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Quantum walk algorithm for element distinctness
  • A. Ambainis
  • Computer Science
    45th Annual IEEE Symposium on Foundations of…
  • 31 October 2003
An O(N/sup k/(k+1)/) query quantum algorithm is given for the generalization of element distinctness in which the authors have to find k equal items among N items.
Quantum lower bounds by quantum arguments
Two new Ω(√N) lower bounds on computing AND of ORs and inverting a permutation and more uniform proofs for several known lower bounds which have been previously proven via a variety of different techniques are proved.
Quantum walks on graphs
A lower bound on the possible speed up by quantum walks for general graphs is given, showing that quantum walks can be at most polynomially faster than their classical counterparts.
Coins make quantum walks faster
The result improves on a previous bound for quantum local search by Aaronson and Ambainis and generalizes the result to 3 and more dimensions where the walk yields the optimal performance of <i>O</i>(√N) and gives several extensions of quantum walk search algorithms and generic expressions for its performance for general graphs.
1-way quantum finite automata: strengths, weaknesses and generalizations
This work constructs a 1-way QFA that is exponentially smaller than any equivalent classical (even randomized) finite automaton, and thinks that this construction may be useful for design of other space-efficient quantum algorithms.
One-dimensional quantum walks
A quantum analog of the symmetric random walk, which the authors call the Hadamard walk, is analyzed, which has position that is nearly uniformly distributed in the range after steps, in sharp contrast to the classical random walk.
Quantum search of spatial regions
An 0(/spl radic/n)-qubit communication protocol for the disjointness problem is given, which improves an upper bound of Hoyer and de Wolf and matches a lower bound of Razborov.
Upper Bound on Communication Complexity of Private Information Retrieval
This work constructs a scheme for private information retrieval with k databases and communication complexity O(n 1/(2k−1) ), where n is the number of databases and O is the communication complexity.
Private quantum channels
It is shown that in order to transmit n qubits privately, 2n bits of shared private key are necessary and sufficient and may be viewed as the quantum analogue of the classical one-time pad encryption scheme.