Quantum Queries on Permutations with a Promise

  title={Quantum Queries on Permutations with a Promise},
  author={Rusins Freivalds and Kazuo Iwama},
This paper studies quantum query complexities for deciding (exactly or with probability 1.0) the parity of permutations of n numbers, 0 through n *** 1. Our results show quantum mechanism is quite strong for this non-Boolean problem as it is for several Boolean problems: (i) For n = 3, we need a single query in the quantum case whereas we obviously need two queries deterministically. (ii) For even n , n /2 quantum queries are sufficient whereas we need n *** 1 queries deterministically. (iii… 

Quantum vs . deterministic queries on permutations

A group of 5-permutations is presented such that the deterministic query complexity is 4 and the quantum Query complexity is 2, and the best proved advantage of quantum query algorithms is the result in [12].

Quantum Queries on Permutations

The best proved advantage of quantum query algorithms is the result by Iwama/Freivalds where the quantum query complexity is \(m\) but the deterministic query complexity will be \((2m-1)\).

Nondeterministic Query Algorithms

A new alternative definition of nondeterministic query algorithms is presented and the power of this model is demonstrated with an example of computing the Fano plane Boolean function, showing that for this function the difference between deterministic and nond deterministic query complexity is 7 N versus O(3 N).

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The work by the author in a field of quantum algorithms development is reviewed, exact and bounded-error quantum query algorithms for computing Boolean functions are presented, and a query model is applied for computing multivalued functions.

Exact quantum algorithms for promise problems in automata theory

In this note, we show that quantum finite automata can be polynomially more succinct than their classical counterparts for promise problems in case of exact computation. Additionally, in terms of

Ultrametric automata and Turing machines

We introduce a notion of ultrametric automata and Turing machines using p-adic numbers to describe random branching of the process of computation. These automata have properties similar to the



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.

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.

Bounds for small-error and zero-error quantum algorithms

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1-way quantum finite automata: strengths, weaknesses and generalizations

  • A. AmbainisR. Freivalds
  • Computer Science
    Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
  • 1998
This work constructs a 1-way QFA that is exponentially smaller than any equivalent classical (even randomized) finite automaton, and thinks that this construction may be useful for design of other space-efficient quantum algorithms.

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.

Complexity measures and decision tree complexity: a survey

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

String Matching Algorithms and Automata

  • Imre Simon
  • Computer Science
    Results and Trends in Theoretical Computer Science
  • 1994
The structure of finite automata recognizing sets of the form A*p, for some word p, is studied, and the results obtained are used to improve the Knuth-Morris-Pratt string searching algorithm.