Abstract
Physically motivated quantum algorithms for specific nearterm 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, blackbox quantum simulation, and oneway 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 nextgeneration quantum processors.
Introduction
Deep learning is revolutionizing computing^{1} for an everincreasing range of applications, from natural language processing^{2} to particle physics^{3} to cancer diagnosis.^{4} These advances have been made possible by a combination of algorithmic design^{5} and dedicated hardware development.^{6} Quantum computing,^{7} while more nascent, is experiencing a similar trajectory, with a rapidly closing gap between current hardware and the scale required for practical implementation of quantum algorithms. Error rates on individual quantum bits (qubits) have steadily decreased,^{8} and the number and connectivity of qubits have improved,^{9} making the socalled noisy intermediatescale quantum (NISQ) processors^{10} capable of tasks too hard for a classical computer a nearterm prospect. Experimental progress has been met with algorithmic advances^{11} and nearterm quantum algorithms have been developed to tackle problems in combinatorics,^{12} quantum chemistry,^{13} and solidstate physics.^{14} However, it is only recently that the potential for quantum processors to accelerate machine learning has been explored.^{15}
Quantum machine learning algorithms for universal quantum computers have been proposed^{16} and smallscale demonstrations implemented.^{17} Relaxing the requirement of universality, quantum machine learning for NISQ processors has emerged as a rapidly advancing field^{18,19,20,21,22} that may provide a plausible route towards practical quantumenhanced machine learning systems. These protocols typically map features of machine learning algorithms (such as hidden layers in a neural network) directly onto a shallow quantum circuits in a platformindependent manner. In contrast, the work presented here leverages features unique to a particular physical platform.
Although the demonstration of an unambiguous quantum advantage in machine learning is an open question,^{23} an increasing number of results and heuristic arguments indicate quantum systems are well suited to addressing such computational tasks. First, certain classes of nonuniversal quantum processor have been shown to sample from probability distributions that, under plausible complexity theoretic conjectures, cannot be sampled from classically.^{24} For example, ensembles of noninteracting photons (which is a subclass of the architecture presented here) sample from nonclassical distributions even without the optical nonlinearities required for quantum universality.^{25,26} Speculatively, this may enable quantum networks, in certain instances, to surpass classical networks in both generative and recognition tasks.
Second, classical machine learning algorithms typically involve many linear algebraic operations. Existing quantum algorithms have already demonstrated theoretical speedups in problems related to many of the most elementary algebraic operations such as Fourier transforms,^{27} vector inner products,^{28} matrix eigenvalues and eigenvectors,^{29} and linear system solving.^{30} These techniques may form parts of a toolkit enabling quantum machine learning algorithms. Finally, certain physical systems, such as those studied in quantum chemistry, are naturally encoded by quantum information.^{31} Quantum features of these states, such as coherence and entanglement, are naturally exploitable by networks that themselves are inherently quantum. Classical computers on the other hand require an exponential (in, for instance, the number of spin orbitals of a molecule) amount of memory to even encode such states.
In this work, we introduce an architecture that unites the complexity of quantum optical systems with the versatility of neural networks: the quantum optical neural network (QONN). We argue that many of the features that are natural to quantum optics (mode mixing, optical nonlinearity) can directly be mapped to neural networks, and train our system to implement both coherent quantum operations and classical learning tasks, suggesting that our architecture is capable of much of the functionality of both its parent platforms. Moreover, technological advances driven by trends in photonic quantum computing^{32} and the microelectronics industry^{33} offer a plausible route towards largescale, highbandwidth QONNs, all within a CMOS (complementarymetaloxidesemiconductor)compatible platform.
