In this note we study the number of quantum queries required to identify an unknown multilinear polynomial of degree d in n variables over a finite field F-q. Any bounded-error classical algorithm for this task requir...
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In this note we study the number of quantum queries required to identify an unknown multilinear polynomial of degree d in n variables over a finite field F-q. Any bounded-error classical algorithm for this task requires Omega (n(d)) queries to the polynomial. We give an exact quantum algorithm that uses O (n(d - 1)) queries for constant d, which is optimal. In the case q = 2, this gives a quantum algorithm that uses O (n(d - 1)) queries to identify a codeword picked from the binary Reed-Muller code of order d. (C) 2012 Elsevier B.V. All rights reserved.
We study the power of nonadaptive quantumquery algorithms, which are algorithms whose queries to the input do not depend on the result of previous queries. First, we show that any bounded-error nonadaptive quantum qu...
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We study the power of nonadaptive quantumquery algorithms, which are algorithms whose queries to the input do not depend on the result of previous queries. First, we show that any bounded-error nonadaptive quantumquery algorithm that computes a total boolean function depending on n variables must make Omega(n) queries to the input in total. Second, we show that, if there exists a quantum algorithm that uses k nonadaptive oracle queries to learn which one of a set of m boolean functions it has been given, there exists a nonadaptive classical algorithm using O(k logm) queries to solve the same problem. Thus, in the nonadaptive setting, quantum algorithms for these tasks can achieve at most a very limited speed-up over classical query algorithms. (C) 2010 Elsevier B.V. All rights reserved.
We establish a lower bound of Omega(rootn) on the bounded-error quantum query complexity of read-once Boolean functions. The result is proved via an inductive argument, together with an extension of a lower bound meth...
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We establish a lower bound of Omega(rootn) on the bounded-error quantum query complexity of read-once Boolean functions. The result is proved via an inductive argument, together with an extension of a lower bound method of Ambainis. Ambainis' method involves viewing a quantum computation as a mapping from inputs to quantum states (unit vectors in a complex inner-product space) which changes as the computation proceeds. Initially the mapping is constant (the state is independent of the input). If the computation evalutes the function f then at the end of the computation the two states associated with any f-distinguished pair of inputs (having different f values) are nearly orthogonal. Thus the inner product of their associated states must have changed from 1 to nearly 0. For any set off-distinguished pairs of inputs, the sum of the inner products of the corresponding pairs of states must decrease significantly during the computation, By deriving an upper bound on the decrease in such a sum, during a single step, for a carefully selected set of input pairs, one can obtain a lower bound on the number of steps. We extend Ambainis' bound by considering general weighted sums off-distinguished pairs. We then prove our result for read-once functions by induction on the number of variables, where the induction step involves a careful choice of weights depending on f to optimize the lower bound attained. (C) 2004 Elsevier Inc. All rights reserved.
In this paper we give tight quantum query complexity bounds of some important linear algebra problems. We prove circle minus(n(2)) quantumquery bounds for verify the determinant, rank, matrix inverse and the matrix p...
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In this paper we give tight quantum query complexity bounds of some important linear algebra problems. We prove circle minus(n(2)) quantumquery bounds for verify the determinant, rank, matrix inverse and the matrix power problem. (C) 2008 Elsevier B.V. All rights reserved.
We show that quantum query complexity satisfies a strong direct product theorem. This means that computing k copies of a function with fewer than k times the quantum queries needed to compute one copy of the function ...
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We show that quantum query complexity satisfies a strong direct product theorem. This means that computing k copies of a function with fewer than k times the quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in k. For a boolean function f, we also show an XOR lemma-computing the parity of k copies of f with fewer than k times the queries needed for one copy implies that the advantage over random guessing will be exponentially small. We do this by showing that the multiplicative adversary method, which inherently satisfies a strong direct product theorem, characterizes bounded-error quantum query complexity. In particular, we show that the multiplicative adversary bound is always at least as large as the additive adversary bound, which is known to characterize bounded-error quantum query complexity.
We study the quantum query complexity of constant-sized subgraph containment. Such problems include determining whether a n-vertex graph contains a triangle, clique or star of some size. For a general subgraph H with ...
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We study the quantum query complexity of constant-sized subgraph containment. Such problems include determining whether a n-vertex graph contains a triangle, clique or star of some size. For a general subgraph H with k vertices, we show that H containment can be solved with quantum query complexity O(n(2-2/h-g(H))), with g(H) a strictly positive function of H. This is better than (O) over tilde (n(2) (2/k)) by Magniez et al. This result is obtained in the learning graph model of Belovs.
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 bee...
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ISBN:
(纸本)0769520405
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 Omega(M-1.321...). This is the first superlinear separation between polynomial degree and quantum query complexity. The lower bound is shown by a generalized version of the quantum adversary method. (c) 2005 Elsevier Inc. All rights reserved.
We show that quantum query complexity satisfies a strong direct product theorem. This means that computing k copies of a function with fewer than k times the quantum queries needed to compute one copy of the function ...
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ISBN:
(纸本)9781467316637
We show that quantum query complexity satisfies a strong direct product theorem. This means that computing k copies of a function with fewer than k times the quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in k. For a boolean function f, we also show an XOR lemma-computing the parity of k copies of f with fewer than k times the queries needed for one copy implies that the advantage over random guessing will be exponentially small. We do this by showing that the multiplicative adversary method, which inherently satisfies a strong direct product theorem, characterizes bounded-error quantum query complexity. In particular, we show that the multiplicative adversary bound is always at least as large as the additive adversary bound, which is known to characterize bounded-error quantum query complexity.
It is known since the work of [1] that for any permutation symmetric function f, the quantum query complexity is at most polynomially smaller than the classical randomized querycomplexity, more precisely that R(f) = ...
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The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit red...
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The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof goes by extending the polynomial method from Boolean functions to quantum state generation problems. In the process, the bound is even strengthened. We then show that this extended polynomial method is a special case of the multiplicative adversary method with an adversary matrix that is independent of the function. This new result therefore provides insight on the reason why in some cases the adversary method is stronger than the polynomial method. It also reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplicative adversary method.
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