56,448 research outputs found
Quantum Probabilistic Subroutines and Problems in Number Theory
We present a quantum version of the classical probabilistic algorithms
la Rabin. The quantum algorithm is based on the essential use of
Grover's operator for the quantum search of a database and of Shor's Fourier
transform for extracting the periodicity of a function, and their combined use
in the counting algorithm originally introduced by Brassard et al. One of the
main features of our quantum probabilistic algorithm is its full unitarity and
reversibility, which would make its use possible as part of larger and more
complicated networks in quantum computers. As an example of this we describe
polynomial time algorithms for studying some important problems in number
theory, such as the test of the primality of an integer, the so called 'prime
number theorem' and Hardy and Littlewood's conjecture about the asymptotic
number of representations of an even integer as a sum of two primes.Comment: 9 pages, RevTex, revised version, accepted for publication on PRA:
improvement in use of memory space for quantum primality test algorithm
further clarified and typos in the notation correcte
A Quantum Computational Learning Algorithm
An interesting classical result due to Jackson allows polynomial-time
learning of the function class DNF using membership queries. Since in most
practical learning situations access to a membership oracle is unrealistic,
this paper explores the possibility that quantum computation might allow a
learning algorithm for DNF that relies only on example queries. A natural
extension of Fourier-based learning into the quantum domain is presented. The
algorithm requires only an example oracle, and it runs in O(sqrt(2^n)) time, a
result that appears to be classically impossible. The algorithm is unique among
quantum algorithms in that it does not assume a priori knowledge of a function
and does not operate on a superposition that includes all possible states.Comment: This is a reworked and improved version of a paper originally
entitled "Quantum Harmonic Sieve: Learning DNF Using a Classical Example
Oracle
Quantum computing classical physics
In the past decade quantum algorithms have been found which outperform the
best classical solutions known for certain classical problems as well as the
best classical methods known for simulation of certain quantum systems. This
suggests that they may also speed up the simulation of some classical systems.
I describe one class of discrete quantum algorithms which do so--quantum
lattice gas automata--and show how to implement them efficiently on standard
quantum computers.Comment: 13 pages, plain TeX, 10 PostScript figures included with epsf.tex;
for related work see http://math.ucsd.edu/~dmeyer/research.htm
Obtaining the Quantum Fourier Transform from the Classical FFT with QR Decomposition
We present the detailed process of converting the classical Fourier Transform
algorithm into the quantum one by using QR decomposition. This provides an
example of a technique for building quantum algorithms using classical ones.
The Quantum Fourier Transform is one of the most important quantum subroutines
known at present, used in most algorithms that have exponential speed up
compared to the classical ones. We briefly review Fast Fourier Transform and
then make explicit all the steps that led to the quantum formulation of the
algorithm, generalizing Coppersmith's work.Comment: 12 pages, 1 figure (generated within LaTeX). To appear in Journal of
Computational and Applied Mathematic
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