Through numerical simulation and analysis, we apply our architecture to a number of key quantum information science protocols. We benchmark the QONN by designing quantum optical gates where circuit decompositions are already known. Next, we show that our system can learn to simulate other quantum systems using only a limited set of input/output state pairs, generalizing what it learns to previously unseen inputs. We demonstrate this learning on both Ising and Bose–Hubbard Hamiltonians. We then introduce and test a new quantum optical autoencoder protocol for compression of quantum information, with applications in quantum communications and quantum networks. This again relies on the ability to train our systems using a subset of possible inputs. Next, we apply our system to a classical machine learning controls task, balancing an inverted pendulum, by a reinforcement learning approach. Finally, we train the QONN to implement a oneway quantum repeater, whose physical implementation was, until now, unknown. Our results may find application both as an important technique for designing nextgeneration quantum optical systems and as a versatile experimental platform for nearterm optical quantum information processing and machine learning. Moreover, machine learning protocols for NISQ processors typically operate on quantum states for which there are no clear classical analog. Similarly, the QONN may be able to perform inference on quantum optical states, such as those generated by molecular systems^{34} or states within a quantum network.^{35}
In prototypical neural networks (see Fig. 1a) an input vector \(\vec x \in {\Bbb R}^n\) is passed through multiple layers of: (1) linear transformation, that is, a matrix multiplication \(W(\theta _i).\vec{x}\) parameterized by weights θ_{i} at layer i, and (2) nonlinear operations \(\sigma (\vec x)\), which are singlesite nonlinear functions sometimes parameterized by biases \(\vec b_i\) (typically referred to as the perceptron or neuron, see Fig. 1a, inset for two examples: the rectifying neuron and the sigmoid neuron). The goal of the neural network is to optimize the parameter sets {θ_{i}} and {b_{i}} to realize a particular input–output function \(f(\vec x) = y\). The power of neural networks lies in the fact that when trained over a large data set \(\{ \vec x_i\}\), this often highly nonlinear functional relationship is generalizable to a large vector set to which the network was not exposed during training. For example, in the context of cancer diagnosis, the input vectors may be grayscale values of pixels of an image of a cell, and the output may be a twodimensional vector that corresponds to the binary label of the cell as either a benign or malignant.^{36} Once the network is trained, it may categorize with high probability new, unlabelled, images of cells as either “benign” or “malignant”.
Results
Architecture
As shown in Fig. 1b, input data to our QONN is encoded as photonic Fock states i_{j} (corresponding to i photons in the j^{th} optical mode), which for n photons in m modes is described by a \(\left( {\begin{array}{*{20}{c}} {n + m  1} \\ m \end{array}} \right)\)dimensional complex vector of unit magnitude. As we will show, leveraging the full Fock space may be advantageous for training certain classes of QONN. The linear circuit is described by an mmode linear optical unitary \(U(\vec \theta )\) parameterized by a vector \(\vec \theta\) of m(m − 1) phases shifts \(\theta _i \in \left( {0,2\pi } \right]\) via the encoding of Reck et al.^{37} The nonlinear layer ∑ comprises singlemode Kerr interactions in the monochromatic approximation, applying a phase that is quadratic in the number of photons present.^{38} For a given interaction strength ϕ, this unitary can be expressed as \(\Sigma \left( \phi \right) = \mathop {\sum}\nolimits_{n = 0}^\infty {e^{in(n  1)\phi /2}} n\rangle \langle n\). The full system comprising N layers is therefore
where \(\vec \Theta\) is a Nm(m − 1)dimensional vector and the strength of the nonlinearity is typically fixed as ϕ = π. Finally, photonnumberresolving detectors are used to measure the photon number at each output. We consider number resolution without loss of generality as the socalled threshold detectors (vacuum, or not) can always be made nondeterministically number resolving via beamsplitters and multiple detectors.^{39} We use the results of this measurement, along with a training set of K desired input/output pairs \(\left\{ {\left {\psi _{{\mathrm{in}}}^i} \right\rangle \to \left {\psi _{{\mathrm{out}}}^i} \right\rangle } \right\}_{i = 1}^K\), to construct a cost function
that is variationally minimized over \(\vec \Theta\) to find a target transformation (up to an unobservable global phase). In the Supplementary Information S2 we show that the QONN architecture is also capable of implementing classical optical neural networks,^{40} and may therefore benefit from advances in this rapidly growing field.^{41}
We distinguish between two approaches to training: in situ and in silico. The in situ approach directly optimizes the quantum optical processor and measurements are made via single photon detectors at the end of the circuit. One aim is to optimize figures of merit that can be estimated with a number of measurements that scales polynomially with the photon number (as opposed to full quantum process tomography).^{42} If the target state is accessible, the overlap can be estimated with the addition of a controlledSWAP operation, which is related to the Hong–Ou–Mandel effect in quantum optics.^{43} Efficient fidelity proxies provide another route towards estimating salient features of quantum states without reconstruction of the full density matrix.^{44} Moreover, the in situ approach may enable a form of error mitigation by routing quantum information around faulty hardware.^{45} In contrast, the in silico approach simulates the QONN on a digital classical computer and keeps track of the full quantum state internal to the system. Simulations will therefore be limited in scale, but may help guide the design of, say, quantum gates where the optimal decomposition is not already known, or as an ansatz for the in situ approach. In the Methods, we detail the computational techniques used in this work.
