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arXiv:2110.12379 · Physical Review A

Variational quantum algorithm for Gaussian discrete solitons and their boson sampling

Claudio Conti

Published Jul 22, 2022• 3 citations
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Abstract

In the context of quantum information, highly nonlinear regimes, such as those supporting solitons, are marginally investigated. We miss general methods for quantum solitons, although they can act as entanglement generators or as self-organized quantum processors. We develop a computational approach that uses a neural network as a variational ansatz for quantum solitons in an array of waveguides. By training the resulting phase space quantum machine-slearning model, we find different soliton solutions, varying the number of particles and interaction strength. We consider Gaussian states that enable measuring the degree of entanglement and sampling the probability distribution of many-particle events. We also determine the probability of generating particle pairs and unveil that soliton bound states emit correlated pairs. These results may have a role in boson sampling with nonlinear systems and in quantum processors for entangled nonlinear waves.

DOI
10.1103/physreva.106.013518

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In the context of quantum information, highly nonlinear regimes, such as those supporting solitons, are marginally investigated. We miss general methods for quantum solitons, although they can act as entanglement generators or as self-organized quantum processors. We develop a computational approach that uses a neural network as a variational ansatz for quantum solitons in an array of waveguides. By training the resulting phase space quantum machine-slearning model, we find different soliton solutions, varying the number of particles and interaction strength. We consider Gaussian states that enable measuring the degree of entanglement and sampling the probability distribution of many-particle events. We also determine the probability of generating particle pairs and unveil that soliton bound states emit correlated pairs. These results may have a role in boson sampling with nonlinear systems and in quantum processors for entangled nonlinear waves.
We use a neural network variational ansatz to compute Gaussian quantum discrete solitons in an array of waveguides described by the quantum discrete nonlinear Schroedinger equation. By training the quantum machine learning model in the phase space, we find different quantum soliton solutions varying the number of particles and interaction strength. The use of Gaussian states enables measuring the degree of entanglement and the boson sampling patterns. We compute the probability of generating different particle pairs when varying the soliton features and unveil that bound states of discrete solitons emit correlated pairs of photons. These results may have a role in boson sampling experiments with nonlinear systems and in developing quantum processors to generate entangled many-photon nonlinear states.
Abstract We use neural networks to represent the characteristic function of many-body Gaussian states in the quantum phase space. By a pullback mechanism, we model transformations due to unitary operators as linear layers that can be cascaded to simulate complex multi-particle processes. We use the layered neural networks for non-classical light propagation in random interferometers, and compute boson pattern probabilities by automatic differentiation. This is a viable strategy for training Gaussian boson sampling. We demonstrate that multi-particle events in Gaussian boson sampling can be optimized by a proper design and training of the neural network weights. The results are potentially useful to the creation of new sources and complex circuits for quantum technologies.
Gaussian boson sampling (GBS) is a near-term platform for photonic quantum computing. Applications have been developed that rely on directly programming GBS devices, but the ability to train and optimize circuits has been a key missing ingredient for developing new algorithms. In this work, we derive analytical gradient formulas for the GBS distribution, which can be used for training devices using standard methods based on gradient descent. We introduce a parametrization of the distribution that allows the gradient to be estimated by sampling from the same device that is being optimized. In the case of training using a Kullback-Leibler divergence or log-likelihood cost function, we show that gradients can be computed classically, leading to fast training. We illustrate these results with numerical experiments in stochastic optimization and unsupervised learning. As a particular example, we introduce the variational Ising solver, a hybrid algorithm for training GBS devices to sample ground states of a classical Ising model with high probability.
Can near-term gate model based quantum processors offer quantum advantage for practical applications in the pre-fault tolerance noise regime? A class of algorithms which have shown some promise in this regard are the so-called classical-quantum hybrid variational algorithms. Here we develop a low-depth quantum algorithm to generative neural networks using variational quantum circuits. We introduce a method which employs the quantum approximate optimization algorithm as a subroutine in order produce then sample low-energy distributions of Ising Hamiltonians. We sample these states to train neural networks and demonstrate training convergence for numerically simulated noisy circuits with depolarizing errors of rates of up to 4%4\%4%.
We show how phase-space simulations of quantum states in a linear photonic network permit the verification of measurable probabilities and entanglement. We compare our predictions with recent Gaussian boson sampling experiments of Zhong et al. These use squeezed inputs and efficient ``on-off'' detectors, with up to 76th-order measured coincidence counts in the data. We introduce a general definition of grouped ``on-off'' detection probabilities for this purpose. The positive-PPP phase-space method is used to compute any grouped or marginal click probabilities. Additional decoherence is included to obtain agreement between theory and experiment. The only limitation in estimating grouped probabilities is the computational sampling error, which is similar in magnitude to the experimental sampling error. The results obtained and graphed here are from first-order up to 16 000th-order grouped count probabilities. However, any order between these is also computable. We extend these results to include grouped probabilities with multidimensional outcomes that have a polynomial number of points. We also analyze quadrature detection experiments and show how to simulate genuine multipartite entanglement using Wigner phase-space methods.
The eigenvalue problem of quantum many-body systems is a fundamental and challenging subject in condensed-matter physics because the dimension of the Hilbert space (and hence the required computational memory and time) grows exponentially as the system size increases. A few numerical methods have been developed for some specific systems but may not be applicable in others. Here we propose a general numerical method, random sampling neural networks (RSNNs), to utilize the pattern recognition technique for the random sampling matrix elements of an interacting many-body system via a self-supervised learning approach. Several exactly solvable one-dimensional models, including the Ising model with a transverse field, the Fermi-Hubbard model, and the spin-12XXZ\frac{1}{2}\phantom{\rule{4pt}{0ex}}XXZ21​XXZ model, are used to test the applicability of RSNN. Pretty high accuracy (>96>96%>96) of energy spectra, magnetization, critical exponents, etc. can be obtained within the strongly correlated regime or near the quantum phase-transition point, even the corresponding RSNN models are trained in the weakly interacting regime. After including data augmentation and transfer learning methods, RSNN can be further applied to systems of larger sizes with much less training data if only pretrained in a smaller system. Our results demonstrate the possibility to combine the existing numerical methods and the RSNN to explore quantum many-body problems in a much wider parameter space.