arXiv : Determining probability density functions with adiabatic quantum computing

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

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Preprint
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)
Imprint	2023-03-20
Number of pages	7
Note	7 pages, 3 figures
Subject category	hep-ph ; Particle Physics - Phenomenology ; quant-ph ; General Theoretical Physics
Abstract	A reliable determination of probability density functions from data samples is still a relevant topic in scientific applications. In this work we investigate the possibility of defining an algorithm for density function estimation using adiabatic quantum computing. Starting from a sample of a one-dimensional distribution, we define a classical-to-quantum data embedding procedure which maps the empirical cumulative distribution function of the sample into time dependent Hamiltonian using adiabatic quantum evolution. The obtained Hamiltonian is then projected into a quantum circuit using the time evolution operator. Finally, the probability density function of the sample is obtained using quantum hardware differentiation through the parameter shift rule algorithm. We present successful numerical results for predefined known distributions and high-energy physics Monte Carlo simulation samples.
Other source	Inspire
Copyright/License	preprint: (License: arXiv nonexclusive-distrib 1.0)

$Example of initial and final state of the algorithm. The $N_{\rm train}$ blue points are the training set selected from a gaussian mixture sample, whose empirical CDF is represented by the black line. The random initialization of the adiabatic evolution leads to the initial sequence of energies (yellow line). After a training time, the evolution is closer to the training set (red line).$ $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.$ $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.$ Show more plots

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