A limiting factor to the exploitation of continuous variables in quantum computing, in particular in the field of quantum machine learning, has been the hindrances of computing the derivatives of a model. The constraint occurs when applying the parameter-shift rule to a photonic circuit, and the number of non-Gaussian gates exceeds a logarithmic trend with respect to the length of the circuit. This condition occurs when dealing with most nonlinear functions. Indeed, it hampers subroutines, such as backpropagation, which are paramount for machine learning and, most significantly, the computation of the derivatives of the model itself. To overcome this hurdle, we propose a method to compute derivatives of functions, implemented by continuous variable circuits, directly through photonic quantum circuits. Unlike the parameter-shift rule, this method does not fail when the number of non-Gaussian gates scales more than logarithmically against the depth of the circuit. We exploit the fact that computing the derivative of a wave function is equivalent to applying the momentum operator, which in the position basis acts as its derivative. The action of this operator is physically embodied by photon-added coherent states. We prove that, up to the second order of derivation, the number of operations does not scale with the depth of the circuit, but only with the order of derivation. We showcase numerical examples of derivatives of elementary functions parametrized on simulated photonic quantum circuits. Thanks to this method, we unleash an embodiment of quantum neural networks on photonic quantum computing platforms.
A limiting factor to the exploitation of continuous variables in quantum computing, in particular in the field of quantum machine learning, has been the hindrances of computing the derivatives of a model. The constraint occurs when applying the parameter-shift rule to a photonic circuit, and the number of non-Gaussian gates exceeds a logarithmic trend with respect to the length of the circuit. This condition occurs when dealing with most nonlinear functions. Indeed, it hampers subroutines, such as backpropagation, which are paramount for machine learning and, most significantly, the computation of the derivatives of the model itself. To overcome this hurdle, we propose a method to compute derivatives of functions, implemented by continuous variable circuits, directly through photonic quantum circuits. Unlike the parameter-shift rule, this method does not fail when the number of non-Gaussian gates scales more than logarithmically against the depth of the circuit. We exploit the fact that computing the derivative of a wave function is equivalent to applying the momentum operator, which in the position basis acts as its derivative. The action of this operator is physically embodied by photon-added coherent states. We prove that, up to the second order of derivation, the number of operations does not scale with the depth of the circuit, but only with the order of derivation. We showcase numerical examples of derivatives of elementary functions parametrized on simulated photonic quantum circuits. Thanks to this method, we unleash an embodiment of quantum neural networks on photonic quantum computing platforms.
We present a method for gradient computation in quantum algorithms implemented on linear optical quantum computing platforms. While parameter-shift rules have become a staple in qubit gate-based quantum computing for calculating gradients, their direct application to photonic platforms has been hindered by the infinite dimensionality of phase-shift operators in Fock space. We introduce a photonic parameter-shift rule that overcomes this limitation, providing an exact formula for gradient computation in linear optical quantum processors. Our method scales linearly with the number of input photons and utilizes the same parametrized photonic circuit with shifted parameters for each evaluation. This advancement bridges a crucial gap in photonic quantum computing, enabling efficient gradient-based optimization for variational quantum algorithms on near-term photonic quantum processors. We demonstrate the efficacy of our approach through numerical simulations in quantum chemistry and generative modeling tasks, showing superior optimization performance as well as robustness to noise from finite sampling and photon distinguishability compared to other gradient-based and gradient-free methods.
Abstract We show how techniques from machine learning and optimization can be used to find circuits of photonic quantum computers that perform a desired transformation between input and output states. In the simplest case of a single input state, our method discovers circuits for preparing a desired quantum state. In the more general case of several input and output relations, our method obtains circuits that reproduce the action of a target unitary transformation. We use a continuous-variable quantum neural network as the circuit architecture. The network is composed of several layers of optical gates with variable parameters that are optimized by applying automatic differentiation using the TensorFlow backend of the Strawberry Fields photonic quantum computer simulator. We demonstrate the power and versatility of our methods by learning how to use short-depth circuits to synthesize single photons, Gottesman–Kitaev–Preskill states, NOON states, cubic phase gates, random unitaries, cross-Kerr interactions, as well as several other states and gates. We routinely obtain high fidelities above 99% using short-depth circuits, typically consisting of a few hundred gates. The circuits are obtained automatically by simply specifying the target state or gate and running the optimization algorithm.
