Quantum Algorithms for Evaluating Min-MaxTrees

@inproceedings{Cleve2008QuantumAF,
  title={Quantum Algorithms for Evaluating Min-MaxTrees},
  author={Richard Cleve and Dmitry Gavinsky and David L. Yonge-Mallo},
  booktitle={TQC},
  year={2008}
}
We present a bounded-error quantum algorithm for evaluating Min - Max trees with $N^{\frac{1}{2}+o(1)}$ queries, where N is the size of the tree and where the allowable queries are comparisons of the form [x j k ]. This is close to tight, since there is a known quantum lower bound of $\Omega(N^{\frac{1}{2}})$. 
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References

SHOWING 1-10 OF 12 REFERENCES
Discrete-Query Quantum Algorithm for NAND Trees
TLDR
It is pointed out that the algorithm given by Edward Farhi, Jeffrey Goldstone, and Sam Gutmann can be converted into an algorithm using N^[1/2 + o(1)] queries in the conventional (discrete) quantum query model.
Every NAND formula of size N can be evaluated in time N^{1/2+o(1)} on a quantum computer
For every NAND formula of size N, there is a bounded-error N^{1/2+o(1)}-time quantum algorithm, based on a coined quantum walk, that evaluates this formula on a black-box input. Balanced, or
Every NAND formula on N variables can be evaluated in time O(N^{1/2+eps})
TLDR
It follows that the (2 − ε)-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.
A lower bound on the quantum query complexity of read-once functions
  • H. Barnum, M. Saks
  • Computer Science, Mathematics
    Electron. Colloquium Comput. Complex.
  • 2002
A Quantum Algorithm for the Hamiltonian NAND Tree
TLDR
A quantum algorithm for the binary NAND tree problem in the Hamil- tonian oracle model using a continuous time quantum walk with a running time proportional to p N is given.
A fast quantum mechanical algorithm for database search
TLDR
In early 1994, it was demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) .
Strengths and Weaknesses of Quantum Computing
TLDR
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 $o(2^{n/2})$.
Probabilistic Boolean decision trees and the complexity of evaluating game trees
  • M. Saks, A. Wigderson
  • Computer Science
    27th Annual Symposium on Foundations of Computer Science (sfcs 1986)
  • 1986
TLDR
A randomized variant of alphabeta pruning is analyzed, it is shown that it is considerably faster than the deterministic one in worst case, and it is proved optimal for uniform trees.
Tight bounds on quantum searching
TLDR
A lower bound on the efficiency of any possible quantum database searching algorithm is provided and it is shown that Grover''s algorithm nearly comes within a factor 2 of being optimal in terms of the number of probes required in the table.
A nearly optimal discrete query quantum algorithm for evaluating NAND formulas
We present an O(\sqrt{N}) discrete query quantum algorithm for evaluating balanced binary NAND formulas and an O(N^{{1/2}+O(\frac{1}{\sqrt{\log N}})}) discrete query quantum algorithm for evaluating
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