36 research outputs found
Certainty and Uncertainty in Quantum Information Processing
This survey, aimed at information processing researchers, highlights
intriguing but lesser known results, corrects misconceptions, and suggests
research areas. Themes include: certainty in quantum algorithms; the "fewer
worlds" theory of quantum mechanics; quantum learning; probability theory
versus quantum mechanics.Comment: Invited paper accompanying invited talk to AAAI Spring Symposium
2007. Comments, corrections, and suggestions would be most welcom
Learning DNFs under product distributions via {\mu}-biased quantum Fourier sampling
We show that DNF formulae can be quantum PAC-learned in polynomial time under
product distributions using a quantum example oracle. The best classical
algorithm (without access to membership queries) runs in superpolynomial time.
Our result extends the work by Bshouty and Jackson (1998) that proved that DNF
formulae are efficiently learnable under the uniform distribution using a
quantum example oracle. Our proof is based on a new quantum algorithm that
efficiently samples the coefficients of a {\mu}-biased Fourier transform.Comment: 17 pages; v3 based on journal version; minor corrections and
clarification
The geometry of quantum learning
Concept learning provides a natural framework in which to place the problems
solved by the quantum algorithms of Bernstein-Vazirani and Grover. By combining
the tools used in these algorithms--quantum fast transforms and amplitude
amplification--with a novel (in this context) tool--a solution method for
geometrical optimization problems--we derive a general technique for quantum
concept learning. We name this technique "Amplified Impatient Learning" and
apply it to construct quantum algorithms solving two new problems: BATTLESHIP
and MAJORITY, more efficiently than is possible classically.Comment: 20 pages, plain TeX with amssym.tex, related work at
http://www.math.uga.edu/~hunziker/ and http://math.ucsd.edu/~dmeyer
Quantum Algorithms for Learning and Testing Juntas
In this article we develop quantum algorithms for learning and testing
juntas, i.e. Boolean functions which depend only on an unknown set of k out of
n input variables. Our aim is to develop efficient algorithms:
- whose sample complexity has no dependence on n, the dimension of the domain
the Boolean functions are defined over;
- with no access to any classical or quantum membership ("black-box")
queries. Instead, our algorithms use only classical examples generated
uniformly at random and fixed quantum superpositions of such classical
examples;
- which require only a few quantum examples but possibly many classical
random examples (which are considered quite "cheap" relative to quantum
examples).
Our quantum algorithms are based on a subroutine FS which enables sampling
according to the Fourier spectrum of f; the FS subroutine was used in earlier
work of Bshouty and Jackson on quantum learning. Our results are as follows:
- We give an algorithm for testing k-juntas to accuracy that uses
quantum examples. This improves on the number of examples used
by the best known classical algorithm.
- We establish the following lower bound: any FS-based k-junta testing
algorithm requires queries.
- We give an algorithm for learning -juntas to accuracy that
uses quantum examples and
random examples. We show that this learning algorithms is close to optimal by
giving a related lower bound.Comment: 15 pages, 1 figure. Uses synttree package. To appear in Quantum
Information Processin
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