A Quantum Algorithm for the Hamiltonian NAND Tree

@article{Farhi2008AQA,
  title={A Quantum Algorithm for the Hamiltonian NAND Tree},
  author={Edward Farhi and Jeffrey Goldstone and Sam Gutmann},
  journal={Theory Comput.},
  year={2008},
  volume={4},
  pages={169-190}
}
We give a quantum algorithm for the binary NAND tree problem in the Hamil- tonian oracle model. The algorithm uses a continuous time quantum walk with a running time proportional to p N. We also show a lower bound of W( p N) for the NAND tree problem in the Hamiltonian oracle model. 
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References

SHOWING 1-10 OF 17 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. Expand
Lower Bounds on Quantum Query Complexity
TLDR
This paper discusses here what quantum computers cannot do, and specifically how to prove limits on their computational power. Expand
Quantum Algorithms for Evaluating Min-MaxTrees
TLDR
A bounded-error quantum algorithm for evaluating Min - Max trees with N = 1 + o(1) queries, where N is the size of the tree and the allowable queries are comparisons of the form [x j k ]. Expand
Span-program-based quantum algorithm for evaluating formulas
TLDR
A quantum algorithm for evaluating formulas over an extended gate set, including all two- and three-bit binary gates (e.g., NAND, 3-majority), generalizes the optimal quantum AND-OR formula evaluation algorithm and is optimal for evaluating the balanced ternary majority formula. Expand
ANY AND-OR FORMULA OF SIZE N CAN BE EVALUATED IN TIME N1/2+o(1) ON A QUANTUM COMPUTER
Consider the problem of evaluating an AND-OR formula on an N-bit black-box input. We present a bounded-error quantum algorithm that solves this problem in time N 1/2+o(1) . In particular,Expand
Quantum computation and decision trees
Many interesting computational problems can be reformulated in terms of decision trees. A natural classical algorithm is to then run a random walk on the tree, starting at the root, to see if theExpand
Any AND-OR Formula of Size N Can Be Evaluated in Time N1/2+o(1) on a Quantum Computer
TLDR
A bounded-error quantum algorithm that solves the problem of evaluating an AND-OR formula on an N-bit black-box input in time and in particular, approximately balanced formulas can be evaluated in O(\sqrt{N}) queries, which is optimal. Expand
A lower bound on the quantum query complexity of read-once functions
  • H. Barnum, M. Saks
  • Computer Science, Mathematics
  • Electron. Colloquium Comput. Complex.
  • 2002
TLDR
A lower bound of Ω(√n) is established on the bounded-error quantum query complexity of read-once Boolean 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. Expand
Hamiltonian Oracles
Hamiltonian oracles are the continuum limit of the standard unitary quantum oracles. In this limit, the problem of finding the optimal query algorithm can be mapped into the problem of findingExpand
Any AND-OR Formula of Size N can be Evaluated in time N^{1/2 + o(1)} on a Quantum Computer
TLDR
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. Expand
...
1
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...