Quantum Algorithms for Evaluating Min-MaxTrees

  title={Quantum Algorithms for Evaluating Min-MaxTrees},
  author={Richard Cleve and Dmitry Gavinsky and David L. Yonge-Mallo},
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 $\Omega(N^{\frac{1}{2}})$. 
A Quantum Algorithm for the Hamiltonian NAND Tree
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 Branch-and-Bound Algorithm and its Application to the Travelling Salesman Problem
A quantum branch-and-bound algorithm based on the general scheme of the branch- and-bound method and the quantum nested searching algorithm is proposed and proved to be several times more efficient than the classical algorithm.
Intricacies of quantum computational paths
This work discusses how to adapt generic quantum search procedures, namely quantum random walks and Grover’s algorithm, in order to obtain computational paths and compares these approaches in the context of tree graphs, finding the different scenarios that are better suited for each approach.
Quantum computation beyond the circuit model
The quantum circuit model is the most widely used model of quantum computation. It provides both a framework for formulating quantum algorithms and an architecture for the physical construction of
Quantum walks: a comprehensive review
This paper has reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.
Quantum Algorithm for Lexicographically Minimal String Rotation
The proposed quantum query algorithm for LMSR has average-case query complexity, and is shown to be asymptotically optimal up to a polylogarithmic factor, compared with its $\Omega\left(\sqrt{n/\log n}\right)$ lower bound.
Quantum Slide and NAND Tree on a Photonic Chip
The construction of quantum slide is proposed, where a propagating Gaussian wave-packet can be generated deterministically along a properly-engineered chain, and the optical NAND-tree is capable of solving computational problems with a total of four input bits, based on the femtosecond laser 3D direct-writing technique on a photonic chip.
Simulation of Quantum Walks using HPC
Hiperwalk is a new simulator of the main quantum walk models using high-performance computing (HPC) and will be able simulate continuous-time quantum walks and Szegedy’s quantum walks.
Challenges of adiabatic quantum evaluation of NAND trees
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.
On quantum algorithms
It is shown how most known quantum algorithms, including quantum algorithms for factorising and counting, may be cast in this manner.


Discrete-Query Quantum Algorithm for NAND Trees
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})
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
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
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
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
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
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