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arXiv:quant-ph/0409033 · International Journal of Quantum Information

QUANTUM COMPUTATION BASED PROBABILITY DENSITY FUNCTION ESTIMATION

Ferenc Balázs, Sándor Imre

Published Dec 16, 2004• 0 citations
arXivHTMLPDF

Abstract

Signal processing techniques will lean on blind methods in the near future, where no redundant, resource allocating information will be transmitted through the channel. To achieve a proper decision, however, it is essential to know at least the probability density function (PDF), which to estimate is classically a time consumpting and/or less accurate hard task that may make decisions to fail. This paper describes the design of a quantum assisted PDF estimation method also by way of an example, which promises to achieve the exact PDF by proper setting of parameters in a very rapid way.

DOI
10.1142/s0219749905000578

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Signal processing techniques will lean on blind methods in the near future, where no redundant, resource allocating information will be transmitted through the channel. To achieve a proper decision, however, it is essential to know at least the probability density function (PDF), which to estimate is classically a time consumpting and/or less accurate hard task that may make decisions to fail. This paper describes the design of a quantum assisted PDF estimation method also by way of an example, which promises to achieve the exact PDF by proper setting of parameters in a very rapid way.
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This thesis investigates three distinct facets of the theory of quantum information. The first two, quantum state estimation and quantum process estimation, are closely related and deal with the question of how to estimate the classical parameters in a quantum mechanical model. The third attempts to bring quantum theory as close as possible to classical theory through the formalism of quasi-probability. \n \nBuilding a large scale quantum information processor is a significant challenge. First, we require an accurate characterization of the dynamics experienced by the device to allow for the application of error correcting codes and other tools for implementing useful quantum algorithms. The necessary scaling of computational resources needed to characterize a quantum system as a function of the number of subsystems is by now a well studied problem (the scaling is generally exponential). However, irrespective of the computational resources necessary to just write-down a classical description of a quantum state, we can ask about the experimental resources necessary to obtain data (measurement complexity) and the computational resources necessary to generate such a characterization (estimation complexity). These problems are studied here and approached from two directions. \n \nThe first problem we address is that of quantum state estimation. We apply high-level decision theoretic principles (applied in classical problems such as, for example, universal data compression) to the estimation of a qubit state. We prove that quantum states are more difficult to estimate than their classical counterparts by finding optimal estimation strategies. These strategies, requiring the solution to a difficult optimization problem, are difficult to implement in practise. Fortunately, we find estimation algorithms which come close to optimal but require far fewer resources to compute. Finally, we provide a classical analog of this quantum mechanical problem which reproduces, and gives intuitive explanations for, many of its features, such as why adaptive tomography can quadratically reduce its difficulty. \n \nThe second method for practical characterization of quantum devices takes is applied to the problem of quantum process estimation. This differs from the above analysis in two ways: (1) we apply strong restrictions on knowledge of various estimation and control parameters (the former making the problem easier, the latter making it harder); and (2) we consider the problem of designing future experiments based on the outcomes of past experiments. We show in test cases that adaptive protocols can exponentially outperform their off-line counterparts. Moreover, we adapt machine learning algorithms to the problem which bring these experimental design methodologies to realm of experimental feasibility. \n \nIn the final chapter we move away from estimation problems to show formally that a classical representation of quantum theory is not tenable. This intuitive conclusion is formally borne out through the connection to quasi-probability -- where it is equivalent to the necessity of negative probability in all such representations of quantum theory. In particular, we generalize previous no-go theorems to arbitrary classical representations of quantum systems of arbitrary dimension. We also discuss recent progress in the program to identify quantum resources for subtheories of quantum theory and operational restrictions motivated by quantum computation.
Probability density function (PDF) methods have been very useful in describing many physical aspects of turbulent mixing. In applications of these methods, modeled PDF transport equations are commonly simulated via classical Monte Carlo techniques, which provide estimates of moments of the PDF at arbitrary accuracy. In this work, recently developed techniques in quantum computing and quantum enhanced measurements (quantum metrology) are used to construct a quantum algorithm that accelerates the computation of such estimates. This quantum algorithm provides a quadratic speedup over classical Monte Carlo methods in terms of the number of repetitions needed to achieve the desired precision. This paper illustrates the power of this algorithm by considering a binary scalar mixing process modeled by means of the coalescence/dispersion (C/D) closure. The equation is first simulated using classical Monte Carlo methods, where error estimates for the computation of central moments are provided. Then the quantum algorithm for this problem is simulated by sampling from the same probability distribution as that of the output of a quantum computer, and it is shown that significantly fewer resources are required to achieve the same precision. The results demonstrate potential applications of future quantum computers for simulation of turbulent mixing, and large classes of related problems.
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This paper investigates the application of quantum entanglement to improve the estimation of probability density within complex systems. The core claim posits the design of a quantum algorithm, termed "Quantum Entangled Probability," that leverages entanglement to accelerate and enhance predictive modeling capabilities. Current probabilistic methods face significant computational challenges, particularly when dealing with intricate systems exhibiting high dimensionality and complex dependencies. Quantum computing, through its unique properties of superposition and entanglement, offers the potential to overcome these limitations, providing a pathway to more accurate and efficient predictions. This research explores the theoretical foundations and practical implementation of this novel approach, focusing on its potential impact on diverse fields requiring precise probabilistic modeling. The paper details the Quantum Entangled Probability algorithm, its underlying principles, and preliminary experimental results demonstrating its efficacy in a simulated environment.