Hardware implementation
A number of the key components of QONNs are readily implementable using stateoftheart integrated quantum photonics. First, matrix multiplication can be realized across optical modes (where each mode contains a complex electric field component) via arrays of beamsplitters and programmable phase shifts.^{37,46} In the lossless case, an nmode optical circuit comprising n(n − 1) component implements an arbitrary n × n single particle unitary operation (which can also be used for classical neural networks),^{47} and a ndimensional nonunitary operation can always be embedded across a 2nmode optical circuit.^{48} Advances in integrated optics^{49} have enabled the implementation of such circuits for applications in quantum computation,^{50} quantum simulation,^{51} and classical optical neural networks.^{40} Second, optical nonlinearities are a core component of many classical^{52} and quantum^{53} optical computing architectures. Single photon coherent nonlinearities can be implemented via measurement,^{54} interaction with threelevel atoms^{55} or superconducting materials,^{56} and through alloptical phenomena such as the Kerr effect.^{57} Notably, promising progress has been made towards solidstate waveguidebased nonlinearities.^{58} Third, superconducting nanowire single photon detectors (SNSPDs) enable ultraefficient single photon readout, either via lowloss outcoupling^{59} to a dedicated highefficiency detection system^{60} or through the direct integration of SNSPDs on chip.^{61} Moreover, advances in electronic readout have made it is possible to scale SNSPDs across many channels and with photonnumber resolution.^{62} While incorporating these technologies into a single scalable system is an outstanding challenge, hybrid integration techniques provide a path towards combining otherwise incompatible material platforms.^{63}
In this work, we focus on discrete variable QONNs due to the maturity of the technology platform, but note that continuous variable implementations are also promising.^{64} Our discrete variable architecture can naturally be mapped to other platforms that manipulate bosonic modes such as ultracold atoms,^{65} superconducting cavities,^{66} or phononic modes in trapped ions.^{67} In each of these platforms, significant progress has been made towards reconfigurable linear mode transformations^{68} to compliment preexisting ultrastrong nonlinearities, thus making bosonic quantum simulators excellent candidates for nearterm QONNs.
Benchmarking
As a first step in validating our architecture, we ensure it can learn elementary quantum tasks such as quantum state preparation, measurement, and quantum gates. We chose Bellstate projection/generation, Greenberger–Horne–Zeilinger (GHZ) state generation, and the implementation of the controlled NOT (CNOT) gate as representative of typical optical quantum information tasks. As described in the Methods section, in each of these cases the training set represents the full basis set for the quantum operation of interest, and successful training tells us something about the expressivity of our architecture.
We trained QONNs of increasing layer depth from N = 2 → 10 with ϕ = π. As shown in Fig. 2, at short layer depths the optimization frequently terminates early, finding a nonoptimal local minima. We observe similar behavior for all of the studied tasks. Most notable here is the behavior of the optimization as the layer count increases: just like a classical neural network, as we increase the layer depth, it becomes consistently easy to find a local minimum that is close to the global minimum. This demonstrates the utility of deep networks: while a single layer may be sufficient to implement, for exampe, a CNOT gate, with deep networks we can reliably discover a configuration that yields the correct operation. For more complex operations, where the smalllayernumber implementation may be difficult to find or simply not exist, this gives hope that we can still reliably train a deep network to perform the task. While the inputs and desired outputs are restricted to the dualrail basis, we have verified that at intermediate layers the joint state of the photons span the entire Fock space, which is a unique feature of photonic systems. In the Supplementary Information S3 we examine the tradeoff between nonlinear interaction strength and gate fidelity.
