We introduce and analyze a novel quantum machine learning model motivated by convolutional neural networks. Our quantum convolutional neural network (QCNN) makes use of only O(log(N )) variational parameters for input sizes of N qubits, allowing for its efficient training and implementation on realistic, near-term quantum devices. The QCNN architecture combines the multi-scale entanglement renormalization ansatz and quantum error correction. We explicitly illustrate its potential with two examples. First, QCNN is used to accurately recognize quantum states associated with 1D symmetry-protected topological phases. We numerically demonstrate that a QCNN trained on a small set of exactly solvable points can reproduce the phase diagram over the entire parameter regime and also provide an exact, analytical QCNN solution. As a second application, we utilize QCNNs to devise a quantum error correction scheme optimized for a given error model. We provide a generic framework to simultaneously optimize both encoding and decoding procedures and find that the resultant scheme significantly outperforms known quantum codes of comparable complexity. Finally, potential experimental realization and generalizations of QCNNs are discussed.Machine learning based on neural networks has recently provided significant advances for many practical applications 1 . In physics, one natural application involves the study of quantum many-body systems, where the extreme complexity of many-body states often makes theoretical analysis intractable. This has led to a number of recent works using machine learning to study properties of quantum systems 2-7 , using physical concepts to interpret machine learning 8,9 , or using quantum computers to enhance conventional machine learning tasks 10-13 .In this work, motivated by the progress towards realizing quantum information processors 14-17 , we bridge these approaches by proposing a quantum circuit model inspired by machine learning and illustrating its success for two important classes of quantum many-body problems. The first class of problems we consider is quantum phase recognition (QPR), which asks whether a given input quantum state ρ in belongs to a particular quantum phase of matter. Critically, in contrast to many existing schemes based on tensor network descriptions 18-20 , we assume ρ in is prepared in a physical system without direct access to its classical description. The second class, quantum error correction (QEC) optimization, asks for an optimal QEC code for a given, a priori unknown error model such as dephasing or potentially correlated depolarization in realistic experimental settings.The highly complex and intrinsically quantum nature of these problems makes them particularly difficult to solve using existing classical and quantum machine learning techniques. While conventional machine learning with large-scale neural networks can successfully solve analogous classical problems such as image recognition or improving classical error correction 1 , the exponentially large many-body...

We present quantum algorithms to efficiently perform discriminant analysis for dimensionality reduction and classification over an exponentially large input data set. Compared with the best-known classical algorithms, the quantum algorithms show an exponential speedup in both the number of training vectors M and the feature space dimension N. We generalize the previous quantum algorithm for solving systems of linear equations (2009 Phys. Rev. Lett. 103 150502) to efficiently implement a Hermitian chain product of k trace-normalized N ×N Hermitian positive-semidefinite matrices with time complexity of ( ( )) O N log . Using this result, we perform linear as well as nonlinear Fisher discriminant analysis for dimensionality reduction over M vectors, each in an N-dimensional feature space, in time ( ( ) )  O p MN polylog 3 , where ò denotes the tolerance error, and p is the number of principal projection directions desired. We also present a quantum discriminant analysis algorithm for data classification with time complexity ( ( ) )  O MN log 3 .

We present an exactly solvable lattice Hamiltonian to realize gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also present the parallel algebraic/categorical structure of gapped boundaries. arXiv:1707.04564v3 [cond-mat.str-el]