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Register a new userA Theoretical Framework for Learning from Quantum Data
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Over decades traditional information theory of source and channel coding advances toward learning and effective extraction of information from data. We propose to go one step further and offer a theoretical foundation for learning classical patterns from quantum data. However, there are several roadblocks to lay the groundwork for such a generalization. First, classical data must be replaced by a density operator over a Hilbert space. Hence, deviated from problems such as state tomography, our samples are i.i.d density operators. The second challenge is even more profound since we must realize that our only interaction with a quantum state is through a measurement which -- due to no-cloning quantum postulate -- loses information after measuring it. With this in mind, we present a quantum counterpart of the well-known PAC framework. Based on that, we propose a quantum analogous of the ERM algorithm for learning measurement hypothesis classes. Then, we establish upper bounds on the quantum sample complexity quantum concept classes.
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Distributed quantum information processing is essential for building quantum networks and enabling more extensive quantum computations. In this regime, several spatially separated parties share a multipartite quantum system, and the most natural set of operations are Local Operations and Classical Communication (LOCC). As a pivotal part in quantum information theory and practice, LOCC has led to many vital protocols such as quantum teleportation. However, designing practical LOCC protocols is challenging due to LOCCs intractable structure and limitations set by near-term quantum devices. Here we introduce LOCCNet, a machine learning framework facilitating protocol design and optimization for distributed quantum information processing tasks. As applications, we explore various quantum information tasks such as entanglement distillation, quantum state discrimination, and quantum channel simulation. We discover novel protocols with evident improvements, in particular, for entanglement distillation with quantum states of interest in quantum information. Our approach opens up new opportunities for exploring entanglement and its applications with machine learning, which will potentially sharpen our understanding of the power and limitations of LOCC.
The main results on quantum walk search are scattered over different, incomparable frameworks, most notably the hitting time framework, originally by Szegedy, the electric network framework by Belovs, and the MNRS framework by Magniez, Nayak, Roland and Santha. As a result, a number of pieces are currently missing. For instance, the electric network framework allows quantum walks to start from an arbitrary initial state, but it only detects marked elements. In recent work by Ambainis et al., this problem was resolved for the more restricted hitting time framework, in which quantum walks must start from the stationary distribution. We present a new quantum walk search framework that unifies and strengthens these frameworks. This leads to a number of new results. For instance, the new framework not only detects, but finds marked elements in the electric network setting. The new framework also allows one to interpolate between the hitting time framework, which minimizes the number of walk steps, and the MNRS framework, which minimizes the number of times elements are checked for being marked. This allows for a more natural tradeoff between resources. Whereas the original frameworks only rely on quantum walks and phase estimation, our new algorithm makes use of a technique called quantum fast-forwarding, similar to the recent results by Ambainis et al. As a final result we show how in certain cases we can simplify this more involved algorithm to merely applying the quantum walk operator some number of times. This answers an open question of Ambainis et al.
The entropy of a quantum system is a measure of its randomness, and has applications in measuring quantum entanglement. We study the problem of measuring the von Neumann entropy, $S(rho)$, and Renyi entropy, $S_alpha(rho)$ of an unknown mixed quantum state $rho$ in $d$ dimensions, given access to independent copies of $rho$. We provide an algorithm with copy complexity $O(d^{2/alpha})$ for estimating $S_alpha(rho)$ for $alpha<1$, and copy complexity $O(d^{2})$ for estimating $S(rho)$, and $S_alpha(rho)$ for non-integral $alpha>1$. These bounds are at least quadratic in $d$, which is the order dependence on the number of copies required for learning the entire state $rho$. For integral $alpha>1$, on the other hand, we provide an algorithm for estimating $S_alpha(rho)$ with a sub-quadratic copy complexity of $O(d^{2-2/alpha})$. We characterize the copy complexity for integral $alpha>1$ up to constant factors by providing matching lower bounds. For other values of $alpha$, and the von Neumann entropy, we show lower bounds on the algorithm that achieves the upper bound. This shows that we either need new algorithms for better upper bounds, or better lower bounds to tighten the results. For non-integral $alpha$, and the von Neumann entropy, we consider the well known Empirical Young Diagram (EYD) algorithm, which is the analogue of empirical plug-in estimator in classical distribution estimation. As a corollary, we strengthen a lower bound on the copy complexity of the EYD algorithm for learning the maximally mixed state by showing that the lower bound holds with exponential probability (which was previously known to hold with a constant probability). For integral $alpha>1$, we provide new concentration results of certain polynomials that arise in Kerov algebra of Young diagrams.
We introduce TensorFlow Quantum (TFQ), an open source library for the rapid prototyping of hybrid quantum-classical models for classical or quantum data. This framework offers high-level abstractions for the design and training of both discriminative and generative quantum models under TensorFlow and supports high-performance quantum circuit simulators. We provide an overview of the software architecture and building blocks through several examples and review the theory of hybrid quantum-classical neural networks. We illustrate TFQ functionalities via several basic applications including supervised learning for quantum classification, quantum control, simulating noisy quantum circuits, and quantum approximate optimization. Moreover, we demonstrate how one can apply TFQ to tackle advanced quantum learning tasks including meta-learning, layerwise learning, Hamiltonian learning, sampling thermal states, variational quantum eigensolvers, classification of quantum phase transitions, generative adversarial networks, and reinforcement learning. We hope this framework provides the necessary tools for the quantum computing and machine learning research communities to explore models of both natural and artificial quantum systems, and ultimately discover new quantum algorithms which could potentially yield a quantum advantage.
Let $H$ be a fixed $k$-vertex graph with $m$ edges and minimum degree $d >0$. We use the learning graph framework of Belovs to show that the bounded-error quantum query complexity of determining if an $n$-vertex graph contains $H$ as a subgraph is $O(n^{2-2/k-t})$, where $ t = max{frac{k^2- 2(m+1)}{k(k+1)(m+1)}, frac{2k - d - 3}{k(d+1)(m-d+2)}}$. The previous best algorithm of Magniez et al. had complexity $widetilde O(n^{2-2/k})$.
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