108 research outputs found
Evolutionary Algorithms for Quantum Computers
In this article, we formulate and study quantum analogues of randomized search heuristics, which make use of Grover search (in Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pp. 212-219. ACM, New York, 1996) to accelerate the search for improved offsprings. We then specialize the above formulation to two specific search heuristics: Random Local Search and the (1+1)Evolutionary Algorithm. We call the resulting quantum versions of these search heuristics Quantum Local Search and the (1+1)Quantum Evolutionary Algorithm. We conduct a rigorous runtime analysis of these quantum search heuristics in the computation model of quantum algorithms, which, besides classical computation steps, also permits those unique to quantum computing devices. To this end, we study the six elementary pseudo-Boolean optimization problems OneMax, LeadingOnes, Discrepancy, Needle, Jump, and TinyTrap. It turns out that the advantage of the respective quantum search heuristic over its classical counterpart varies with the problem structure and ranges from no speedup at all for the problem Discrepancy to exponential speedup for the problem TinyTrap. We show that these runtime behaviors are closely linked to the probabilities of performing successful mutations in the classical algorithms
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
Effects of noise on quantum error correction algorithms
It has recently been shown that there are efficient algorithms for quantum
computers to solve certain problems, such as prime factorization, which are
intractable to date on classical computers. The chances for practical
implementation, however, are limited by decoherence, in which the effect of an
external environment causes random errors in the quantum calculation. To combat
this problem, quantum error correction schemes have been proposed, in which a
single quantum bit (qubit) is ``encoded'' as a state of some larger number of
qubits, chosen to resist particular types of errors. Most such schemes are
vulnerable, however, to errors in the encoding and decoding itself. We examine
two such schemes, in which a single qubit is encoded in a state of qubits
while subject to dephasing or to arbitrary isotropic noise. Using both
analytical and numerical calculations, we argue that error correction remains
beneficial in the presence of weak noise, and that there is an optimal time
between error correction steps, determined by the strength of the interaction
with the environment and the parameters set by the encoding.Comment: 26 pages, LaTeX, 4 PS figures embedded. Reprints available from the
authors or http://eve.physics.ox.ac.uk/QChome.htm
Simulating 0+1 Dimensional Quantum Gravity on Quantum Computers: Mini-Superspace Quantum Cosmology and the World Line Approach in Quantum Field Theory
Quantum computers are a promising candidate to radically expand computational
science through increased computing power and more effective algorithms. In
particular quantum computing could have a tremendous impact in the field of
quantum cosmology. The goal of quantum cosmology is to describe the evolution
of the Universe through the Wheeler-DeWitt equation or path integral methods
without having to first formulate a full theory of quantum gravity. The quantum
computer provides an advantage in this endeavor because it can perform path
integrals in Lorentzian space and does not require constructing contour
integrations in Euclidean gravity. Also quantum computers can provide
advantages in systems with fermions which are difficult to analyze on classical
computers. In this study, we first employed classical computational methods to
analyze a Friedmann-Robertson-Walker mini-superspace with a scalar field and
visualize the calculated wave function of the Universe for a variety of
different values of the spatial curvature and cosmological constant. We them
used IBM's Quantum Information Science Kit Python library and the variational
quantum eigensolver to study the same systems on a quantum computer. The
framework can also be extended to the world line approach to quantum field
theory.Comment: 5 pages, 4 figure
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