Hamiltonian simulation
While the results thus far benchmark the training of the QONN, a critical feature of any learning system is that it can generalize to states on which it has not been trained. To assess generalization, we apply the QONN to the task of quantum simulation, whereby a wellcontrolled system in the laboratory \(S(\vec \Theta )\) is programmed over parameters \(\vec \Theta\) to mimic the evolution of a quantum system of interest described by the Hamiltonian \(\hat H\). In particular, we train our QONN on K sets of input/output states \(\{ \psi _{{\mathrm{in}}}^i\rangle \}\) \(\{ \psi _{{\mathrm{out}}}^i\rangle \}\) related by the Hamiltonian of interest \(\left {\psi _{{\mathrm{out}}}^i} \right\rangle = {\mathrm{exp}}(  i\hat Ht)\left {\psi _{{\mathrm{in}}}^i} \right\rangle\), and test it on new states which it has not been exposed to.
As a first test we look at the Ising model (see Methods section), which is optically implemented via a dualrail encoding with m = 2n, where ↑〉 ≡ 10〉_{12} and ↓〉 ≡ 01〉_{12}. For the n = 2 spin case, we train the QONN on a training set of 20 random twophoton states and test it on 50 different states. We empirically determine that for a wide range of J/B values (with t = 1), a threelayer QONN reliably converges to an optimum. In Fig. 3a we vary the interaction strength J/B and plot the probability of finding a particular spin configuration given an initialization state ↑↑〉. Critically, this input state is not in set of states for which the QONN was trained. We also train our QONN for the n = 3 spin case, reaching an average test error of 10.1%. This higher error in the larger system motivates the need for advanced training methods such as backpropagation^{69} or layerwise training approaches^{70} to efficiently train deeper QONN.
Finally, we look at a Hamiltonian more natural for photons in optical modes, the Bose–Hubbard model, see Methods section for further details. Now, the (n, m) configuration of bosons to be simulated is naturally mapped to an nphoton mmode photonic system.
To benchmark our system, we look at the number of layers required to express a (2, 4) strongly interacting Bose–Hubbard model on a square lattice (Fig. 3b, inset). Figure 3b shows that increasing the number of layers reduces the error on the test set, suggesting that deeper networks can express a richer class of quantum functions (i.e. Hamiltonians); a concept familiar in classical deep neural networks.^{71} Choosing five layers to give a reasonable tradeoff between error (~1%) and computational tractability, we vary the interaction strength in the the range U/t_{hop} ∈ [−20, 20]. Across all numerical simulations we achieve a mean test error of 2.9 ± 1.3% (error given by the standard deviation in 22 simulations).
While our analysis has focused on Hamiltonians that exist in nature, the approach itself is very general: mimicking input–output configurations given access to a reduced set of input–output pairs from some family of quantum states. This may find application in learning representations of quantum systems where circuit decompositions are unknown, or finding compiled implementations of known circuits.
Quantum optical autoencoder
Photons play a critical role in virtually all quantum communication and quantum networking protocols, either as information carriers themselves or to mediate interactions between longlived atomic memories.^{72} However, such schemes are exponentially sensitive to loss: given a channel transmissivity η and the number of photons n required to encode a message, the probability of successful transmission scales as η^{n}. Reducing the photon number while maintaining the information content, therefore, exponentially increases the communication rate. In the following we use the QONN as a quantum autoencoder to learn a compressed representation of quantum states. This compressed representation could be used, for example, to more efficiently and reliably exchange information between physically separated quantum nodes.^{35}
Quantum autoencoders have been proposed as a general technique for encoding, or compressing, a family of states on n qubits to a lowerdimensional kqubit manifold called the latent space.^{73,74} Similar to classical autoencoders, a quantum autoencoder learns to generalize from a small training set T and is able to compress states from the family that it has not previously seen. As well as applications in quantum communication and quantum memory, it has recently been proposed as a subroutine to augment variational algorithms in finding more efficient devicespecific ansatzes.^{75} In contrast, the quantum optical autoencoder encodes input states in the Fock basis. Moreover, even if optical input states are encoded in the dualrail qubit basis, the autoencoder may learn a compression onto a noncomputational Fock basis latent space.
