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Quantum interpolation of polynomilas

Daniel M. Kane, Samuel Kutin

Published Jan 1, 2011• 7 citations
Publisher

Abstract

Can a quantum computer efficiently interpolate polynomials? We consider black-boxalgorithms that seek to learn information about a polynomial f from input/output pairs(xi, f(xi)). We define a more general class of (d, S)-independent function properties,where, outside of a set S of exceptions, knowing d input values does not help one predictthe answer. There are essentially two strategies to computing such a function: queryd + 1 random input values, or search for one of the |S| exceptions. We show that, up toconstant factors, we cannot beat these two approaches.

DOI
10.5555/2011383.2011390

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•Apr 25, 2018•1 citation
Can a quantum computer efficiently interpolate polynomials? We consider black-boxalgorithms that seek to learn information about a polynomial f from input/output pairs(xi, f(xi)). We define a more general class of (d, S)-independent function properties,where, outside of a set S of exceptions, knowing d input values does not help one predictthe answer. There are essentially two strategies to computing such a function: queryd + 1 random input values, or search for one of the |S| exceptions. We show that, up toconstant factors, we cannot beat these two approaches.
Can a quantum computer efficiently interpolate polynomials? We consider black-box algorithms that seek to learn information about a polynomial fff from input/output pairs (xi,f(xi))(x_i, f(x_i))(xi​,f(xi​)). We define a more general class of \emph{(d,S)(d,S)(d,S)-independent} function properties, where, outside of a set SSS of exceptions, knowing ddd input values does not help one predict the answer. There are essentially two strategies to computing such a function: query d+1d+1d+1 random input values, or search for one of the ∣S∣|S|∣S∣ exceptions. We show that, up to constant factors, we cannot beat these two approaches.
The nature of the search Suppose you know that exactly one n -bit integer satisfies a certain condition, and suppose you have a black-boxed subroutine that acts on the N = 2 n different n -bit integers, outputting 1 if the integer satisfies the condition and 0 otherwise. In the absence of any other information, to find the special integer you can do no better with a classical computer than to apply the subroutine repeatedly to different random numbers until you hit on the special one. If you apply it to M different integers the probability of your finding the special number is M/N . You must test ½ N different integers to have a 50% chance of success. If, however, you have a quantum computer with a subroutine that performs such a test, then you can find the special integer with a probability that is very close to 1 when N is large, using a method that calls the subroutine a number of times no greater than. This very general capability of quantum computers was discovered by Lov Grover, and goes under the name of Grover's search algorithm . Shor's period-finding algorithm and Grover's search algorithm, together with their various modifications and extensions, constitute the two masterpieces of quantum-computational software. One can think of Grover's black-boxed subroutine in various ways. The subroutine might perform a mathematical calculation to determine whether the input integer is the special one. Here is a simple example.
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.
This thesis studies strengths and weaknesses of quantum computers. In the first part we present three contributions to quantum algorithms. 1) consider a search space of N elements. One of these elements is "marked" and our goal is to find this. We describe a quantum algorithm to solve this problem using essentially sqrt{N} queries and other operations, improving over the gate count of Grover's algorithm. 2) We give a succinct characterization of quantum algorithms in terms of polynomials, and develop a new technique for showing upper and lower bounds on quantum query complexity based on this. 3) One generic technique used to compute the minimum of a given function is "gradient descent". We present a quantum gradient-calculation algorithm and a gradient-optimization algorithm that is quadratically faster than classical algorithms. In the second part of this thesis we look at quantum learning theory. 1) We survey quantum learning theory, describing the main results known for three models of learning, using classical as well as quantum data: exact learning from membership queries, the probably approximately correct (PAC) learning model, and the agnostic learning model. 2) We then consider if a quantum learner can PAC-learn a given concept class from fewer quantum examples. We give a negative answer by showing that quantum examples are not more powerful than classical random examples, both for PAC learning and for agnostic learning.