35 research outputs found
Frame functions in finite-dimensional Quantum Mechanics and its Hamiltonian formulation on complex projective spaces
This work concerns some issues about the interplay of standard and geometric
(Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in
the projective space. Our analysis relies upon the notion and the properties of
so-called frame functions, introduced by A.M. Gleason to prove his celebrated
theorem. In particular, the problem of associating quantum state with positive
Liouville densities is tackled from an axiomatic point of view, proving a
theorem classifying all possible correspondences. A similar result is
established for classical observables representing quantum ones. These
correspondences turn out to be encoded in a one-parameter class and, in both
cases, the classical objects representing quantum ones result to be frame
functions. The requirements of covariance and (convex) linearity play a
central r\^ole in the proof of those theorems. A new characterization of
classical observables describing quantum observables is presented, together
with a geometric description of the -algebra structure of the set of
quantum observables in terms of classical ones.Comment: 32 pages, no figure, fixed some coefficients and added some comments
and references, accepted for publication in Int. J. Geom. Methods. Mod. Phy
A geometrization of quantum mutual information
It is well known that quantum mechanics admits a geometric formulation on the
complex projective space as a Kahler manifold. In this paper we consider the
notion of mutual information among continuous random variables in relation to
the geometric description of a composite quantum system introducing a new
measure of total correlations that can be computed in terms of Gaussian
integrals.Comment: 8 page
An efficient geometric approach to quantum-inspired classifications
Optimal measurements for the discrimination of quantum states are useful tools for classification problems. In order to exploit the potential of quantum computers, feature vectors have to be encoded into quantum states represented by density operators. However, quantum-inspired classifiers based on nearest mean and on Helstrom state discrimination are implemented on classical computers. We show a geometric approach that improves the efficiency of quantum-inspired classification in terms of space and time acting on quantum encoding and allows one to compare classifiers correctly in the presence of multiple preparations of the same quantum state as input. We also introduce the nearest mean classification based on Bures distance, Hellinger distance and Jensen-Shannon distance comparing the performance with respect to well-known classifiers applied to benchmark datasets
A general learning scheme for classical and quantum Ising machines
An Ising machine is any hardware specifically designed for finding the ground
state of the Ising model. Relevant examples are coherent Ising machines and
quantum annealers. In this paper, we propose a new machine learning model that
is based on the Ising structure and can be efficiently trained using gradient
descent. We provide a mathematical characterization of the training process,
which is based upon optimizing a loss function whose partial derivatives are
not explicitly calculated but estimated by the Ising machine itself. Moreover,
we present some experimental results on the training and execution of the
proposed learning model. These results point out new possibilities offered by
Ising machines for different learning tasks. In particular, in the quantum
realm, the quantum resources are used for both the execution and the training
of the model, providing a promising perspective in quantum machine learning.Comment: 25 pages, 9 figure
A quantum k-nearest neighbors algorithm based on the Euclidean distance estimation
The k-nearest neighbors (k-NN) is a basic machine learning (ML) algorithm, and several quantum versions of it, employing different distance metrics, have been presented in the last few years. Although the Euclidean distance is one of the most widely used distance metrics in ML, it has not received much consideration in the development of these quantum variants. In this article, a novel quantum k-NN algorithm based on the Euclidean distance is introduced. Specifically, the algorithm is characterized by a quantum encoding requiring a low number of qubits and a simple quantum circuit not involving oracles, aspects that favor its realization. In addition to the mathematical formulation and some complexity observations, a detailed empirical evaluation with simulations is presented. In particular, the results have shown the correctness of the formulation, a drop in the performance of the algorithm when the number of measurements is limited, the competitiveness with respect to some classical baseline methods in the ideal case, and the possibility of improving the performance by increasing the number of measurements