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International Journal of General Systems

Quantum computation techniques for gauging reliability of interval and fuzzy data

Luc Longpré, Christian Servín, Владик Крейнович

Published Nov 24, 2010• 4 citations
Publisher

Abstract

In traditional interval computations, we assume that the interval data correspond to guaranteed interval bounds, and that fuzzy estimates provided by experts are correct. In practice, measuring instruments are not 100% reliable, and experts are not 100% reliable, we may have estimates which are ‘way off’, intervals which do not contain the actual values at all. Usually, we know the percentage of such outlier un-reliable measurements. However, it is desirable to check that the reliability of the actual data is indeed within the given percentage. The problem of checking (gauging) this reliability is, in general, NP-hard; in reasonable cases, there exist feasible algorithms for solving this problem. In this paper, we show that quantum computation techniques can drastically speed up the computation of reliability of the given data.

DOI
10.1080/03081079.2010.510247

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In traditional interval computations, we assume that the interval data correspond to guaranteed interval bounds, and that fuzzy estimates provided by experts are correct. In practice, measuring instruments are not 100% reliable, and experts are not 100% reliable, we may have estimates which are ‘way off’, intervals which do not contain the actual values at all. Usually, we know the percentage of such outlier un-reliable measurements. However, it is desirable to check that the reliability of the actual data is indeed within the given percentage. The problem of checking (gauging) this reliability is, in general, NP-hard; in reasonable cases, there exist feasible algorithms for solving this problem. In this paper, we show that quantum computation techniques can drastically speed up the computation of reliability of the given data.
In traditional interval computations, we assume that the interval data corresponds to guaranteed interval bounds, and that fuzzy estimates provided by experts are correct. In practice, measuring instruments are not 100% reliable, and experts are not 100% reliable, we may have estimates which are "way off", intervals which do not contain the actual values at all. Usually, we know the percentage of such outlier un-reliable measurements. However, it is desirable to check that the reliability of the actual data is indeed within the given percentage. The problem of checking (gauging) this reliability is, in general, NP-hard; in reasonable cases, there exist feasible algorithms for solving this problem. In this paper, we show that quantum computations techniques can drastically speed up the computation of reliability of given data.
In traditional interval computations, we assume that the interval data corresponds to guaranteed interval bounds, and that fuzzy estimates provided by experts are correct. In practice, measuring instruments are not 100% reliable, and experts are not 100% reliable, we may have estimates which are \way ofi, intervals which do not contain the actual values at all. Usually, we know the percentage of such outlier un-reliable measurements. However, it is desirable to check that the reliability of the actual data is indeed within the given percentage. The problem of checking (gauging) this reliability is, in general, NP-hard; in reasonable cases, there exist feasible algorithms for solving this problem. In this paper, we show that quantum computations techniques can drastically speed up the computation of reliability of given data.
Abstract In many real‐life situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. To estimate y, we find some easier‐to‐measure quantities x1, … , xn which are related to y by a known relation y = f(x1, … , xn). Measurements are never 100% accurate; hence, the measured values $ \tilde {x} _{i} $ are different from xi, and the resulting estimate $ \tilde {y} = f( \tilde {x} _{1} \ldots , \tilde {x} _{n}) $ is different from the desired value y = f(x1, … , xn). How different can it be? Traditional engineering approach to error estimation in data processing assumes that we know the probabilities of different measurement errors $ \Delta x_i {\mathop {=} \limits ^{\rm def}} \tilde x _i - x_i $. In many practical situations, we only know the upper bound Δi for this error; hence, after the measurement, the only information that we have about xi is that it belongs to the interval $ {\bf x}_i {\mathop {=} \limits ^{\rm def}} [ \tilde x _i - \Delta _i, \tilde x_i + \Delta _i] $. In this case, it is important to find the range y of all possible values of y = f(x1, … , xn) when xi ∈ xi. We start the paper with a brief overview of the computational complexity of the corresponding interval computation problems: Most of the related problems turn out to be, in general, at least NP‐hard. In this paper, we show how the use of quantum computing can speed up some computations related to interval and probabilistic uncertainty. We end the paper with speculations on whether (and how) “hypothetic” physical devices can compute NP‐hard problems faster than in exponential time. Most of the paper's results were first presented at NAFIPS'2003 [30]. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
In data processing, we input the results xi of measuring easy-to-measure quantities xi and use these results to find estimates y = f (x1, . . ., xn) for difficult-to-measure quantities y which are related to xi by a known relation y = f(x1, . . ., xn). Due to measurement inaccuracy, the measured values xi are, in general, different from the (unknown) actual values xi of the measured quantities, hence the result y of data processing is different from the actual value of the quantity y.In many practical situations, we only know the bounds Δi on the measurement errors Δxi [EQUATION] xi - xi. In such situations, we only know that the actual value xi belongs to the interval [xi - Δi, xi + Δi], and we want to know the range of possible values of y. The corresponding problems of interval computations are NP-hard, so solving these problems may take an unrealistically long time. One way to speed up computations is to use quantum computing, and quantum versions of interval computations algorithms have indeed been developed.In many practical situations, we also know some constraints on the possible values of the directly measured quantities x1, . . ., xn. In such situations, we must combine interval techniques with constraint satisfaction techniques. It is therefore desirable to extend quantum interval algorithms to such combinations. As a first step towards this combination, in this paper, we consider quantum algorithms for discrete constraint satisfaction problems.