Quantum mean embedding of probability distributions

Abstract

The kernel mean embedding of probability distributions is commonly used in machine learning as an injective mapping from distributions to functions in an infinite dimensional Hilbert space. It allows us, for example, to define a distance measure between probability distributions, called maximum mean discrepancy (MMD). In this work we propose to represent probability distributions in a pure quantum state of a system that is described by an infinite dimensional Hilbert space. This enables us to work with an explicit representation of the mean embedding, whereas classically one can only work implicitly with an infinite dimensional Hilbert space through the use of the kernel trick. We show how this explicit representation can speed up methods that rely on inner products of mean embeddings and discuss the theoretical and experimental challenges that need to be solved in order to achieve these speedups.

The Inductive Bias of Quantum Kernels

“…In recent years, much attention has been dedicated to studies of how small and noisy quantum devices [1] could be used for near term applications to showcase the power of quantum computers. Besides fundamental demonstrations [2], potential applications that have been discussed are in quantum chemistry [3], discrete optimization [4] and machine learning (ML) [5][6][7][8][9][10][11][12].…”

“…There are two ways to that: the first are so-called Quantum Neural Networks (QNN) or parametrized quantum circuits [5][6][7] which can be trained via gradient based optimization [5,[19][20][21][22][23]. The second approach is to use a predefined way of encoding the data in the quantum system and defining a quantum kernel as the inner product of two quantum states [7][8][9][10][11]. These two approaches essentially provide a parametric and a non-parametric path to quantum machine learning, which are closely related to each other [11].…”

Smooth input preparation for quantum and quantum-inspired machine learning

“…In recent years, there has been substantial interest in algorithms based on "quantum linear algebra", where quantum states are used to represent vectors with exponentially large dimensions, which are manipulated by large matrices representing quantum operations . A particularly fruitful domain of applying such methods is quantum machine learning, where quantum algorithms promise for exponential or high-degree polynomial improvements on computational bottlenecks, or enhancement in model expressiveness due to the ability to construct kernels using access to exponentially large Hilbert spaces (Aïmeur et al 2006;Lloyd et al 2013;2014b;Schuld et al 2014;Dunjko and Briegel 2018;Schuld and Killoran 2019;Tiwari et al 2020;Dehdashti et al 2020;Kübler et al 2019;Havlíček et al 2019). Many of these methods leverage quantum algorithms for solving linear systems that trace back to the seminal result of Harrow, Hassidim and Lloyd (2009).…”

Ancilla-Error-Transparent Controlled Beam Splitter Gate

“…In the most general context, the pair of systems {A, B} could be either discrete variable (qubits) or continuous variable (bosonic modes). In the discrete variable context, the cSWAP is a non-Clifford gate (a member of the second level of the Clifford hierarchy) with important applications for universal quantum computation, including machine learning [5][6][7], for routing quantum information through a quantum-controlled switching network to create a quantum random access memory (QRAM) [8][9][10][11][12][13][14], and, in general, for state preparation of quantum non-Gaussian entanglement [15,16] and carrying out the 'swap test' for the purity of a quantum state [17,18], computing the Renyi entropy [19] or the overlap of two different quantum states for quantum fingerprinting [20] and other verification purposes [17,21], and a variety of related tasks [22][23][24][25][26][27][28].…”