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- Chapter 250 Quantum Classical Hybrid Machine Learning
With the fast development of quantum technology, the size of quantum systems we can digitally manipulate and analogly probe increase drastically. In order to have a better control and understanding of the quantum hardware, an important task is to characterize the interaction, i.e., to learn the Hamiltonian, which determines both static or dynamic properties of the system. Conventional Hamiltonian learning methods either require costly process tomography or adopt impractical assumptions, such as prior information of the Hamiltonian structure and the ground or thermal states of the system. In this work, we present a practical and efficient Hamiltonian learning method that circumvents these limitations. The proposed method can efficiently learn any Hamiltonian that is sparse on the Pauli basis using only short time dynamics and local operations without any information of the Hamiltonian or preparing any eigenstates or thermal states. The method has scalable complexity and vanishing failure probability regarding the qubit number. Meanwhile, it is free from state preparation and measurement error and robust against a certain amount of circuit and shot noise. We numerically test the scaling and the estimation accuracy of the method for transverse field Ising Hamiltonian with random interaction strengths and molecular Hamiltonians, both with varying sizes. All these results verify the practicality and efficacy of the method, paving the way for a systematic understanding of large quantum systems.
One of the central foundational questions of physics is to identify what makes a system quantum as opposed to classical. One seminal notion of classicality of a quantum system is the existence of a non-contextual hidden variable model as introduced in the early work by Bell, Kochen and Specker. In quantum optics, the non-negativity of the Wigner function is a ubiquitous notion of classicality. In this work we establish an equivalence between these two concepts. In particular, we show that any non-contextual hidden variable model for Gaussian quantum optics has an alternative non-negative Wigner function description. Conversely, it was known that the Wigner representation provides a non-negative non-contextual description of Gaussian quantum optics. It follows that contextuality and Wigner negativity are equivalent notions of non-classicality and equivalent resources for this quantum subtheory. In particular, both contextuality and Wigner negativity are necessary for a computational speed-up of quantum Gaussian optics. At the technical level, our result holds true for any subfamily of Gaussian measurements that include homodyne measurements, i.e., measurements of standard quadrature observables.
Quantum Neural Networks (QNNs) with random structures have poor trainability due to the exponentially vanishing gradient as the circuit depth and the qubit number increase. This result leads to a general belief that a deep QNN will not be feasible. In this work, we provide the first viable solution to the vanishing gradient problem for deep QNNs with theoretical guarantees. Specifically, we prove that for circuits with controlled-layer architectures, the expectation of the gradient norm can be lower bounded by a value that is independent of the qubit number and the circuit depth. Our results follow from a careful analysis of the gradient behaviour on parameter space consisting of rotation angles, as employed in almost any QNNs, instead of relying on impractical 2-design assumptions. We explicitly construct examples where only our QNNs are trainable and converge, while others in comparison cannot.
We present a new approach to the study of equilibrium properties in many-body quantum physics. Our method takes inspiration from Density Matrix Quantum Monte Carlo and incorporates new crucial features. First of all, the dynamics is transferred to the Laplace representation where an exact equation can be derived and solved using a simulation-step that, unlike most Monte Carlo methods, is not a priori physically bounded. Moreover, the spawning events are formulated in terms of two-process stochastic unravellings of quantum master equations, a formalism that is particularly useful when working with density matrices. And last, this is equivalent to an interaction picture, where the free part is integrated exactly and the convergence rate can be greatly increased if the interaction parameter is small. We benchmark our method by applying it to two case-studies in condensed matter physics, show its accuracy and further discuss its efficiency.
Deep learning is one of the most successful and far-reaching strategies used in machine learning today. However, the scale and utility of neural networks is still greatly limited by the current hardware used to train them. These concerns have become increasingly pressing as conventional computers quickly approach physical limitations that will slow performance improvements in years to come. For these reasons, scientists have begun to explore alternative computing platforms, like quantum computers, for training neural networks. In recent years, variational quantum circuits have emerged as one of the most successful approaches to quantum deep learning on noisy intermediate scale quantum devices. We propose a hybrid quantum-classical neural network architecture where each neuron is a variational quantum circuit. We empirically analyze the performance of this hybrid neural network on a series of binary classification data sets using a simulated universal quantum computer and a state of the art universal quantum computer. On simulated hardware, we observe that the hybrid neural network achieves roughly 10% higher classification accuracy and 20% better minimization of cost than an individual variational quantum circuit. On quantum hardware, we observe that each model only performs well when the qubit and gate count is sufficiently small.
We propose a new kernel that quantifies success for the task of computing a core-periphery partition for an undirected network. Finding the associated optimal partitioning may be expressed in the form of a quadratic unconstrained binary optimization (QUBO) problem, to which a state-of-the-art quantum annealer may be applied. We therefore make use of the new objective function to (a) judge the performance of a quantum annealer, and (b) compare this approach with existing heuristic core-periphery partitioning methods. The quantum annealing is performed on the commercially available D-Wave machine. The QUBO problem involves a full matrix even when the underlying network is sparse. Hence, we develop and test a sparsified version of the original QUBO which increases the available problem dimension for the quantum annealer. Results are provided on both synthetic and real data sets, and we conclude that the QUBO/quantum annealing approach offers benefits in terms of optimizing this new quantity of interest.
Recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as general function approximators. In this work, we propose a quantum-classical deep network structure to enhance classical CNN model discriminability. The convolutional layer uses linear filters to scan the input data. Moreover, we build PQC, which is a more potent function approximator, with more complex structures to capture the features within the receptive field. The feature maps are obtained by sliding the PQCs over the input in a similar way as CNN. We also give a training algorithm for the proposed model. The hybrid models used in our design are validated by numerical simulation. We demonstrate the reasonable classification performances on MNIST and we compare the performances with models in different settings. The results disclose that the model with ansatz in high expressibility achieves lower cost and higher accuracy.
Similar to its classical version, quantum Markovian evolution can be either time-discrete or time-continuous. Discrete quantum Markovian evolution is usually modeled with completely-positive trace-preserving maps while time-continuous one is often specified with operators referred to as “Lindbladians”. Here we address the following question: Being given a quantum map, can we find a Lindbladian which generates an evolution identical — when monitored at discrete instances of time — to the one induced by the map? It was demonstrated that the problem of getting the answer to this question can be reduced to an NP-complete (in the dimension NN of the Hilbert space the evolution takes place in) problem. We approach this question from a different perspective by considering a variety of Machine Learning (ML) methods and trying to estimate their potential ability to give the correct answer. As a test bed, we use a single-qubit model for which the answer can be obtained by using the reduction procedure. The main outcome of our experiment is that some of the used ML schemes can be trained to match the correct answer with surprisingly high accuracy. This gives us hope that, by implementing the ML methodology, we could get beyond the limit set by the complexity of the reduced problem.