524 research outputs found
On Quantum Algorithms
Quantum computers use the quantum interference of different computational
paths to enhance correct outcomes and suppress erroneous outcomes of
computations. In effect, they follow the same logical paradigm as
(multi-particle) interferometers. We show how most known quantum algorithms,
including quantum algorithms for factorising and counting, may be cast in this
manner. Quantum searching is described as inducing a desired relative phase
between two eigenvectors to yield constructive interference on the sought
elements and destructive interference on the remaining terms.Comment: 15 pages, 8 figure
On Quantum Algorithms for Noncommutative Hidden Subgroups
Quantum algorithms for factoring and discrete logarithm have previously been
generalized to finding hidden subgroups of finite Abelian groups. This paper
explores the possibility of extending this general viewpoint to finding hidden
subgroups of noncommutative groups. We present a quantum algorithm for the
special case of dihedral groups which determines the hidden subgroup in a
linear number of calls to the input function. We also explore the difficulties
of developing an algorithm to process the data to explicitly calculate a
generating set for the subgroup. A general framework for the noncommutative
hidden subgroup problem is discussed and we indicate future research
directions.Comment: 13 pages, no figures, LaTeX2
Physics and computer science: quantum computation and other approaches
This is a position paper written as an introduction to the special volume on
quantum algorithms I edited for the journal Mathematical Structures in Computer
Science (Volume 20 - Special Issue 06 (Quantum Algorithms), 2010)
A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions
The degrees of polynomials representing or approximating Boolean functions
are a prominent tool in various branches of complexity theory. Sherstov
recently characterized the minimal degree deg_{\eps}(f) among all polynomials
(over the reals) that approximate a symmetric function f:{0,1}^n-->{0,1} up to
worst-case error \eps: deg_{\eps}(f) = ~\Theta(deg_{1/3}(f) +
\sqrt{n\log(1/\eps)}). In this note we show how a tighter version (without the
log-factors hidden in the ~\Theta-notation), can be derived quite easily using
the close connection between polynomials and quantum algorithms.Comment: 7 pages LaTeX. 2nd version: corrected a few small inaccuracie
Effects of Imperfections on Quantum Algorithms: A Software Engineering Perspective
Quantum computers promise considerable speedups over classical approaches,
which has raised interest from many disciplines. Since any currently available
implementations suffer from noise and imperfections, achieving concrete
speedups for meaningful problem sizes remains a major challenge. Yet,
imperfections and noise may remain present in quantum computing for a long
while. Such limitations play no role in classical software computing, and
software engineers are typically not well accustomed to considering such
imperfections, albeit they substantially influence core properties of software
and systems.
In this paper, we show how to model imperfections with an approach tailored
to (quantum) software engineers. We intuitively illustrate, using numerical
simulations, how imperfections influence core properties of quantum algorithms
on NISQ systems, and show possible options for tailoring future NISQ machines
to improve system performance in a co-design approach.
Our results are obtained from a software framework that we provide in form of
an easy-to-use reproduction package. It does not require computer scientists to
acquire deep physical knowledge on noise, yet provide tangible and intuitively
accessible means of interpreting the influence of noise on common software
quality and performance indicators
Single-Step Quantum Search Using Problem Structure
The structure of satisfiability problems is used to improve search algorithms
for quantum computers and reduce their required coherence times by using only a
single coherent evaluation of problem properties. The structure of random k-SAT
allows determining the asymptotic average behavior of these algorithms, showing
they improve on quantum algorithms, such as amplitude amplification, that
ignore detailed problem structure but remain exponential for hard problem
instances. Compared to good classical methods, the algorithm performs better,
on average, for weakly and highly constrained problems but worse for hard
cases. The analytic techniques introduced here also apply to other quantum
algorithms, supplementing the limited evaluation possible with classical
simulations and showing how quantum computing can use ensemble properties of NP
search problems.Comment: 39 pages, 12 figures. Revision describes further improvement with
multiple steps (section 7). See also
http://www.parc.xerox.com/dynamics/www/quantum.htm
- …