Springer : Determining probability density functions with adiabatic quantum computing

Robbiati, Matteo; Cruz-Martinez, Juan M.; Carrazza, Stefano

doi:10.1007/s42484-024-00228-2

Article
Report number	arXiv:2303.11346 ; TIF-UNIMI-2023-9 ; CERN-TH-2023-042
Title	Determining probability density functions with adiabatic quantum computing
Author(s)	Robbiati, Matteo (CERN ; Milan U. ; INFN, Milan) ; Cruz-Martinez, Juan M. (CERN) ; Carrazza, Stefano (CERN ; Milan U. ; INFN, Milan ; Technol. Innovation Inst., UAE)
Publication	2025-01-04
Imprint	2023-03-20
Number of pages	11
In:	Quantum Machine Intelligence 7 (2025) 5
DOI	10.1007/s42484-024-00228-2
Subject category	hep-ph ; Particle Physics - Phenomenology ; quant-ph ; General Theoretical Physics
Abstract	The two main approaches to quantum computing are gate-based computation and analog computation, which are polynomially equivalent in terms of complexity, and they are often seen as alternatives to each other. In this work, we present a method for fitting one-dimensional probability distributions as a practical example of how analog and gate-based computation can be used together to perform different tasks within a single algorithm. In particular, we propose a strategy for encoding data within an adiabatic evolution model, which accomodates the fitting of strictly monotonic functions, as it is the cumulative distribution function of a dataset. Subsequently, we use a Trotter-bounded procedure to translate the adiabatic evolution into a quantum circuit in which the evolution time t is identified with the parameters of the circuit. This facilitates computing the probability density as derivative of the cumulative function using parameter shift rules.
Copyright/License	preprint: (License: arXiv nonexclusive-distrib 1.0) publication: © 2025 The Author(s) (License: CC-BY-4.0)

$Top row: comparisons between the true CDF and the result of training an adiabatic hamiltonian to follow the CDF from the sampled distribution. The exact CDF is represented by a discontinuous black line while the simulated circuit results are represented by a blue (realistic) and red (ideal) continuous lines. In grey an arbitrary binning of the data that has been used to train the circuit. Bottom row: comparison between the original PDF and the result of the derivative of the trained circuit. Once again the true result is represented by a discontinuous black, the data histogram is shown in grey and the exact simulation of the circuit is represented as a red continuous line. As regarding the realistic simulations, we show two different continuous lines, corresponding to different values of shots used for evaluating the expectation value of the target observable. In particular, we show in orange and blue the results obtained executing the circuit respectively $N_{\rm shots} = 2\cdot 10^4$ and $N_{\rm shots} = 2\cdot 10^5$ times. All af these results are shown in the form of a ratio between the target true law and the QML estimations in the lowest part of the figures. While representing the PDFs, only one realistic simulation is drawn ($N_{\rm shots} = 2\cdot 10^5$). All the realistic simulations curves are represented together with a $1\sigma$ confidence belt calculated using $N_{\rm runs}=20$ experiments.$ $Example of Quantum Adiabatic Machine Learning (QAML). We select $N_{\rm train}$ data from a sample of points picked up from a Gamma distribution. The training labels (blue points) lay on the empirical CDF of the sample (black line). The untrained sequence of $\braket{\mathcal{O}}$ (yellow line) is compared with the values after the QAML training (red line). The scheduling function is the one presented in Eq.~\eqref{eq:scheduling_ansatz} with $p=12$.$ Show more plots

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Notice créée le 2023-03-22, modifiée le 2025-02-27

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