Quantum Queries on Permutations

  title={Quantum Queries on Permutations},
  author={Taisia Mischenko-Slatenkova and Alina Vasilieva and Ilja Kucevalovs and Rusins Freivalds},
K. Iwama and R. Freivalds considered query algorithms where the black box contains a permutation. Since then several authors have compared quantum and deterministic query algorithms for permutations. It turns out that the case of \(n\)-permutations where \(n\) is an odd number is difficult. There was no example of a permutation problem where quantization can save half of the queries for \((2m+1)\)-permutations if \(m\ge 2\). Even for \((2m)\)-permutations with \(m\ge 2\), the best proved… 

Characterizations of symmetrically partial Boolean functions with exact quantum query complexity

The optimal exact quantum query complexity for a generalized Deutsch-Jozsa problem is proved, the symmetrically partial Boolean functions are characterized and an algorithm is provided to determine the degree of any symmetrally partial Boolean function.

Exact Quantum 1-Query Algorithms and Complexity

This paper obtains a new result that a partial Boolean function with exact quantum 1-query complexity is constructed and it cannot be computed by D–J algorithm.

Revisiting Deutsch-Jozsa algorithm

Characterizations of promise problems with exact quantum query complexity

It is proved that any symmetrical (and partial) Boolean function f has exact quantum 1-query complexity if and only if $f$ can be computed by the Deutsch-Jozsa algorithm.



Quantum Queries on Permutations with a Promise

There is a nontrivial promise such that if the authors impose that promise to the input of size n = 4m , then they need only two quantum queries, while at least 2m + 2 ( = n /2 + 2) deterministic queries are necessary.

Quantum lower bounds by polynomials

This work examines the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model and gives asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings.

Average-Case Quantum Query Complexity

It is shown that for average-case complexity under the uniform distribution, quantum algorithms can be exponentially faster than classical algorithms and under non-uniform distributions the gap can even be super-exponential.

Bounds for small-error and zero-error quantum algorithms

We present a number of results related to quantum algorithms with small error probability and quantum algorithms that are zero-error. First, we give a tight analysis of the trade-offs between the

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.

Polynomial degree vs. quantum query complexity

  • A. Ambainis
  • Computer Science
    44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.
  • 2003
The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f and this lower bound is shown by a new, more general version of quantum adversary method.

Languages Recognizable by Quantum Finite Automata

For the most popular definition of the QFA, the class of languages recognizable by a QFA is not closed under union or any other binary Boolean operation where both arguments are significant.

Algebraic Factoring and Geometry Proving

It is explained how geometric theorems may be proved by using irreducible zero decomposition for which algebraic factoring is necessary to help understand the ambiguity of a theorem and prove it even if its algebraic formulation does not precisely correspond to the geometric statement.

Quantum algorithms revisited

Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. A common pattern underpinning quantum