As a choice of a family of states, and one which is relevant to quantum chemistry on NISQ processors, we consider the set of ground states of molecular hydrogen, H_{2}, in the STO3G minimal basis set,^{76} mapped from their fermionic representation into qubits via the Jordan–Wigner transformation.^{77} Ground states in this qubit basis have the form ψ(i)〉 = α(i)0011〉_{L} + β(i)1100〉_{L}, where i is the bond length of the ground state. The qubits themselves are represented in a dualrail encoding thus the network consists of n = 4 photons in m = 8 optical modes. We note that the set of states {ψ_{i}〉} are no longer related by a single unitary transformation as in previous sections.
The goal of the quantum optical autoencoder S is for all states in the training set ψ_{i}〉 ∈ K, satisfying
for some twomode state \(\left {\psi _i^C} \right\rangle\) in the latent space. The quantum autoencoder can therefore be seen as an algorithm that systematically disentangles n − k qubits from the set of input states and sets them to a fixed reference state (e.g., \(\left 0 \right\rangle _{\mathrm{L}}^{ \otimes n  k}\)). For this reason, the fidelity of the reference state will be used a proxy for the fidelity of the decoded state.
To train a quantum autoencoder one should choose a circuit architecture with general enough operations to compress the input states, but few enough parameters to train the network efficiently. As shown in Fig. 4a, b, we test three training schemes for the QONN autoencoder: (1) locally structured training (Fig. 4a, blue): sequentially optimizing twolayer QONNs to disentangle a single qubit at each stage, where each subsequent stage acts only on a reduced qubit subspace. This approach is followed by a final global refinement step after all layers have been individually trained; (2) globally structured training (Fig. 4a, orange): where the above layer structure is trained simultaneously rather than sequentially; and (3) globally unstructured training (Fig. 4b, green): where a sixlayer system acting on all four qubits is trained.
The optimization was performed using an implementation of MLSL^{78} (also available in the NLopt library), which is a global optimization algorithm that explores the cost function landscape with a sequence of local optimizations (in this case BOBYQA) from carefully chosen starting points, using a heuristic to avoid local optima that have already been found. Our training states are the set of four ground states of H_{2} corresponding to bond lengths of 0.5, 1.0, 1.5, and 2.0 Å. Both the global and iterative optimizations performed comparably. However, we note that the the iterative approach could potentially be made more efficient if more stringent convergence criteria were introduced. The unstructured optimization achieved a lower fidelity; however, it is unclear from our data whether the iterative approach would have better scaling or accuracy than the global optimization in an asymptotic setting.
Quantum reinforcement learning
To demonstrate the utility of QONNs for classical machine learning tasks, and to show that they continue to generalize in that setting, we examine a standard reinforcement learning problem: that of trying to balance an inverted pendulum.^{79} Classical deep reinforcement learning uses a policy network, that is, a network that takes an observation vector as input and outputs a probability distribution over the space of allowed actions. This probability vector is then sampled to choose an action, a new observation is taken, and the process repeats. As the output from a QONN is inherently a probability distribution, policy networks are a natural application. See Methods for further details.
We simulate a cart moving on a onedimensional frictionless track, with a pole on a hinge attached to its top (see Fig. 5c, inset). At the beginning of the simulation, the cart is initialized to a random position, with the pole at a random angle. At each time step, the neural network receives four values, the position of the cart x, its velocity \(\dot x\), the angle of the pole with respect to the track θ, and the time derivative of that angle \(\dot \theta\). From those four values, it determines whether to apply a force of unit magnitude either in the +x or −x directions; those are the only two options. Each run of the simulation continues until a boundary condition in x, θ, or t (t_{max} = 300) is reached (i.e., the cart runs into the edge of the track or the pole falls over). The number of time steps before failure is the fitness of that run; we want to make this as large as possible.
As shown in Fig. 5a, b we demonstrate training using two different input encodings. First, we directly encode the four observation values x, \(\dot x\), θ, and \(\dot \theta\) onto four dualrail qubits. Second, we encode these four values onto a uniform quantum state over two qubits, a type of quantum random access memory (QRAM) encoding. While it is unknown in general how to efficiently encode a given state into a QRAM, this numerical simulation demonstrates that these networks are capable of learning from general, highly entangled, quantum states, not just those with direct classical analogs.
