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Quantum Optical Neural Networks Quantum optical neural networks Steinbrecher, Gregory R.; Olson, Jonathan P.; Englund, Dirk; Carolan, Jacques Published in: npj Quantum Information DOI: 10.1038/s41534-019-0174-7 Publication date: 2019 Document version Publisher's PDF, also known as Version of record Document license: CC BY Citation for published version (APA): Steinbrecher, G. R., Olson, J. P., Englund, D., & Carolan, J. (2019). Quantum optical neural networks. npj Quantum Information, 5, [60]. https://doi.org/10.1038/s41534-019-0174-7 Download date: 30. sep.. 2021 www.nature.com/npjqi ARTICLE OPEN Quantum optical neural networks Gregory R. Steinbrecher1, Jonathan P. Olson2, Dirk Englund 1 and Jacques Carolan1 Physically motivated quantum algorithms for specific near-term quantum hardware will likely be the next frontier in quantum information science. Here, we show how many of the features of neural networks for machine learning can naturally be mapped into the quantum optical domain by introducing the quantum optical neural network (QONN). Through numerical simulation and analysis we train the QONN to perform a range of quantum information processing tasks, including newly developed protocols for quantum optical state compression, reinforcement learning, black-box quantum simulation, and one-way quantum repeaters. We consistently demonstrate that our system can generalize from only a small set of training data onto inputs for which it has not been trained. Our results indicate that QONNs are a powerful design tool for quantum optical systems and, leveraging advances in integrated quantum photonics, a promising architecture for next-generation quantum processors. npj Quantum Information (2019) 5:60 ; https://doi.org/10.1038/s41534-019-0174-7 INTRODUCTION distributions that, under plausible complexity theoretic conjec- 24 Deep learning is revolutionizing computing1 for an ever-increasing tures, cannot be sampled from classically. For example, range of applications, from natural language processing2 to ensembles of non-interacting photons (which is a subclass of particle physics3 to cancer diagnosis.4 These advances have been the architecture presented here) sample from non-classical made possible by a combination of algorithmic design5 and distributions even without the optical nonlinearities required for 25,26 dedicated hardware development.6 Quantum computing,7 while quantum universality. Speculatively, this may enable quantum more nascent, is experiencing a similar trajectory, with a rapidly networks, in certain instances, to surpass classical networks in closing gap between current hardware and the scale required for both generative and recognition tasks. practical implementation of quantum algorithms. Error rates on Second, classical machine learning algorithms typically involve individual quantum bits (qubits) have steadily decreased,8 and the many linear algebraic operations. Existing quantum algorithms number and connectivity of qubits have improved,9 making the have already demonstrated theoretical speedups in problems so-called noisy intermediate-scale quantum (NISQ) processors10 related to many of the most elementary algebraic operations such 27 28 capable of tasks too hard for a classical computer a near-term as Fourier transforms, vector inner products, matrix eigenva- 29 30 prospect. Experimental progress has been met with algorithmic lues and eigenvectors, and linear system solving. These advances11 and near-term quantum algorithms have been techniques may form parts of a toolkit enabling quantum machine developed to tackle problems in combinatorics,12 quantum learning algorithms. Finally, certain physical systems, such as those chemistry,13 and solid-state physics.14 However, it is only recently studied in quantum chemistry, are naturally encoded by quantum that the potential for quantum processors to accelerate machine information.31 Quantum features of these states, such as learning has been explored.15 coherence and entanglement, are naturally exploitable by net- Quantum machine learning algorithms for universal quantum works that themselves are inherently quantum. Classical compu- computers have been proposed16 and small-scale demonstrations ters on the other hand require an exponential (in, for instance, the implemented.17 Relaxing the requirement of universality, quan- number of spin orbitals of a molecule) amount of memory to even tum machine learning for NISQ processors has emerged as a encode such states. rapidly advancing field18–22 that may provide a plausible route In this work, we introduce an architecture that unites the towards practical quantum-enhanced machine learning systems. complexity of quantum optical systems with the versatility of These protocols typically map features