than the experimental data.

Universal Linear Optics for Ultra-Fast Neuromorphic Silicon Photonics Towards fJ/MAC and TMAC/sec/mm2 Engines

The field of neuromorphic photonics has been projected to comprise the next-generation Neural Network platform, expected to lead to remarkable advances in compute energy- and area-efficiency metrics. Herein, we review the performance of state-of-the-art neuromorphic photonic demonstrators, summarizing the impact of the circuit architecture and employed weight technology on the system credentials in terms of scalability, energy- and footprint-efficiency. We provide an overview of a recently demonstrated photonic crossbar multi-port interferometer, holding significant insertion loss, technology versatility and robustness advantages over all state-of-the-art linear optical layouts. This novel linear optical circuit architecture is then transferred onto an integrated silicon photonic (SiPho) platform, realizing a single-column crossbar and selecting SiGe electro-absorption modulators (EAM) for both its fan in and weighting stages. This single-neuron coherent SiPho prototype is then experimentally benchmarked on the MNIST dataset, allowing for record-high compute-rates up to 50 GHz/axon. The classification accuracy was studied for compute rates ranging between 16-50 GHz, revealing 99.03% accuracy at 16 GHz that reduces by only 3.79% at 50 GHz. Finally, we investigate the interdependence of compute rate, bit resolution and energy efficiency, in photonic accelerator layouts and benchmark our proposed Xbar architecture against state-of-the-art photonic and electronic accelerators. The analysis reveals an energy- and footprint-efficiency of 54 fJ/MAC and 1.54 TMAC/s/mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , respectively, shaping in this way a promising roadmap for next-generation neuromorphic accelerators.

Scheme for Universal High-Dimensional Quantum Computation with Linear Optics.

Photons are natural carriers of high-dimensional quantum information, and, in principle, can benefit from higher quantum information capacity and noise resilience. However, schemes to generate the resources required for high-dimensional quantum computing have so far been lacking in linear optics. Here, we show how to generate GHZ states in arbitrary dimensions and numbers of photons using linear optical circuits described by Fourier transform matrices. Combining our results with recent schemes for qudit Bell measurements, we show that universal linear optical quantum computing can be performed in arbitrary dimensions.

Linear optical circuits characterization by means of thermal field correlation measurement

We present a novel technique for linear optical circuits characterization, based on thermal field correlation measurements. Unlike the other known methods for multichannel interferometers measurements, proposed technique is robust to the input and output phase fluctuations and does not require any single-photon sources and detectors. This method has been tested in a numerical experiment and has proven to be effective even in the presence of noise.

Efficient Trainability of Linear Optical Modules in Quantum Optical Neural Networks

The existence of “barren plateau landscapes” for generic discrete-variable quantum neural networks, which obstructs efficient gradient-based optimization of cost functions defined by global measurements, would be surprising in the case of generic linear optical modules in quantum optical neural networks due to the tunability of the intensity of continuous variable states and the relevant unitary group having exponentially smaller dimension. We demonstrate that coherent light in m modes can be generically compiled efficiently if the total intensity scales sublinearly with m, and extend this result to cost functions based on homodyne, heterodyne, or photon detection measurement statistics, and to noisy cost functions in the presence of attenuation. We further demonstrate efficient trainability of m mode linear optical quantum circuits for variational mean field energy estimation of positive quadratic Hamiltonians for input states that do not have energy exponentially vanishing with m.

Hierarchical Quantum Network using Hybrid Entanglement

The advent of a new kind of entangled state known as hybrid entangled state, i.e., entanglement between different degrees of freedom, makes it possible to perform various quantum computational and communication tasks with lesser amount of resources. Here, we aim to exploit the advantage of these entangled states in communication over quantum networks. Unfortunately, the entanglement shared over the network deteriorates due to its unavoidable interaction with surroundings. Thus, an entanglement concentration protocol is proposed to obtain a maximally entangled hybrid Omega-type state from the corresponding non-maximally entangled states. The advantage of the proposed entanglement concentration protocol is that it is feasible to implement this protocol with linear optical components and present technology. The corresponding linear optical quantum circuit is provided for experimental realizations, while the success probability of the concentration protocol is also reported. Thereafter, we propose an application of maximally entangled hybrid state in the hierarchical quantum teleportation network by performing information splitting using Omega-type state, which is also the first hierarchical quantum communication scheme in the hybrid domain so far. The present hybrid entangled state has advantage in circumventing Pauli operations on the coherent state by polarization rotation of single qubit, which can be performed with lesser errors.

Enhancement of amplitude-squared squeezing of light with the SU(3) multiport beam splitters

We study the effect of a tritter, a linear optical circuit of three beam splitters following SU(3) unitary group transformations having three inputs and three output ports, on the ‘Amplitude-Squared Squeezing (ASS)’. We study the variation of ASS with different parameters of the tritter and find that there are a number of conditions where ASS is enhanced at the output of the tritter when we inject light beam exhibiting ASS at one of the input port and light beams in coherent state at another two input ports of the tritter.