Both encodings are performed by first compressing each of the four observation variables into γ_{j} ∈ [0, π/2] (j ∈ {1, …, 4}). For the direct encoding, each qubit q_{j} is set to cos(γ_{j})0〉_{L} + sin(γ_{j})1〉_{L}. For the QRAM encoding, the state over the two input qubits is set to \(\frac{1}{4}\left( {e^{i\gamma _1}\left {00} \right\rangle _{\mathrm{L}} + e^{i\gamma _2}\left {01} \right\rangle _{\mathrm{L}} + e^{i\gamma _3}\left {10} \right\rangle _{\mathrm{L}} + e^{i\gamma _4}\left {11} \right\rangle _{\mathrm{L}}} \right)\). Finally, the QRAM encoding is given an ancilla qubit to act as phase reference.
We use this qubit encoding only for ease of encoding; after this point, we no longer regard the photons as qubits and simply measure the output state, potentially increasing the computational power of the system. In both systems, we picked the arbitrary measure of “number of photons in mode 1” vs. “number of photons in mode 2”: if the number of photons in the first mode exceeds the number in the second mode, we apply a force in the −x direction; otherwise, we apply a force in the +x direction. Finally, we train these networks using an evolutionary strategies method.^{80}
In Fig. 5c we show the results of five training cycles (each with different starting conditions) using a sixlayer QONN. For each cycle, we use a batch size of 100 to determine the approximate gradient, and average the fitness over 80 distinct runs of the network at each \(\vec \Theta\) we evaluate. Hyperparameters (layer depth, batch size, and averaging group) were tuned using linear sweeps. These values apply to both the direct and QRAM encodings. Fitness increases with training generation, meaning the QONN consistently learns to balance the pole for longer times as generation increases; that is, it generalizes examples it has previously seen to new instances of the problem.
To crosscheck our performance we trained equivalently sized classical networks, that is, fourneuron, sixlayer networks with constant width. Hidden layers had ReLu neurons, while the final layer was a single sigmoid neuron to generate a probability p ∈ 0, 1) of applying force in the −x direction. We used the same training strategy for the classical networks as for the QONNs and observed a comparable performance, with a mean fitness after 1000 generations in the classical case of 37.1 compared with 61.9 for the directly encoded QONN and 136.1 for the QRAM encoded QONN. The direct encoding took ~5000 generations to reach a comparable fitness as the QRAM. Both networks can likely be optimized, and one should be cautious in directly comparing the classical and quantum results. Notwithstanding, this exploratory work demonstrates that quantum systems can learn on physically relevant data, and future directions will seek to leverage uniquely quantum properties such as superposition for batch learning.^{81}
Oneway quantum repeaters
Finally, to demonstrate both the flexibility of the QONN platform and the advantages of codesigning both the architecture and the physical platform, we demonstrate the realization of oneway quantum repeaters. The goal of a oneway quantum repeater is equivalent to that of forward error correction in classical communications: to distribute the information over several symbols in such a way that even if errors occur, the original information can still be recovered. In quantum optics the primary error mechanism is loss; therefore, one should encode a single qubit of information across n photons such that if m ≤ k photons are lost (for a kloss tolerant code), the state can be repaired without round trip communications between the sender and the receiver (see Fig. 6a). Loss correction techniques are critical both for quantum communications over distance^{82} and protecting qubits in photonic quantum computing schemes.^{83}
In this work, we focus on a recent proposal for unitary oneway quantum repeaters, which do not require measurements or quantum memories.^{84} While it can be shown that Hamiltonians for oneway repeaters exist, the question of how to realize these with physical components remains open. Here we train the QONN architecture to implement such quantum repeater schemes, demonstrating the utility of physically realizable variational quantum architectures.
We consider the twomode code
which is robust against single photon loss. It can be shown that for an input state ψ〉_{L} = α0〉_{L} + β1〉_{L}, the loss of a single photon can be corrected by a system \(\hat S\), which coherently performs the map
Mathematically, \(\hat S\left[ {\hat a_1\rho \hat a_1^\dagger } \right]\hat S^\dagger = \rho\) and \(\hat S\left[ {\hat a_2\rho \hat a_2^\dagger } \right]\hat S^\dagger = \rho\), where ρ = ψ〉_{LL}〈ψ.
