275 research outputs found
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
Quantum algorithms for testing Boolean functions
We discuss quantum algorithms, based on the Bernstein-Vazirani algorithm, for
finding which variables a Boolean function depends on. There are 2^n possible
linear Boolean functions of n variables; given a linear Boolean function, the
Bernstein-Vazirani quantum algorithm can deterministically identify which one
of these Boolean functions we are given using just one single function query.
The same quantum algorithm can also be used to learn which input variables
other types of Boolean functions depend on, with a success probability that
depends on the form of the Boolean function that is tested, but does not depend
on the total number of input variables. We also outline a procedure to futher
amplify the success probability, based on another quantum algorithm, the Grover
search
Learning with Errors is easy with quantum samples
Learning with Errors is one of the fundamental problems in computational
learning theory and has in the last years become the cornerstone of
post-quantum cryptography. In this work, we study the quantum sample complexity
of Learning with Errors and show that there exists an efficient quantum
learning algorithm (with polynomial sample and time complexity) for the
Learning with Errors problem where the error distribution is the one used in
cryptography. While our quantum learning algorithm does not break the LWE-based
encryption schemes proposed in the cryptography literature, it does have some
interesting implications for cryptography: first, when building an LWE-based
scheme, one needs to be careful about the access to the public-key generation
algorithm that is given to the adversary; second, our algorithm shows a
possible way for attacking LWE-based encryption by using classical samples to
approximate the quantum sample state, since then using our quantum learning
algorithm would solve LWE
A Survey of Quantum Learning Theory
This paper surveys quantum learning theory: the theoretical aspects of
machine learning using quantum computers. We describe the main results known
for three models of learning: exact learning from membership queries, and
Probably Approximately Correct (PAC) and agnostic learning from classical or
quantum examples.Comment: 26 pages LaTeX. v2: many small changes to improve the presentation.
This version will appear as Complexity Theory Column in SIGACT News in June
2017. v3: fixed a small ambiguity in the definition of gamma(C) and updated a
referenc
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