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Signal Processing

Great-length wavelets on quantum computing platform: Applications and fast approximations

Guangsheng Ma, Ziwei Zhou

Published Mar 13, 2024• 1 citation
Publisher
DOI
10.1016/j.sigpro.2024.109472

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The Burt–Adelson wavelet decomposition was used to analyze the Monte Carlo spin dynamics of a one‐dimensional Ising ring. Burt–Adelson wavelets were used because they select for spin domains and domain walls. For wavelet amplitudes cni(t), we computed mean‐square fluctuations 〈[cni(t)]2〉, spatial correlation functions 〈cni(t)cni+a(t)〉, and time correlation functions 〈cni(t)cni(t+τ)〉. The simulations are in excellent agreement with analytic calculations. The temperature and decomposition order dependencies of the static correlation functions are readily explained by the wavelet support L and the system’s correlation length ξ. High‐order (long‐wavelength) wavelet time correlation functions decay exponentially. Low‐order wavelet time correlation functions have an approximate two‐exponential decay, with a fast temperature‐independent relaxation corresponding to the random walk of domain edges and a slow, temperature‐dependent relaxation corresponding to domain creation and annihilation. At intermediate temperatures the slow decay rate satisfies the scaling relationship Γ−1∼〈[cn(t)]2〉z for z=1.80. © 1995 American Institute of Physics.
Time-resolved magnetic sensing is of great importance from fundamental studies to applications in physical and biological sciences. Recently, the nitrogen-vacancy defect center in diamond has been developed as a promising sensor of magnetic fields under ambient conditions. However, methods to reconstruct time-resolved magnetic fields with high sensitivity are not yet fully developed. Here, we propose and demonstrate a sensing method based on spin echo and Haar wavelet transformation. Our method is exponentially faster in reconstructing time-resolved magnetic fields with comparable sensitivity than existing methods. It is also easier to implement in experiments. Furthermore, the wavelet's unique features enable our method to extract information from the whole signal with only part of the measuring sequences. We then explore this feature for a fast detection of simulated nerve impulses. These results will be useful to time-resolved magnetic sensing with quantum probes at nanoscale.
We develop a multiscale approach to estimate high-dimensional probability distributions.Our approach applies to cases in which the energy function (or Hamiltonian) is not known from the start.Using data acquired from experiments or simulations we can estimate the energy function.We obtain a representation of the approximate probability distribution based on a multiscale cascade of conditional probabilities.This representation allows for fast sampling of many-body systems in various domains, from statistical physics to cosmology.Our method-the wavelet-conditional renormalization group (WCRG)-proceeds scale by scale, estimating models for the conditional probabilities of "fast degrees of freedom" conditioned by coarse-grained fields.These probability distributions are modeled by energy functions associated with scale interactions, and are represented in an orthogonal wavelet basis.The WCRG decomposes the microscopic energy function as a sum of interaction energies at all scales and can efficiently generate new samples by going from coarse to fine scales.Near phase transitions, this representation of the approximate probability distribution completely avoids the "critical slowing-down" of direct estimation and sampling algorithms.This is explained theoretically by combining results from RG and wavelet theories, and verified numerically for the Gaussian and φ 4 -field theories.We show that multiscale WCRG energy-based models are more general than local potential models and can capture the physics of complex many-body interacting systems at all length scales.This is demonstrated for weak-gravitational-lensing fields reflecting darkmatter distributions in cosmology, which include long-range interactions with long-tail probability distributions.The WCRG has a large number of potential applications in nonequilibrium systems, where the underlying distribution is not known a priori.Finally, we discuss the connection between the WCRG and deep network architectures.