By photonnumber preservation, \(\hat S\) cannot be unitary on two modes, but \(\hat S\) can be realized as a unitary with additional ancilla. To train the QONN to implement this mapping, we do the following: let {ψ_{i}〉_{L}}_{i} be the set of states \(\left\{ {\left 0 \right\rangle _{\mathrm{L}},\left 1 \right\rangle _{\mathrm{L}},\left( {\left 0 \right\rangle _{\mathrm{L}} + \left 1 \right\rangle _{\mathrm{L}}} \right)/\sqrt 2 ,\left( {\left 0 \right\rangle _{\mathrm{L}}  \left 1 \right\rangle _{\mathrm{L}}} \right)/\sqrt 2 ,\left( {\left 0 \right\rangle _{\mathrm{L}} + i\left 1 \right\rangle _{\mathrm{L}}} \right)/\sqrt 2 ,\left( {\left 0 \right\rangle _{\mathrm{L}}  i\left 1 \right\rangle _{\mathrm{L}}} \right)/\sqrt 2 } \right.\), and \(\sigma _{i,j} = \hat a_j\rho _i\hat a_j^\dagger\). The action of \(\hat S\) on the computational (nonancilla) modes with single photon loss is given by
where ρ_{A} is the input ancilla state. In the lossless case the output is given by
The desired system should be able to correct all inputs that have single photonloss error, and also leave the input undisturbed if there is no photon loss. This corresponds to the map \(\sigma _{i,j}^{({\mathrm{out}})} = \rho _i^{({\mathrm{out}})} = \rho _i \forall i,j\).
Numerically, we calculate a cost function that quantifies the average distance (given by the Hilbert–Schmidt inner product \({\mathrm{Tr}}\left[ {A^\dagger B} \right]\)) between the six photon subtracted states and nonphoton subtracted states, and variationally optimize the QONN. Due to the complexity of the system a backpropagation method was developed and gradientbased optimization methods were used, to achieve efficient and accurate training. Figure 6b plots the average fidelity of the output states against the number of nonlinear layers, reaching numerical precision at 50 layers. In conclusion, the QONN yields an explicit optical construction of a oneway quantum repeater, which was otherwise unknown. We therefore anticipate other physically motivated variational architectures to yield insights which platformindependent approaches cannot.
Discussion
We have proposed an architecture for nearterm quantum optical systems that maps many of the auspicious features of classical neural networks onto the quantum domain. Through numerical simulation and analysis we have applied our QONN to a broad range of quantum information processing tasks, including newly developed protocols such as quantum optical state compression for quantum networking and blackbox quantum simulation. Experimentally, advances in integrated photonics and nanofabrication have enabled monolithically integrated circuits with many thousands of optoelectronic components.^{85} The architecture we present is not limited to the integration of systems with strong single photon nonlinearities and we anticipate our approach will serve as a natural intermediate step^{10} towards largescale photonic quantum technologies. In this intermediate regime, the QONN may learn practical quantum operations with weak or noisy nonlinearities, which are otherwise unsuitable for faulttolerant quantum computing.^{86} The effect of such noise is an important subject for future work.
Future work will likely focus on loss correction techniques, which are also possible in an alloptical context.^{87} Additionally, classical neural networks have benefitted greatly from improved training techniques. For example, in transfer learning, a trained network has its final layer or two removed and new layers added, which are then trained for an entirely new application.^{88} Future work will explore whether similar techniques may be used to more efficiently solve new problems in the QONN architecture, and whether different architectures such as generative adversarial networks can be applied to quantum optics.^{89} Together, our results point towards both a powerful simulation tool for the design of nextgeneration quantum optical systems and a versatile experimental platform for nearterm optical quantum information processing and machine learning.
