We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modifies the input, we use a quantum adversary that runs t...
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We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its state becomes entangled with the superposition over inputs. We bound the number of queries needed to achieve a sufficient entanglement and this implies a lower bound on the number of queries for the computation. Using this method, we prove two new Omega(rootN) lower bounds on computing AND of ORs and inverting a permutation and also provide more uniform proofs for several known lower bounds which have been previously proven via a variety of different techniques. (C) 2002 Elsevier Science (USA).
The complexity of quantum query algorithms computing Boolean functions is related to the degree of the algebraic polynomial representing this Boolean function. Hence to find a Boolean function with quantumquery compl...
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ISBN:
(纸本)1932415718
The complexity of quantum query algorithms computing Boolean functions is related to the degree of the algebraic polynomial representing this Boolean function. Hence to find a Boolean function with quantumquery complexity being smaller than the deterministic query complexity, one needs to find Boolean functions with low degree of the representing polynomial and high deterministic query complexity. We have noticed that the existing examples of such Boolean functions involve usage of combinatorial block designs being Kirkman triple systems. We have constructed new combinatorial block designs related to the Kirkman's schoolgirl problem hoping to use these designs to construct new Boolean functions with a large gap between the quantum and deterministic query complexity.
We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness ( the problem of finding two equal items among N given items), we get an O(N-2/3) que...
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ISBN:
(纸本)0769522289
We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness ( the problem of finding two equal items among N given items), we get an O(N-2/3) queryquantum algorithm. This improves the previous O(N-3/4) quantum algorithm of Buhrman et al. [SIAM J. Comput., 34 ( 2005), pp. 1324-1330] and matches the lower bound of Aaronson and Shi [J. ACM, 51 ( 2004), pp. 595-605]. We also give an O(Nk/(k+1)) queryquantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items.
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