of machine learning neural networks: the quantum optical neural network (QONN). We algorithms (such as hidden layers in a neural network) directly argue that many of the features that are natural to quantum optics onto a shallow quantum circuits in a platform-independent (mode mixing, optical nonlinearity) can directly be mapped to manner. In contrast, the work presented here leverages features neural networks, and train our system to implement both unique to a particular physical platform. coherent quantum operations and classical learning tasks, Although the demonstration of an unambiguous quantum suggesting that our architecture is capable of much of the advantage in machine learning is an open question,23 an functionality of both its parent platforms. Moreover, technological increasing number of results and heuristic arguments indicate advances driven by trends in photonic quantum computing32 and quantum systems are well suited to addressing such computa- the microelectronics industry33 offer a plausible route towards tional tasks. First, certain classes of non-universal quantum large-scale, high-bandwidth QONNs, all within a CMOS (comple- processor have been shown to sample from probability mentary-metal-oxide-semiconductor)-compatible platform. 1Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA and 2Zapata Computing Inc., 501 Massachusetts Avenue, Cambridge, MA 02139, USA Correspondence: Jacques Carolan ([email protected]) Received: 28 March 2019 Accepted: 26 June 2019 Published in partnership with The University of New South Wales G.R. Steinbrecher et al. 2 (a) (b) A Classical Neural Network A Quantum Optical Neural Network Linear Transform Linear Transform Linear TransformLinear Transform Linear Transform Linear Transform Input Hidden Hidden Output Fock Single-Site Single-Site Photon Counting Layer Layer Layer Layer Input State Nonlinearities Nonlinearities Detectors Fig. 1 Quantum optical neural network (QONN). a An example of a classical neural network architecture. Hidden layers are rectified linear units (ReLUs) and the output neuron uses a sigmoid activation function to map the output into the range (0, 1). b An example of our quantum optical neural network (QONN) architecture. Inputs are single photon Fock states. The single-site nonlinearities are given a Kerr-type interaction applying a phase quadratic in the number of photons. Readout is given by photon-number-resolving detectors, which measure the photon number at each output mode Through numerical simulation and analysis, we apply our RESULTS architecture to a number of key quantum information science Architecture protocols. We benchmark the QONN by designing quantum As shown in Fig. 1b, input data to our QONN is encoded as optical gates where circuit decompositions are already known. th photonic Fock states ij (corresponding to i photons in the j Next, we show that our system can learn to simulate other optical mode), which for n photons in m modes is described by a quantum systems using only a limited set of input/output state n þ m À 1 -dimensional complex vector of unit magnitude. As pairs, generalizing what it learns to previously unseen inputs. We m demonstrate this learning on both Ising and Bose–Hubbard we will show, leveraging the full Fock space may be advantageous 1234567890():,; Hamiltonians. We then introduce and test a new quantum optical for training certain classes of QONN. The linear circuit is described ~ autoencoder protocol for compression of quantum information, by an m-mode linear optical unitary UðθÞ parameterized by a with applications in quantum communications and quantum ~ vector θ of m(m − 1) phases shifts θi 2 ð0; 2π via the encoding of networks. This again relies on the ability to train our systems using Reck et al.37 The nonlinear layer ∑ comprises single-mode Kerr a subset of possible inputs. Next, we apply our system to a interactions in the monochromatic approximation, applying a classical machine learning controls task, balancing an inverted phase that is quadratic in the number of photons present.38 For a pendulum, by a reinforcement learning approach. Finally, we train given interactionP strength ϕ, this unitary can be expressed as the QONN to implement a one-way quantum repeater, whose Σ ϕ 1 inðnÀ1Þϕ=2 ðÞ¼ n¼0 e jnihnj. The full system comprising N layers physical implementation was, until now, unknown. Our results is therefore may find application both as an important technique for YN designing next-generation quantum optical systems and as a ~ ~ SðΘÞ¼ ΣðϕÞ:UðθiÞ; (1) versatile experimental platform for near-term optical quantum i information processing and machine learning. Moreover, machine ~Θ − learning protocols for NISQ processors typically operate on where is a Nm(m 1)-dimensional vector
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