Quantum photonic simulators have emerged as indispensable tools for modeling and optimizing quantum photonic circuits, bridging the gap between theoretical models and experimental implementations. This review explores the landscape of photonic quantum simulation, focusing on the transition from Gaussian to non-Gaussian models and the computational challenges associated with simulating large-scale photonic systems. Gaussian states and operations, which enable efficient simulations through covariance matrices and phase-space representations, serve as the foundation for photonic quantum computing. However, non-Gaussian states crucial for universal quantum computation introduce significant computational complexity, requiring advanced numerical techniques such as tensor networks and high-performance GPU acceleration. We evaluate the leading photonic quantum simulators, including Strawberry Fields, Piquasso, QuTiP SimulaQron, Perceval, and QuantumOPtics.jl analyzing their capabilities in handling continuous-variable (CV) and discrete-variable (DV) quantum systems. Special attention is given to hardware-accelerated methods, including GPU-based tensor network approaches, machine learning integration, and hybrid quantum-classical workflows. Furthermore, we investigate noise modeling techniques, such as photon loss and dark counts, and their impact on simulation accuracy. As photonic quantum computing moves toward practical implementations, advancements in high-performance computing (HPC) architectures, such as tensor processing units (TPUs) and system-on-a-chip (SoC) solutions, are accelerating the field. This review highlights emerging trends, challenges, and future directions for developing scalable and efficient photonic quantum simulators.
Continuous-variables (CV) quantum optics is a natural formalism for neural networks (NNs) due to its ability to reproduce the information processing of such trainable interconnected systems. In quantum optics, Gaussian operators induce affine mappings on the quadratures of optical modes while non-Gaussian resources (the challenging piece for physical implementation) originate the nonlinear effects, unlocking quantum analogs of an artificial neuron. This work presents a novel experimentally-feasible framework for continuous-variable quantum optical neural networks (QONNs) developed with available photonic components: coherent states as input encoding, a general Gaussian transformation followed by multi-mode photon subtractions as the processing layer, and homodyne detection as outputs readout. The closed-form expressions of such architecture are derived demonstrating the family of adaptive activations and the quantum-optical neurons that emerge from the amount of photon-subtracted modes, proving that the proposed design satisfies the Universal Approximation Theorem within a single layer. To classically simulate the QONN training, the high-performance QuaNNTO library has been developed based on Wick-Isserlis expansion and Bogoliubov transformations, allowing multi-layer exact expectation values of non-Gaussian states without truncating the infinite-dimensional Hilbert space. Experiments on supervised learning and state-preparation tasks show balanced-resource efficiency with strong expressivity and generalization capabilities, illustrating the potential of the architecture for scalable photonic quantum machine learning and for quantum applications such as complex non-Gaussian gate synthesis.
In the era of noisy intermediate-scale quantum computers, variational quantum algorithms are promising approaches for solving optimization tasks by training parameterized quantum circuits with the aid of classical routines informed by quantum measurements. In this context, photonic platforms based on reconfigurable integrated optics are an ideal candidate for implementing these algorithms. Among various techniques to train variational circuits, the parameter shift rule enables the exact calculation of cost function derivatives efficiently, facilitating gradient descent-based optimization. In this paper, we derive a formulation of the parameter shift rule for computing derivatives and integrals tailored to reconfigurable optical linear circuits and based on the Boson Sampling paradigm. This allows us to naturally embed common types of experimental noise, such as partial distinguishability and mixedness of the states, thus obtaining a resilient approach. Finally, we employ the developed approach to experimentally test variational algorithms with single-photon states processed in a reconfigurable six-mode universal integrated interferometer. Specifically, we apply the photonic parameter shift rules to the variational implementation, on a photonic platform, of both an eigensolver and a Universal- gate.
Parametrized quantum optical circuits are a class of quantum circuits in which the carriers of quantum information are photons and the gates are optical transformations. Classically optimizing these circuits is challenging due to the infinite dimensionality of the photon number vector space that is associated to each optical mode. Truncating the space dimension is unavoidable, and it can lead to incorrect results if the gates populate photon number states beyond the cutoff. To tackle this issue, we present an algorithm that is orders of magnitude faster than the current state of the art, to recursively compute the exact matrix elements of Gaussian operators and their gradient with respect to a parametrization. These operators, when augmented with a non-Gaussian transformation such as the Kerr gate, achieve universal quantum computation. Our approach brings two advantages: first, by computing the matrix elements of Gaussian operators directly, we don't need to construct them by combining several other operators; second, we can use any variant of the gradient descent algorithm by plugging our gradients into an automatic differentiation framework such as TensorFlow or PyTorch. Our results will find applications in quantum optical hardware research, quantum machine learning, optical data processing, device discovery and device design.