Polynomial degree vs. quantum query complexity
Andris Ambainis
- Vol. 72, Iss: 2, pp 220-238
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TLDR
In this paper, the first superlinear separation between polynomial degree and quantum query complexity was shown by a more general version of the quantum adversary method, and the lower bound was shown to be tight.Abstract:
The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. We exhibit a function with polynomial degree M and quantum query complexity (M/sup 1.321.../). This is the first superlinear separation between polynomial degree and quantum query complexity. The lower bound is shown by a new, more general version of quantum adversary method.read more
Citations
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Book
Analysis of Boolean Functions
TL;DR: This text gives a thorough overview of Boolean functions, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry, and includes a "highlight application" such as Arrow's theorem from economics.
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Quantum algorithms for the triangle problem
TL;DR: Two new quantum algorithms are presented that either find a triangle (a copy of K3 ) in an undirected graph or reject if it is triangle free, and they are based on a new design concept of Ambainis.
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Negative weights make adversaries stronger
TL;DR: A stronger version of the adversary method, called ADV+-, was proposed in this article, which goes beyond this principle to make explicit use of the stronger condition that the algorithm actually computes the function.
Journal ArticleDOI
Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range
TL;DR: A general method for proving quantum lower bounds for problems with small range is given, and a better lower bound is obtained on the polynomial degree of the two-level AND-OR tree.
Proceedings ArticleDOI
Any AND-OR Formula of Size N can be Evaluated in time N^{1/2 + o(1)} on a Quantum Computer
TL;DR: It follows that the (2-o(1))th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.
References
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Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
TL;DR: In this paper, the authors considered factoring integers and finding discrete logarithms on a quantum computer and gave an efficient randomized algorithm for these two problems, which takes a number of steps polynomial in the input size of the integer to be factored.
Proceedings ArticleDOI
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TL;DR: In this paper, it was shown that a quantum mechanical computer can solve integer factorization problem in a finite power of O(log n) time, where n is the number of elements in a given integer.
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TL;DR: In this paper, Tchebycheff polynomials and other linear families have been used for approximating least-squares approximations to systems of equations with one unknown solution.
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Strengths and Weaknesses of Quantum Computing
TL;DR: It is proved that relative to an oracle chosen uniformly at random with probability 1 the class $\NP$ cannot be solved on a quantum Turing machine (QTM) in time $o(2^{n/2})$.
Journal ArticleDOI
Complexity measures and decision tree complexity: a survey
Harry Buhrman,Ronald de Wolf +1 more
TL;DR: Several complexity measures for Boolean functions are discussed: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial, and how they give bounds for the decision tree complexity of Boolean functions on deterministic, randomized, and quantum computers.