Methods
Computational techniques
The quantum optics simulations in this work were performed with custom, optimized code written in Python, with performancesensitive sections translated to Cython. The Numba library was used to GPU accelerate some large operations. The most computationally intensive step was the calculation of the multiphoton unitary transform (\(U(\vec \theta _i)\) in Eq. 1) from the single photon unitary. The multiphoton unitary has \(\left( {\begin{array}{*{20}{c}} {n + m  1} \\ n \end{array}} \right)^2\) entries, each of which requires calculating the permanent of an n × n matrix.^{90}
As with classical neural networks, different optimization algorithms perform better for different tasks. We rely on gradientfree optimization techniques that optimize an objective function without an explicitly defined derivative (or one based on finite difference methods), as computing and backpropagating the gradient through the system likely requires knowledge of the internal quantum state of the system, preventing efficient in situ training. While this might be acceptable for designing small systems in simulation (say, designing quantum gates), it does not allow for systems to be variationally trained in situ. We empirically determined that the BOBYQA algorithm^{91} performs well for most applications in terms of speed and accuracy for our QONN, and is available in the NLopt library.^{92} We note that calculation of such a gradient is possible with classical optical neural networks.^{93} For the quantum reinforcement learning simulations, we used our own implementation of evolutionary strategies.^{80} At each stage evolution strategies takes a vector parameterizing the network, generates a population of new vectors by repeatedly perturbing the vector with gaussian noise, and then calculates a fitness for each perturbed vector. The new vector is then the fitnessweighted average of all the perturbed vectors. Evolution strategies does not require backpropagation, in comparison to strategies based on Markov decision processes, making it more suitable for quantum applications.
Hardware and libraries
The computer used to perform these simulations is a custombuilt workstation with a 12core Intel Core i75820K and 64 GB of RAM. The GPU used was an Nvidia Tesla K40. Relevant software versions are: Ubuntu 16.04 LTS, Linux 4.13.039generic #44 16.04.1Ubuntu SMP, Python 2.7.12, NumPy 1.14.1, NLopt 2.4.2, Cython 0.27.3, and Numba 0.40.0.
Benchmarking training
The training set for the Bellstate projector is the full set of Bell states \(\{ \psi _{{\mathrm{in}}}^i\} = \{ \Phi ^ + \rangle ,\Phi ^  \rangle ,\Psi ^ + \rangle ,\Psi ^  \rangle \}\) encoded as dualrail qubits. Our goal is to map these to a set of states distinguishable by single photon detectors, thus we opt for a binary encoding \(\{ \psi _{{\mathrm{out}}}^i\rangle \} = \{ 1010\rangle ,1001\rangle ,0110\rangle ,0101\rangle \}\). A system designed to perform this map can then be run in reverse to generate Bell states from input Fock states. The CNOT gate uses a full input–output basis set with \(\{ \psi _{{\mathrm{in}}}^i\rangle \} = \{ 1010\rangle ,1001\rangle ,0110\rangle ,0101\rangle \}\) and \(\{ \psi _{{\mathrm{out}}}^i\rangle \} = \{ 1010\rangle ,1001\rangle ,0101\rangle ,0110\rangle \}\). For the GHZ generator we select just a single input–output configuration \(\{ \psi _{{\mathrm{in}}}^i\rangle \} = \{ 101010\rangle \}\) and \(\{ \psi _{{\mathrm{out}}}^i\rangle \} = \{ (101010\rangle + 010101\rangle )/\sqrt 2 \}\).
Simulated Hamiltonians
The Ising model we simulate is described by the Hamiltonian
where B represents the interaction of each spin with a magnetic field in the x direction, and J is the interaction strength between spins in an orthogonal direction. The Bose–Hubbard model we simulate is described by the Hamiltonian
where \(\hat b_i^\dagger\) \((\hat b_i)\) represents the creation (annihilation) operator in mode i, \(\hat n_i\) the number operator and ω, t_{hop}, and U the onsite potential, the hopping amplitude, and the onsite interaction strength, respectively.
Data availability
The data sets generated during and or analyzed during the current study are available from the corresponding author on reasonable request. The QONN codebase is available at https://github.com/steinbrecher/bosonic.
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Acknowledgements
This work was supported by the AFOSR MURI for Optimal Measurements for Scalable Quantum Technologies (FA95501410052) and by the AFOSR program FA95501610391, supervised by Gernot Pomrenke. G.R.S. acknowledges support from the Facebook Fellowship Program. J.C. is supported by EU H2020 Marie SklodowskaCurie grant number 751016. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Tesla K40 GPU used for this research.
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G.R.S., D.E. and J.C. conceived of the project. The core codebase was built by G.R.S. with contributions from J.C. Numerical experiments were performed by G.R.S., J.P.O. and J.C. All authors contribute to the writing of the manuscript. D.E. and J.C. supervised the project.
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Steinbrecher, G.R., Olson, J.P., Englund, D. et al. Quantum optical neural networks. npj Quantum Inf 5, 60 (2019). https://doi.org/10.1038/s4153401901747
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