• Corpus ID: 15460492

A nearly optimal discrete query quantum algorithm for evaluating NAND formulas

@article{Ambainis2007ANO,
  title={A nearly optimal discrete query quantum algorithm for evaluating NAND formulas},
  author={Andris Ambainis},
  journal={arXiv: Quantum Physics},
  year={2007}
}
  • A. Ambainis
  • Published 26 April 2007
  • Computer Science
  • arXiv: Quantum Physics
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 arbitrary binary NAND formulas. 
View PDF on arXiv
33 Citations
Background Citations
6
Methods Citations
5

Figures from this paper

33 Citations

Citation Type

Citation Type

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 ].
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…
The quantum query complexity of certification
TLDR
It is shown that the zero-error quantum query complexity of evaluating such formulas is O(d^{(k+1)/2}) (again neglecting a logarithmic factor), and the lower bound relies on the fact that the quantum adversary method obeys a direct sum theorem.
ANY AND-OR FORMULA OF SIZE N CAN BE EVALUATED IN TIME N 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 N1/2+o(1) and in particular, approximately balanced formulas can be evaluated in O( √ N) queries, which is optimal.
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.
Faster quantum algorithm for evaluating game trees
TLDR
An O(√n log n)-query quantum algorithm for evaluating size-n AND-OR formulas and a novel tensor-product span program composition method, which generates graphs with vertices corresponding to minimal zero-certificates is given.
Span-Program-Based Quantum Algorithm for Evaluating Unbalanced Formulas
TLDR
A quantum algorithm is given to evaluate formulas over any finite boolean gate set, Provided that the complexities of the input subformulas to any gate differ by at most a constant factor, the algorithm has optimal query complexity.
Challenges of adiabatic quantum evaluation of NAND trees
TLDR
This work focuses on a number of issues ranging from: (1) mapping mechanisms; (2) spectrum analysis and remapping; (3) numerical evaluation of spectrum gaps; and (4) algorithmic procedures, which are then used to provide numerical evidence for the existence of a N2logN2 gap.
Reflections for quantum query algorithms
We show that any boolean function can be evaluated optimally by a quantum query algorithm that alternates a certain fixed, input-independent reflection with a second reflection that coherently…
Reflections for quantum query complexity : The general adversary bound is tight for every boolean function
We show that any boolean function can be evaluated optimally by a bounded-error quantum query algorithm that alternates a certain fixed, input-independent reflection with coherent queries to the…
...
...

References

SHOWING 1-10 OF 24 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 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 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.
Quantum walk algorithm for element distinctness
  • A. Ambainis
  • Computer Science
    45th Annual IEEE Symposium on Foundations of Computer Science
  • 2004
TLDR
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 Search on Bounded-Error Inputs
TLDR
It is shown that quantum amplitude amplification can be made to work also with a bounded-error verifier, and optimal quantum upper bounds of O(√N) queries for all constant-depth AND-OR trees on N variables are obtained.
Quantum lower bounds by quantum arguments
TLDR
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.
Complexity measures and decision tree complexity: a survey
A lower bound on the quantum query complexity of read-once functions
  • H. Barnum, M. Saks
  • Computer Science, Mathematics
    Electron. Colloquium Comput. Complex.
  • 2002
Size-Depth Tradeoffs for Algebraic Formulas
TLDR
It is shown that, for any fixed $\epsilon > 0$, any algebraic formula of size $S$ can be converted into an equivalent formula of depth $O(\log S)$ and size $O(S^{1+\ep silon})$.
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) .
...
...