5,767 research outputs found
A Quantum-inspired Similarity Measure for the Analysis of Complete Weighted Graphs
We develop a novel method for measuring the similarity between complete weighted graphs, which are probed by means of discrete-time quantum walks. Directly probing complete graphs using discrete-time quantum walks is intractable due to the cost of simulating the quantum walk. We overcome this problem by extracting a commute-time minimum spanning tree from the complete weighted graph. The spanning tree is probed by a discrete time quantum walk which is initialised using a weighted version of the Perron-Frobenius operator. This naturally encapsulates the edge weight information for the spanning tree extracted from the original graph. For each pair of complete weighted graphs to be compared, we simulate a discrete-time quantum walk on each of the corresponding commute time minimum spanning trees, and then compute the associated density matrices for the quantum walks. The probability of the walk visiting each edge of the spanning tree is given by the diagonal elements of the density matrices. The similarity between each pair of graphs is then computed using either a) the inner product or b) the negative exponential of the Jensen-Shannon divergence between the probability distributions. We show that in both cases the resulting similarity measure is positive definite and therefore corresponds to a kernel on the graphs. We perform a series of experiments on publicly available graph datasets from a variety of different domains, together with time-varying financial networks extracted from data for the New York Stock Exchange. Our experiments demonstrate the effectiveness of the proposed similarity measures
Quantum ground state isoperimetric inequalities for the energy spectrum of local Hamiltonians
We investigate the relationship between the energy spectrum of a local
Hamiltonian and the geometric properties of its ground state. By generalizing a
standard framework from the analysis of Markov chains to arbitrary
(non-stoquastic) Hamiltonians we are naturally led to see that the spectral gap
can always be upper bounded by an isoperimetric ratio that depends only on the
ground state probability distribution and the range of the terms in the
Hamiltonian, but not on any other details of the interaction couplings. This
means that for a given probability distribution the inequality constrains the
spectral gap of any local Hamiltonian with this distribution as its ground
state probability distribution in some basis (Eldar and Harrow derived a
similar result in order to characterize the output of low-depth quantum
circuits). Going further, we relate the Hilbert space localization properties
of the ground state to higher energy eigenvalues by showing that the presence
of k strongly localized ground state modes (i.e. clusters of probability, or
subsets with small expansion) in Hilbert space implies the presence of k energy
eigenvalues that are close to the ground state energy. Our results suggest that
quantum adiabatic optimization using local Hamiltonians will inevitably
encounter small spectral gaps when attempting to prepare ground states
corresponding to multi-modal probability distributions with strongly localized
modes, and this problem cannot necessarily be alleviated with the inclusion of
non-stoquastic couplings
Entropic Dynamic Time Warping Kernels for Co-evolving Financial Time Series Analysis
In this work, we develop a novel framework to measure the similarity between
dynamic financial networks, i.e., time-varying financial networks.
Particularly, we explore whether the proposed similarity measure can be
employed to understand the structural evolution of the financial networks with
time. For a set of time-varying financial networks with each vertex
representing the individual time series of a different stock and each edge
between a pair of time series representing the absolute value of their Pearson
correlation, our start point is to compute the commute time matrix associated
with the weighted adjacency matrix of the network structures, where each
element of the matrix can be seen as the enhanced correlation value between
pairwise stocks. For each network, we show how the commute time matrix allows
us to identify a reliable set of dominant correlated time series as well as an
associated dominant probability distribution of the stock belonging to this
set. Furthermore, we represent each original network as a discrete dominant
Shannon entropy time series computed from the dominant probability
distribution. With the dominant entropy time series for each pair of financial
networks to hand, we develop a similarity measure based on the classical
dynamic time warping framework, for analyzing the financial time-varying
networks. We show that the proposed similarity measure is positive definite and
thus corresponds to a kernel measure on graphs. The proposed kernel bridges the
gap between graph kernels and the classical dynamic time warping framework for
multiple financial time series analysis. Experiments on time-varying networks
extracted through New York Stock Exchange (NYSE) database demonstrate the
effectiveness of the proposed approach.Comment: Previously, the original version of this manuscript appeared as
arXiv:1902.09947v2, that was submitted as a replacement by a mistake. Now,
that article has been replaced to correct the error, and this manuscript is
distinct from that articl
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