Continuous-variable quantum neural networks

We introduce a general method for building neural networks on quantum computers. The quantum neural network is a variational quantum circuit built in the continuous-variable (CV) architecture, which encodes quantum information in continuous degrees of freedom such as the amplitudes of the electromagnetic field. This circuit contains a layered structure of continuously parameterized gates which is universal for CV quantum computation. Affine transformations and nonlinear activation functions, two key elements in neural networks, are enacted in the quantum network using Gaussian and non-Gaussian gates, respectively. The non-Gaussian gates provide both the nonlinearity and the universality of the model. Due to the structure of the CV model, the CV quantum neural network can encode highly nonlinear transformations while remaining completely unitary. We show how a classical network can be embedded into the quantum formalism and propose quantum versions of various specialized model such as convolutional, recurrent, and residual networks. Finally, we present numerous modeling experiments built with the Strawberry Fields software library. These experiments, including a classifier for fraud detection, a network which generates Tetris images, and a hybrid classical-quantum autoencoder, demonstrate the capability and adaptability of CV quantum neural networks

Learning Real and Boolean Functions: When Is Deep Better Than Shallow

We describe computational tasks - especially in vision - that correspond to compositional/hierarchical functions. While the universal approximation property holds both for hierarchical and shallow networks, we prove that deep (hierarchical) networks can approximate the class of compositional functions with the same accuracy as shallow networks but with exponentially lower VC-dimension as well as the number of training parameters. This leads to the question of approximation by sparse polynomials (in the number of independent parameters) and, as a consequence, by deep networks. We also discuss connections between our results and learnability of sparse Boolean functions, settling an old conjecture by Bengio.This work was supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF 1231216. HNM was supported in part by ARO Grant W911NF-15-1-0385